Determine the $x$ and $y$ components of the forces shown, then determine the magnitude and direction of their resultant.

${F_1}_x = 50.23$ kN | ${F_1}_y = 29.00$ kN ${F_2}_x = -35.36$ kN | ${F_2}_y = 35.36$ kN ${F_3}_x = -17.31 \, \text{ kN}$ | ${F_3}_y = -41.54$ kN ${F_4}_x = 40$ kN |${F_4}_y = 0$ $\textbf{R} = 43.95 \, \text{ kN at 31.27°}$ | [More info.](http://mathalino.com/reviewer/engineering-mechanics/example-001-components-of-a-force)

Find the force acting in all members of the truss shown.

$F_{AB} = 5.56$ kN tension $F_{AE} = 75.56$ kN tension $F_{BC} = 4.45$ kN tension $F_{BE} = 3.34$ kN compression $F_{CD} = 88.87$ kN compression $F_{CE} = 5.57$ kN tension $F_{CF} = 50$ kN tension $F_{DF} = 71.11$ kN tension $F_{FE} = 71.11$ kN tension002-mj [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-mj-01-method-joints)

The structure shown is pinned together at points $A$, $B$, and $C$ and held in equilibrium by the cable $CD$. $A$ load of 12,000 lb is acting at the midpoint of member $AB$, and a load of 8000 lb is applied at point $C$. Determine the reaction at $A$, the internal force in member $BC$, and the tension on cable $CD$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-001-mm-method-members)

Determine the force in members $BC$, $CE$, and $EF$.

$F_{BC} = 4.45$ kN T $F_{EF} = 71.11$ kN T $F_{CE} = 5.55$ kN T [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-001-ms-method-sections)

Compute the $x$ and $y$ components of each of the four forces shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/example-002-components-of-a-force)

The structure is a truss which is pinned to the floor at point $A$, and supported by a roller at point $D$. Determine the force to all members of the truss.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-002-mj-method-joints)

Members $AB$ and $BC$ shown are pinned together at point $B$, and are pinned to the floor at points $A$ and $C$. The structure supports a pulley at point $B$ with which, a person is hoisting a 2.0 kN load. Member $BC$ has a weight of 1.6 kN, which may be considered to act at its center, while $AB$ is made of strong-light material and has negligible weight. Determine the value of the external support reactions at $A$ and $C$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-002-mm-method-members)

The roof truss shown is pinned at point $A$, and supported by a roller at point $H$. Determine the force in member $DG$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-002-ms-method-sections)

Find the force in each member of the truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-003-mj-method-joints)

For the structure shown, determine the reactions at $A$ and $D$ and the internal force in member $CF$.

$A_x = 180$ kN leftward $A_y = 120$ kN downward $D_x = 180$ kN rightward $D_y = 210$ kN upward $F_{CF} = 300$ kN tension [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-003-mm-method-members)

The truss is pinned to the wall at point $F$, and supported by a roller at point $C$. Calculate the force (tension or compression) in members $BC$, $BE$, and $DE$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-003-ms-method-sections)

The truss pinned to the floor at $D$, and supported by a roller at point $A$ is loaded as shown. Determine the force in member $CG$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-004-mj-method-joints)

For the structure shown, members $AD$, $DC$, and ABC are assumed to be solid rigid members; member $ED$ is a cable. For this structure, determine the reaction at $A$, the tension on cable $ED$, and the force in member $DC$.

$T = 1041.67$ kN $A_H = 833.33$ kN $A_V = 275$ kN $F_{CD} = 1001.54$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-004-mm-method-members)

For the truss shown, find the internal force in member $BE$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-004-ms-method-sections)

Find the rectangular components of the force $P$ = 10 kN in the $(x, y)$ and $(u, v)$ directions.

$P_x = 5$ kN $P_y = 8.66$ kN $P_u = 7.66$ kN $P_v = 6.43$ kN [More info.](http://mathalino.com/reviewer/engineering-mechanics/example-005-components-force)

Compute the force in all members of the truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-005-mj-method-joints-0)

For the structure, member $ABC$ which is assumed to be rigid is pinned at $A$ and held in equilibrium by cable $CD$. For this structure, determine the reaction at $A$ and the tension in the cable.

[More info.](http://www.mathalino.com/reviewer/engineering-mechanics/problem-005-mm-method-members)

The structure shown is pinned to the floor at $A$ and $H$. Determine the magnitude of all the support forces acting on the structure and find the force in member $BF$.

$R_H = 400$ kN upward $A_V = 300$ kN downward $A_H = 80$ kN to the left $F_{BF} = 128.06$ kN tension [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-005-mj-method-joints)

The force $P$ of magnitude 50 kN is acting at 215° from the $x$-axis. Find the components of $P$ along the $u$ axis, 157° from $x$, and $v$ axis, negative 69° from $x$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/example-006-components-force)

In the structure shown, all members are assumed to be solid rigid members. The system is pinned to the wall at point $A$ and supported by a roller at point $E$. Calculate the force on member $BD$ and the reactions at $A$ and $E$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-006-fr-analysis-simple-frame)

A block is resting on an incline of slope 5:12 as shown. It is subjected to a force $F$ = 500 N on a slope of 3:4. Determine the components of $F$ parallel and perpendicular to the incline.

$F_x = 253.85$ kN $F_y = -430.77$ kN [More info.](http://mathalino.com/reviewer/engineering-mechanics/example-007-components-force)

In the structure shown, members BCE, and $CD$ are assumed to be solid rigid members. Members $AE$ and $DE$ are cables. For this structure, determine the
reaction at $B$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-007-cb-analysis-cabled-frame)

A force $P$ = 800 N is shown. 1. Find the $y$-component of $P$ with respect to $x$ and $y$ axis. 1. Find the $y'$-component of $P$ with respect to $x'$ and $y'$ axis. 1. Find the $y$-component of $P$ with respect to $x'$ and $y$ axis. 1. Find the $y'$-component of $P$ with respect to $x$ and $y'$ axis.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/008-components-force-different-pairs-axes)

The body on the 30° incline is acted upon by a force $P$ inclined at 20° with the horizontal. If $P$ is resolved into components parallel and perpendicular to incline and the value of the parallel component is 1800 N, compute the value of the perpendicular component and that of $P$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/009-force-given-component-parallel-incline)

The triangular block shown is subjected to the loads $P$ = 1600 lb and $F$ = 600 lb. If $AB$ = 8 in. and $BC$ = 6 in., resolve each load into components normal and tangential to $AC$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/010-components-force-normal-and-tangential-hypotenuse-triangle)

Three ropes are tied to a small metal ring. Three students are pulling, each trying to move the ring in their direction. If we look down from above, the forces and directions they are applying are shown in the figure. Find the net force on the ring due to the three applied forces.

R = 53.79 lb downward to the left at $\theta_x$ = 17.16°. [More info.](http://mathalino.com/reviewer/engineering-mechanics/011-resultant-three-forces-acting-ring)

Find the resultant vector of vectors $\mathbf{A}$ and $\mathbf{B}$ shown in the figure.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/012-resultant-two-velocity-vectors)

Three vectors $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ are shown in the figure. Find the magnitude and directions of one vector that will have the same effect as the three vectors.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/013-resultant-three-forces-angles-greater-90-degree)

$\mathbf{P}$ is directed at an angle $\alpha$ from $x$-axis and the 200 N force is acting at a slope of 5 vertical to 12 horizontal.

1. Find $\mathbf{P}$ and $\alpha$ if the resultant is 500 N to the right along the $x$-axis.
2. Find $\mathbf{P}$ and $\alpha$ if the resultant is 500 N upward to the right with a slope of 3 horizontal to 4 vertical.
3. Find $\mathbf{P}$ and $\alpha$ if the resultant is zero.

a. $P = 324.63$ N, $\alpha = 13.71^\circ$ b. $\alpha = 76.4^\circ$, $P = 490.68$ N c. $P = 200 \, \text{ N at} \, \alpha = 157.38°$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/014-solving-force-given-resultant)

Forces $\mathbf{F}$, $\mathbf{P}$, and $\mathbf{T}$ are concurrent and acting in the direction as shown.

1. Find the value of $F$ and $\alpha$ if $T$ = 450 N, $P$ = 250 N, $\beta$ = 30°, and the resultant is 300 N acting up along the $y$-axis.
2. Find the value of $F$ and $\alpha$ if $T$ = 450 N, $P$ = 250 N, $\beta$ = 30° and the resultant is zero.
3. Find the value of <$\alpha$ and $\beta$ if $T$ = 450 N, $P$ = 250 N, $F$ = 350 N, and the resultant is zero.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/015-solving-force-and-its-angle-and-angle-two-forces-given-resultant)

The magnitude of vertical force $F$ shown is 8000 N. Resolve $F$ into components parallel to the bars $AB$ and $AC$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/016-components-force-parallel-supporting-bars)

If the force $F$ shown in is resolved into components parallel to the bars $AB$ and $BC$, the magnitude of the component parallel to bar $BC$ is 4 kN. What are the magnitudes of $F$ and its component parallel to $AB$?

$F = 7.02$ kN, $F_{AB} = 6.75$ kN [More info.](http://mathalino.com/reviewer/engineering-mechanics/017-computation-force-magnitude-given-component-parallel-frame-member)

Determine the magnitude of force $B$ such that the net moment about point $A$ is zero.

[More info.](\mechanics\examples\images\HW1002-soln.png)

Resolve the forces \\(\textbf{F}_1\\) and \\(\textbf{F}_2\\) into components acting along the $u$ and $v$ axes and determine the magnitudes of the components.

$F_{1u}$ = 205 N, $F_{1v}$ = 160 N [More info.](https://www.youtube.com/watch?v=AGDkAFrBZcI&list=PLmjeuPDHP7zD7ogEGP0aaBDqYDAVu-yT3&index=3)

If the magnitude of the resultant force is to be 500 N, directed along the positive $y$ axis, determine graphically the magnitude of force $\mathbf{F}$ and its direction \\(\theta\\).

$\mathbf{F} = 959.8 N, $\theta$ = 45.2° [More info.](https://www.youtube.com/watch?v=8cyYqzlvSmg))

Determine the moment of force $\mathbf{F}$ about point $A$.

$M_A$ = 2366 N-m. See also [this video.](http://www.engineer4free.com/4/find-the-moment-of-a-force-about-a-point) [More info.](/mechanics/examples/images/HW1005-soln.png)

Find the equivalent resultant force and couple moment acting at $A$ and then the equivalent single force location along beam *AB*.

$F_r =$ 65.9 N @ 49.8° down and to the right $d$ = 0.21 m [More info.](https://www.youtube.com/watch?v=oueKQ5-dJQc)

Knowing that \\(F_1\\) = 200 lb, \\(F_2\\) = 450 lb, and \\(F_3\\) = 300 lb, determine the moment of each couple and the resultant moment acting on the plate.

$M_{net}$ = 950 ft-lb CW [More info.]((https://www.youtube.com/watch?v=-mDTSuIFuNA))

Find: a) the resultant force and moment at $A$ and, b) the reactions at $A$.

$ (M_R)_A = $ 61.072 kN-m $F_A$= 0 [More info.]((https://www.youtube.com/watch?v=TvSrDtee8SQ))

Determine the moment of the 600 lb force about point $O$.

$M$ = 2491 ft-lb [More info.](\mechanics\examples\images\HW1010-soln.png)

A horizontal force $P$ is applied to the handle of the puller. Determine, in terms of force $P$, the tension $T$ in the chain and the magnitude and direction of the force acting on the handle at pin $A$.

$\bar{T} = \text{5.36 P}$ at 20° down and to the right $\bar{A} = \text{4.43 P}$ at 24.4° up and to the left [More info.](\mechanics\examples\images\HW1011-soln.png)

The boat trailer is parked on a 15° slope. The two wheels at $A$ are free to turn, and the hitch $H$ behaves like a pinned connection. Determine the reaction at each wheel and at the hitch.

[More info.](\mechanics\examples\images\HW1012-soln.png)

Use the three force body method to determine the magnitudes of the reactions at wheels $A$ and hitch $H$.

[More info.](\mechanics\examples\images\HW1013-soln.png)

Recognizing that member *AB* is a two-force body, and that member *BC* is a three-force body, use the three-force-body method to determine the magnitude and direction of reaction force at point $C$.

[More info.](\mechanics\examples\images\HW1014-soln.png)

Determine the required magnitude of force $\mathbf{F}$ if the resultant couple moment on the frame is 200 lb-ft clockwise.

$F$ = 221.3 lb [More info.](https://www.youtube.com/watch?v=hYNgGKIKT8w)

Draw a free body diagram of the crowbar, then determine the reactions at smooth contact points $A$, $B$, and $C$ on the bar. Hint: You may want to take advantage of the fact that two unknowns are parallel, rather than trying to find a point where two unknowns intersect.

[More info.](\mechanics\examples\images\HW1016-soln.png)

The floor crane and driver have a combined weight of 1500 lb with a center of gravity at $G$. For the position shown, what is the maximum weight that can be lifted without causing the crane to tip over?

[More info.]()

The gangway for the *Enterprise* weighs 3000 lb and has a center of gravity at $G$. Determine the tension required in cable *CD* in order to lift point $B$ off the dock. Also, determine the $x$- and $y$- components of force at hinge $A$.

[More info.](\mechanics\examples\images\HW1018-soln.png)

Use integration to prove that the centroid of a quarter-circular area with radius $r$ is $\bar{x} = \bar{y} = \frac{4r}{3\pi}$.

[More info.](https://youtu.be/6WB_sBpSuOw)

Identify and circle any errors in the student drawn free body diagrams of bar $AB$, then draw a neat, correct, labeled free body diagram of your own.

[More info.](\mechanics\examples\images\HW1020-soln.png)

Identify and circle any errors in the student drawn free body diagrams of post $CD$, then draw a neat, correct, labeled free body diagram of your own. Assume that both parts have non-negligible weight, and that contact surfaces are smooth.

[More info.](\mechanics\examples\images\HW1021-soln.png)

Identify and circle any errors in the student drawn free body diagrams of bar $AB$, then draw a neat, correct, labeled free body diagram of your own. Assume that bar $AB$ has non-negligible weight.

[More info.](\mechanics\examples\images\HW1022-soln.png)

Use integration to determine the coordinates of the centroid of the area bounded by the two curves.

[More info.](\mechanics\examples\images\HW1023-soln.png)

First, watch this [Khan Academy Video](https://www.khanacademy.org/video/formula-area-polar-graph) where Sal discusses how to integrate using [polar coordinates](http://mathworld.wolfram.com/PolarCoordinates.html). In particular, he shows that the differential element of area in polar coordinates is \\(dA = \frac{1}{2} r^2\, d\theta \\). Then prove, using integration with polar coordinates, that the area of a semi-circle is $\pi r^2$ and that $\bar{y} = \frac{4 r}{3 \pi}$. Hints: 1. The equation of a circle with radius $r$ in polar coordinates is $r(\theta) = r$. 2. The process to find area and centroids is exactly the same as previous problems, only the form of the integrals is different. 3. Finding the area of a circle using polar coordinates is trivially easy. 4. To find $Q_x$ you will need to determine $\bar{y}_{el}$ as a function of $r$ and $\theta$.

[More info.](https://youtu.be/9e6c16jAWXw)

Use integration to show that moment of inertia of a rectangle with base $b$ and height $h$ about a centroidal axis parallel to the base is $\bar{I} = \frac{1}{12} b h^3$

$\bar{I} = \frac{1}{12} b h^3$ [More info.](\mechanics\examples\images\HW1025-soln.png)

Knowing that distance $h$ is 40 mm, determine the moment of inertia of the shape about the $x$ axis. Hint: break the shape up into three rectangles with their bases along the $x$ axis, then add their individual values of $I_x$.

$I_x= 846.7 \times 10^3 \textrm{ mm}^4$ [More info.](\mechanics\examples\images\HW1026-soln.png)

Knowing that distance $h$ is 40 mm, determine the moment of inertia of the shape about the $y$ axis.

$I_y= 206.7 \times 10^3 \textrm{ mm}^4$ [More info.](\mechanics\examples\images\HW1027-soln.png)

Determine the distance $h$ for which $I_x = I_y$ for the shaded area shown. Write a sentence which justifies your answer using a symmetry argument.

$h = 20 \textrm{ mm}$ [More info.](\mechanics\examples\images\HW1028-soln.png)

Find the moment of inertia and radius of gyration with respect to the $x$-axis for the shaded area shown.

$I_x = 209.2 \textrm{ in.}^4 k_x = 2.58 \textrm{ in}$ [More info.](\mechanics\examples\images\HW1029-soln.png)

Find the moment of inertia and radius of gyration with respect to the $y$-axis for the shaded area shown.

$I_y = 1971 \textrm{ in.}^4 k_y = 3.70 \textrm{ in}$ [More info.](\mechanics\examples\images\HW1030-soln.png)

Find the moment of inertia of the shaded area about the $x$- and $y$ axes.

$I_x = 250 \times 10^3 \textrm{ cm}^4 \bar{I}_y = 66.28 \times 10^3 \textrm{ cm}^4$ [More info.](\mechanics\examples\images\HW1031-soln.png)

Find the moment of inertia of the shaded area with respect to the $x$-axis.

$I_x = 13866 \textrm{ in.}^4$ [More info.](\mechanics\examples\images\HW1032-soln.png)

Given the truss and loading shown, identify any zero-force members, and then determine the forces in members $CB$, $CF$, $CD$, and $CG$. Indicate tension or compression.

[More info.](https://youtu.be/IUW7Ab2By58)

Given the truss and loading shown, identify any zero-force members, and then determine the forces in members $CB$, $CF$, $CD$, and $CG$. Use a symmetry argument to determine the forces in the remaining members. Indicate tension or compression.

[More info.](https://youtu.be/IUW7Ab2By58?t=660)

Determine the force that the bolt cutters exert on the bolt at $C$ when 100 N vertical forces are applied at the handles.

$F_c = 19.6 \text{ kN}$ [More info.](https://youtu.be/_QqcKxWJ-5o)

Determine the gripping force on the object at $C$ when a 50 lb force $P$ is applied to the handles as shown.

[More info.]()

The tree feller cuts off large trees near the ground then continues to grasp the trunk. Determine the force in hydraulic cylinder *AB* for the position show if the tree weighs 6000 lb.

[More info.](https://youtu.be/8kdOZ4Ze3iY)

The tractor shovel carries a 500-kg load of soil having a center of mass at $G$. Compute the forces developed in hydraulic cylinders *IJ* and *BC* due to this loading.

[More info.](https://youtu.be/Xzf3I4cfc0M)

Find the reaction forces acting on the frame at $A$ and the internal forces at point $K$ when force $P$ is 90 lb.

[More info.](https://youtu.be/Hk3r5nECMqA)

Determine the coordinates of the centroid of the parabolic spandrel bounded by the function $y(x) = b + kx^2$, and the lines $x =a$ and $y=b$ in terms of constants $a$ and $b$.

[More info.](\mechanics\examples\images\HW1040-soln.png)

Determine the coordinates of the centroid of the parabolic spandrel bounded by the function $y(x) = b + kx^2$, and the positive $x$ and $y$ axes in terms of constants $a$ and $b$.

[More info.](\mechanics\examples\images\HW1041-soln.png)

Determine the coordinates of the centroid of the region bounded by the parabola and the line in terms of constants $a$ and $b$.

[More info.]()

Calculate the moment of inertia with respect to the $x$-axis for the combined shape. Dimensions are in mm.

$I_x = 10.47 \times 10^6 \textrm{ mm}^4$ [More info.](https://youtu.be/-nrjwLBVAwk)

Determine the internal shear force, normal force, and bending moment at point $C,$ just to the left of the concentrated load. Use the standard sign convention for internal forces.

[More info.](https://youtu.be/mYnG4QrxzjQ)

If $w$ = 150 N/m, determine the internal loadings at point $D$. Use standard sign convention for internal loadings.

[More info.](https://youtu.be/qUCOQ-4dNu0)

Determine the normal force, shear force, and bending moment at a section passing through point $D$ of the two member frame.

[More info.](https://youtu.be/vBicIyzDZgs)

Determine $I_x$ for the region in the first quadrant between by the two polar functions: $r_1 = \sin(2\theta)$ and $r_2 = \cos(\theta)$.

[More info.]()

Derive the shear and moment equations for the beam and draw the corresponding shear and bending moment diagrams, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](\mechanics\examples\images\HW1048-soln.png)

Derive the shear and moment equations for the beam and draw the corresponding shear and bending moment diagrams, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](\mechanics\examples\images\HW1049-soln.png)

Cables *AB* and *AC* have the slopes indicated. Determine the angle $\theta$. You should not need a calculator to find the answer.

$θ= 90°$ [More info.](\mechanics\examples\images\HW1050-soln.png)

Determine the measure of angles $θ$ and $ϕ$. You should not need a calculator to find the answer.

$\theta = \phi = 72.5°$ [More info.](\mechanics\examples\images\HW1051-soln.png)

Determine the perpendicular distance $D$. You should not need a calculator to find the answer.

$D = 3$ m [More info.](\mechanics\examples\images\HW1052-soln.png)

If the distance between points $A$ and $D$ is 1.75 m, how long is the cable?

$\ell = 1.27$ m [More info.](\mechanics\examples\images\HW1053-soln.png)

Determine the distance *AC* when angle $\theta$ is 28°.

$AC = 200.3$ mm [More info.](\mechanics\examples\images\HW1054-soln.png)

Determine the measures of angle *BAC*.

$\angle BAC = 73.6°$ mm [More info.](\mechanics\examples\images\HW1055-soln.png)

Draw the shear and bending-moment diagrams for the beam and loading shown.

[More info.](\mechanics\examples\images\HW1056-soln.png)

Draw the shear and bending-moment diagrams for the beam and loading shown.

[More info.](\mechanics\examples\images\HW1057-soln.png)

The strength of the wide flanged rolled section shown is increased by welding a channel to its upper flange. Determine the moment of inertia and radius of gyration of the combined section with respect to its centroidal $x$ and $y$ axes.

$\bar{y} = 27 \text{ mm}$ $\bar{I}_{x'} = 745 \times 10^6 \text{ mm}^4$ $\bar{I}_{y} = 91.4 \times 10^6 \text{ mm}^4$ [More info.](\mechanics\examples\images\L24H-3soln.png)

Assuming clockwise moments as positive, compute the moment of force $F$ = 200 N and force $P$ = 165 N about points $A$, $B$, $C$, and $D$.

Moment of force F: $M_A = 180 $ N-m $M_B = -288 $ N-m $M_C = -180 $ N-m $M_D = -108 $ N-m Moment of force P: $M_A = 0$ $M_B = 41.186 $ N-m $M_C = 164.746 $ N-m $M_D = -164.746 $ N-m [More info.](http://mathalino.com/reviewer/engineering-mechanics/226-moment-force-about-different-points)

Two forces $P$ and $Q$ pass through a point $A$ which is 4 m to the right of and 3 m above a moment center $O$. Force $P$ is 890 N directed up to the right at 30° with the horizontal and force $Q$ is 445 N directed up to the left at 60° with the horizontal. Draw a sketch of the situation and determine the moment of the resultant of these two forces about point $O$.

$M_O = 1676.74 \, \text{ N}\cdot\text{m}$ (counterclockwise). [More info.](http://mathalino.com/reviewer/engineering-mechanics/227-moment-resultant-force-about-point-o)

Without computing the magnitude of the resultant, compute where the resultant of the forces shown intersects the $x$ and $y$ axes.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/228-intercepts-resultant-force)

Find the $y$-coordinate of point $A$ so that the 361-lb force, directed up and to the right with a 2:3 slope, will have a clockwise moment of 400 ft-lb about $O$. Also determine the $x$ and $x$ intercepts of the line of action of the force.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/229-y-coordinate-point-application-force)

A force $P$ passing through points $A$ and $B$ has a clockwise moment of 300 ft-lb about $O$. Compute the value of $P$.

$P = 50\sqrt{5} \, \text{ lb}$ down to the right from $A$ to $B$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/231-force-p-producing-clockwise-moment-about-origin)

The moment of a certain force $F$ is 180 ft·lb clockwise about $O$ and 90 ft·lb counterclockwise about $B$. If its moment about $A$ is zero, determine the force.

$F$ = 75 lb downward to the right at $\theta_x$ = 36.87° and $x$-intercept at (4, 0) [More info.](http://mathalino.com/reviewer/engineering-mechanics/232-moment-force-about-points-o-and-b)

The line of action of force $P$ intersects the $x$ axis at 4 ft to the right of $O$. If its moment about $A$ is 170 ft·lb counterclockwise and its moment about $B$ is 40 ft·lb clockwise, determine its $y$ intercept.

$y$ intercept of force P is (0, -8/3) [More info.](http://mathalino.com/reviewer/engineering-mechanics/233-force-creating-counterclockwise-and-clockwise-moments)

A parallel force system acts on the lever shown. Determine the magnitude and position of the resultant.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/236-computation-resultant-parallel-forces-acting-lever)

Four parallel forces act on a rocker arm which is free to rotate about point $O$. Determine the resultant of the four parallel forces.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/237-finding-resultant-parallel-forces-acting-both-sides-rocker-arm)

The beam $AB$ supports a load which varies an intensity of 220 N/m to 890 N/m. Calculate the magnitude and position of the resultant load.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/238-finding-resultant-trapezoidal-loading)

The 16-ft wing of an airplane is subjected to a lift which varies from zero at the tip to 360 lb per ft at the fuselage according to $w = 90x^{1/2}$ lb per ft where $x$ is measured from the tip. Compute the resultant and its location from the wing tip.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/239-resultant-lift-wing-airplane)

The shaded area represents a steel plate of uniform thickness. $A$ hole of 4-in. diameter has been cut in the plate. Locate the center of gravity the plate. *Hint:* The weight of the plate is equivalent to the weight of the original plate minus the weight of material cut away. Represent the original plate weight of plate by a downward force acting at the center of the 10 × 14 in. rectangle. Represent the weight of the material cut away by an upward force acting at the center of the circle. Locate the position of the resultant of these two forces with respect to the left edge and bottom of the plate.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/240-how-locate-centroid-metal-plate-circular-hole)

Locate the amount and position of the resultant of the loads acting on the Fink truss.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/241-finding-resulatnt-vertical-forces-acting-fink-truss)

Find the value of $P$ and $F$ so that the four forces shown produce an upward resultant of 300 lb acting at 4 ft from the left end of the bar.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/242-finding-unknown-two-forces-given-resultant)

The resultant of three parallel loads (one is missing in the figure) is 13.6 kg acting up at 3 m to the right of $A$. Compute the magnitude and position of the missing load.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/243-finding-magnitude-and-position-missing-force)

A couple consists of two vertical forces of 60 lb each. One force acts up through $A$ and the other acts down through $D$. Transform the couple into an equivalent couple having horizontal forces acting through $E$ and $F$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/245-couple-box)

Determine the resultant moment about point $A$ of the system of forces shown in. Each square is 1 ft on a side.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/246-system-couples-and-forces-it)

The three-step pulley shown is subjected to the given couples. Compute the value of the resultant couple. Also determine the forces acting at the rim of the middle pulley that are required to balance the given system.

$C_R = $760 lb-in counterclockwise The resultant couple is composed of two 63.3 lb forces. [More info.](http://mathalino.com/reviewer/engineering-mechanics/resultent-couples-3-step-pulley)

To close a gate valve it is necessary to exert two forces of 60 lb at opposite sides of a handwheel 3 ft in diameter. Through an accident the wheel is broken and the valve must be closed by a thrusting bar through a slot in the valve stem and exerting a force 4 ft out from the center. Determine the force required and draw a free-body diagram of the bar.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/248-handwheel-replaced-lever-close-gate-vale)

The diagram represents the top view of a speed reducer which is geared for a four to one reduction in speed. The torque input at the horizontal shaft $C$ is 100 lb·ft. The torque output at the horizontal shaft $D$, because of the speed reduction, is 400 lb·ft. Compute the torque reaction at the mounting bolts $A$ and $B$ holding the reducer to the floor. Hint: The torque reaction is caused by the unbalanced torque, which is a couple.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/249-reactions-bolts-speed-reducer-gear-box)

The cantilever truss shown carries a vertical load of 10.8 kN. The truss is supported by bearing at $A$ and $B$ which exert the forces $A_v$, $A_h$, and $B_h$. The four forces shown constitute two couples which must have opposite moment effects to prevent movement of the truss. Determine the magnitude of the supporting forces.

$A_v =$ 10.8 kN upward $A_h =$ 16.2 kN to the left $B_h =$ 16.2 kN to the right [More info.](http://mathalino.com/reviewer/engineering-mechanics/250-support-reactions-cantilever-truss)

A vertical force $P$ at $A$ and another vertical force $F$ at $B$ in produce a resultant of 100 lb down at $D$ and a counterclockwise couple $C$ of 200 lb·ft. Find the magnitude and direction of forces $P$ and $F$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/251-two-forces-producing-couple-and-force)

A force system consists of a clockwise couple of 480 N·m plus a 240 N force directed up to the right through the origin of $x$ and $y$ axes at $\theta_x$ = 30°. Replace the given system by an equivalent single force and compute the intercepts its line of action with the $x$ and $y$ axes.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/252-equivalent-single-force-force-and-couple)

A system of forces reduces to downward vertical force 400 lb through $A$ plus a counterclockwise couple of 800 lb·ft. Determine the single force that will produce an equivalent effect.

400 lb straight down, 2 feet to the right of the $y$ axis. [More info.](http://mathalino.com/reviewer/engineering-mechanics/253-254-equivalent-single-force-replace-force-and-couple)

A short compression member carries an eccentric load $P$ = 200 lb situated 2 in. from the axis of the member, as shown in the figure. In strength of materials it is learned that the internal stresses are determined from the equivalent axial load and couple into which $P$ may be resolved. Determine the equivalent axial load and couple.

200 lb down, plus 400 in-lb clockwise moment. [More info.](http://mathalino.com/reviewer/engineering-mechanics/255-equivalent-loads-compression-member-eccentric-load)

Replace the system of forces acting on the frame by a resultant $R$ at $A$ and a couple acting horizontally through $B$ and $C$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/257-horizontal-couple-equivalent-vertical-forces)

Replace the system of forces shown by an equivalent force through $O$ and a couple acting through $A$ and $B$. Solve if the forces of the couple are (a) horizontal and (b) vertical.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/258-equivalent-horizontal-and-vertical-couples)

The effect of a certain non-concurrent force system is defined by the following data: $\Sigma F_x$ = +90 kN, $\Sigma F_y$ = -60 kN, and $\Sigma M_O$ = 360 kN·m counterclockwise. Determine the point at which the resultant intersects the $x$-axis.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problems-260-261-resultant-non-concurrent-force-system)

In a certain non-concurrent force system it is found that $\Sigma F_x$ = -80 lb,$\Sigma F_y$ = +160 lb, and $\Sigma M_O$ = 480 lb·ft in a counterclockwise sense. Determine the point at which the resultant intersects the $y$-axis.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problems-260-261-resultant-non-concurrent-force-system)

Determine completely the resultant of the forces acting on the step pulley shown.

$\mathbf{R}$ = 1254.89 lb downward to the right at $\theta_x$ = 44.21° and passes through the axle. [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-262-resultant-non-concurrent-force-system)

Determine the resultant of the force system shown and its $x$ and $y$ intercepts.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-263-resultant-non-concurrent-force-system)

Completely determine the resultant with respect to point $O$ of the force system shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-264-resultant-non-concurrent-force-system)

Compute the resultant of the three forces shown. Locate its intersection with $x$ and $y$ axes.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-265-resultant-non-concurrent-force-system)

Determine the resultant of the three forces acting on the dam and locate its intersection with the base $AB$. For good design, this intersection should occur within the middle third of the base. Does it?

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-266-resultant-non-concurrent-force-system)

The Howe roof truss shown carries the given loads. The wind loads are perpendicular to the inclined members. Determine the magnitude of the resultant, its inclination with the horizontal, and where it intersects $AB$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-267-resultant-non-concurrent-force-system)

The resultant of four forces, of which three are shown, is a couple of 480 lb·ft clockwise in sense. If each square is 1 ft on a side, determine the fourth force completely.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-268-resultant-non-concurrent-force-system)

Repeat Prob. 268 is the resultant is 390 lb directed down to the right at a slope of 5 to 12 passing through point $A$. Also determine the $x$ and $y$ intercepts of the missing force $F$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-269-resultant-non-concurrent-force-system)

The three forces shown are required to cause a horizontal resultant acting through point $A$. If $F$ = 316 lb, determine the values of $P$ and $T$. Hint: Apply $M_R = \Sigma M_B$ to determine $R$, then $M_R =\Sigma M_C$ to find $P$, and finally $M_R = \Sigma M_D$ or $R_y= \Sigma F_y$ to compute $T$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-270-resultant-non-concurrent-force-system)

The three forces create a vertical resultant acting through point $A$. If $T$ is known to be 361 lb, compute the values of $F$ and $P$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-271-resultant-non-concurrent-force-system)

The cable and boom shown support a load of 600 lb. Determine the tensile force $T$ in the cable and the compressive for $C$ in the boom.

$T =$ 439.24 lb $C =$ 537.94 lb [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-308-equilibrium-concurrent-force-system)

A cylinder weighing 400 lb is held against a smooth incline by means of the weightless rod $AB$ as shown. Determine the forces $P$ and $N$ exerted on the cylinder by the rod and the incline.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-309-equilibrium-concurrent-force-system)

A 300-lb box is held at rest on a smooth plane by a force $P$ inclined at an angle $\theta$ with the plane as shown. If $\theta$ = 45°, determine the value of $P$ and the normal pressure $N$ exerted by the plane.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-310-311-equilibrium-concurrent-force-system)

If the value of $P$ is 180 lb. Determine the angle $\theta$ at which it must be inclined with the smooth plane to hold 300-lb box in equilibrium.

$\theta =$ 33.56° [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-310-311-equilibrium-concurrent-force-system)

Determine the magnitude of $P$ and $F$ necessary to keep the concurrent force system shown in equilibrium.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-3112-equilibrium-concurrent-force-system)

The free body diagram represents the concurrent force system acting at a joint of a bridge truss. Determine the value of $P$ and $F$ to maintain equilibrium of the forces.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-313-equilibrium-concurrent-force-system)

The five forces shown are in equilibrium. Compute the values of $P$ and $F$.

$F$ = -12.63 kN $P$ = -5.31 kN [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-314-equilibrium-concurrent-force-system)

The 300-lb force and the 400-lb force shown are to be held in equilibrium by a third force $F$ acting at an unknown angle $\theta$ with the horizontal. Determine the values of $F$ and $\theta$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-315-equilibrium-concurrent-force-system)

Determine the values of $\alpha$ and $\theta$ so that the forces shown will be in equilibrium.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-316-equilibrium-concurrent-force-system)

The system of knotted cords shown in support the indicated weights. Compute the tensile force in each cord.

$C$ = 400 lb $D$ = 207.06 lb $B$ = 914.16 lb $A$ = 846.41 lb [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-317-equilibrium-concurrent-force-system)

Three bars, hinged at $A$ and $D$ and pinned at $B$ and $C$ as show, form a four-link mechanism. Determine the value of $P$ that will prevent motion.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-318-equilibrium-concurrent-force-system)

Cords are loop around a small spacer separating two cylinders each weighing 400 lb and pass over a frictionless pulleys to weights of 200 lb and 400 lb. Determine the angle $\theta$ and the normal pressure $N$ between the cylinders and the smooth horizontal surface.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-319-equilibrium-concurrent-force-system)

The Fink truss shown is supported by a roller at $A$ and a hinge at $B$. The given loads are normal to the inclined member. Determine the reactions at $A$ and $B$. Hint: Replace the loads by their resultant.

$R_A$ = 4618.80 lb $R_B$ = 4618.80 lb 30° CW from negative $x$-axis. [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-322-equilibrium-force-system)

The truss shown is supported by a hinge at $A$ and a roller at $B$. $A$ load of 20 kN is applied at $C$. Determine the reactions at $A$ and $B$. The truss may be treated as a rigid body.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-323-equilibrium-force-system)

A wheel of 10-in radius carries a load of 1000 lb, as shown in the figure. (a) Determine the horizontal force $P$ applied at the center which is necessary to start the wheel over a 5-in. block. Also find the reaction at the block. (b) If the force $P$ may be inclined at any angle with the horizontal, determine the minimum value of $P$ to start the wheel over the block; the angle $P$ makes with the horizontal; and the reaction at the block.

**Part (a)** $P$ = 1732.05 lb $R$ = 2000 lb **Part (b)** $\theta$ = 60° $P_{min}$ = 866.02 lb $R$ = 500 lb [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-323-equilibrium-three-force-system)

Determine the amount and direction of the smallest force $P$ required to start the wheel over the block. What is the reaction at the block?

$\alpha$ = 71.41° $P_{min}$ = 1895.65 lb $R$ = 637.59 lb [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-325-equilibrium-three-force-system)

The cylinders have the indicated weights and dimensions. Assuming smooth contact surfaces, determine the reactions at $A$, $B$, $C$, and $D$ on the cylinders.

$R_A$ = 347.39 kN $R_B$ = 600 kN $R_C$ = 400.85 kN $R_D$ = 347.39 kN [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-326-equilibrium-force-system)

Forces $P$ and $F$ acting along the bars shown maintain equilibrium of pin $A$. Determine the values of $P$ and $F$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-327-equilibrium-force-system)

Two weightless bars pinned together as shown support a load of 35 kN. Determine the forces $P$ and $F$ acting respectively along bars $AB$ and $AC$ to maintain equilibrium of pin $A$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-328-equilibrium-force-system)

Two cylinders $A$ and $B$, weighing 100 lb and 200 lb respectively, are connected by a rigid rod curved parallel to the smooth cylindrical surface shown. Determine the angles $\alpha$ and $\beta$ that define the position of equilibrium.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-328-equilibrium-force-system-0)

Determine the reactions for the beam shown.

$R_1$ = 1580 lb $R_2$ = 520 lb [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-332-equilibrium-parallel-force-system)

Determine the reactions $R_1$ and $R_2$ of the beam loaded with a concentrated load of 1600 lb and a load varying from zero to an intensity of 400 lb per ft.

$R_1$ = 1900 lb $R_2$ = 2100 lb [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-333-equilibrium-parallel-force-system)

Determine the reactions for the beam loaded as shown.

$R_1$ = 23.4 kN $R_2$ = 29.1 kN [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-334-equilibrium-parallel-force-system)

The roof truss is supported by a roller at $A$ and a hinge at $B$. Find the values of the reactions.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-335-equilibrium-parallel-force-system)

The cantilever beam shown is built into a wall 2 ft thick so that it rests against points $A$ and $B$. The beam is 12 ft long and weighs 100 lb per ft. Find the reactions at $A$ and $B$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-336-equilibrium-parallel-force-system)

The upper beamis supported at $D$ and a roller at $C$ which separates the upper and lower beams. Determine the values of the reactions at $A$, $B$, $C$, and $D$. Neglect the weight of the beams.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-337-equilibrium-parallel-force-system)

The two 12-ft beams shownare to be moved horizontally with respect to each other and load $P$ shifted to a new position on $CD$ so that all three reactions are equal. How far apart will $R_2$ and $R_3$ then be? How far will $P$ be from D?

$P$ is 8 ft to the left of $D$. $R_2$ and $R_3$ are 6 ft apart. [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-338-equilibrium-parallel-force-system)

The differential chain hoist shown consists of two concentric pulleys rigidly fastened together.The pulleys form two sprockets for an endless chain looped over them in two loops. In one loop is mounted a movable pulley supporting a load $W$. Neglecting friction, determine the maximum load $W$ that can just be raised by a pull $P$ supplied as shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-339-equilibrium-parallel-force-system)

For the system of pulleys shown, determine the ratio of $W$ to $P$ to maintain equilibrium. Neglect axle friction and the weights of the pulleys.

$W/P$ = 9 [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-340-341-equilibrium-parallel-force-system)

If each pulley weighs 36 kg and $W$ = 720 kg, find $P$ to maintain equilibrium.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-340-341-equilibrium-parallel-force-system)

The wheel loads on a jeep are given. Determine the distance $x$ so that the reaction of the beam at $A$ is twice as great as the reaction at $B$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-342-equilibrium-parallel-force-system)

The weight $W$ of a traveling crane is 20 tons acting as shown. To prevent the crane from tipping to the right when carrying a load $P$ of 20 tons, a counterweight $Q$ is used. Determine the value and position of $Q$ so that the crane will remain in equilibrium both when the maximum load $P$ is applied and when the load $P$ is removed.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-343-equilibrium-parallel-force-system)

A boom $AB$ is supported in a horizontal position by a hinge $A$ and a cable which runs from $C$ over a small pulley at $D$ as shown. Compute the tension $T$ in the cable and the horizontal and vertical components of the reaction at $A$. Neglect the size of the pulley at $D$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-346-equilibrium-non-concurrent-force-system)

A boom $AB$ is supported in by a hinge $A$ and a cable which runs from $C$ over a small pulley at $D$ as shown. Compute the tension $T$ in the cable and the horizontal and vertical components of the reaction at $A$ if the cable pulls the boom $AB$ into a position at which it is inclined at 30° above the horizontal. The loads remain vertical. Neglect the size of the pulley at $D$.

$T = 216.51$ lb $A_H = 108.25$ lb $A_V = 112.50$ lb [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-347-equilibrium-non-concurrent-force-system)

The frame shown is supported in pivots at $A$ and $B$. Each member weighs 5 kN/m. Compute the horizontal reaction at $A$ and the horizontal and vertical components of the reaction at $B$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-348-equilibrium-non-concurrent-force-system)

The truss shown is supported on roller at $A$ and hinge at $B$. Solve for the components of the reactions.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-349-equilibrium-non-concurrent-force-system)

Compute the total reactions at $A$ and $B$ for the truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-350-equilibrium-non-concurrent-force-system)

The beam shown is supported by a hinge at $A$ and a roller on a 1 to 2 slope at $B$. Determine the resultant reactions at $A$ and $B$.

$R_B = 33.54$ kN $R_A = 18.03 \, \text{ kN}\text{ up to the right at }33.69^\circ$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-351-equilibrium-non-concurrent-force-system)

A pulley 4 ft in diameter and supporting a load 200 lb is mounted at $B$ on a horizontal beam as shown. The beam is supported by a hinge at $A$ and rollers at $C$. Neglecting the weight of the beam, determine the reactions at $A$ and $C$.

$R_C = 50$ lb $R_A = 180.27 \, \text{ lb} \text{ up to the right at } 16.1^\circ \text{ from horizontal.}$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-352-equilibrium-non-concurrent-force-system)

Compute the total reactions at $A$ and $B$ on the truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-354-equilibrium-non-concurrent-force-system)

Determine the reactions at $A$ and $B$ on the Fink truss shown. Members $CD$ and $FG$ are respectively perpendicular to $AE$ and $BE$ at their midpoints.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-355-equilibrium-non-concurrent-force-system)

The cantilever truss shown is supported by a hinge at $A$ and a strut $BC$. Determine the reactions at $A$ and $B$.

$R_B = 34.64$ kN $R_A = 20$ kN up to the right at 60° from horizontal. [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-356-equilibrium-non-concurrent-force-system)

The uniform rod weighs 420 lb and has its center of gravity at $G$. Determine the tension in the cable and the reactions at the smooth surfaces at $A$ and $B$.

$R_B = 254.56$ lb $T = 180$ lb $R_A = 240$ lb [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-357-equilibrium-non-concurrent-force-system)

A bar $AE$ is in equilibrium under the action of the five forces shown. Determine $P$, $R$, and $T$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-358-equilibrium-non-concurrent-force-system)

A 4-m bar of negligible weight rests in a horizontal position on the smooth planes shown. Compute the distance $x$ at which load $T$ = 10 kN should be placed from point $B$ to keep the bar horizontal.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-359-equilibrium-non-concurrent-force-system)

Referring to Problem 359, what value of $T$ acting at $x$ = 1 m from $B$ will keep the bar horizontal.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-360-equilibrium-non-concurrent-force-system)

Referring to Problem 359, if $T$ = 30 kN and $x$ = 1 m, determine the angle θ at which the bar will be inclined to the horizontal when it is in a position of equilibrium.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-361-equilibrium-non-concurrent-force-system)

Determine the force in each bar of the truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-403-method-joints)

Derive the shear and moment equations for the beam and draw the corresponding shear and bending moment diagrams, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[See](/images/403-load-shear-moment-diagrams.png) [More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-403-shear-and-moment-diagrams)

Determine the forces in the members of the roof truss shown.

$AB = 450$ N (C) $AC = 389.71$ N (T) $BC = 450$ N (T) $BD = 900$ N (C) $CD = 389.71$ N (T) [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-403-method-joints-0)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-404-shear-and-moment-diagrams)

Determine the force in each bar of the truss shown caused by lifting the 120 kN load at a constant velocity of 8 m per sec. What change in these forces, if any, results from placing the roller support at $D$ and the hinge support at A?

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-405-method-joints)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-405-shear-and-moment-diagrams)

The cantilever truss is hinged at $D$ and $E$. Find the force in each member.

$AB = 2000 \textrm{ N (T)}$ $AC = 1732.05 \textrm{ N (C)}$ $BC = 866.02 \textrm{ N (C)}$ $BD = 2500 \textrm{ N (T)}$ $CD = 2020.72 \textrm{ N (T)}$ $CE = 3175.42 \textrm{ N (C)}$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-406-cantilever-truss-method-joints)

Write shear and moment equations for the beam and draw shear and moment diagrams. Also, specify values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam, and neglect the mass of the beam in each problem.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-406-shear-and-moment-diagrams)

In the cantilever truss shown, compute the force in members $AB$, $BE$, and $DE$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-407-cantilever-truss-method-joints)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-407-shear-and-moment-diagrams)

Compute the force in each member of the Warren truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-407-warren-truss-method-joints)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-408-shear-and-moment-diagrams)

Determine the force in members $AB$, $BD$, $BE$, and $DE$ of the Howe roof truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-409-howe-roof-truss-method-joints)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-409-shear-and-moment-diagrams)

Determine the force in each member of the Pratt roof truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-410-pratt-roof-truss-method-joints)

Derive the shear and moment equations for the beam and draw the corresponding shear and bending moment diagrams, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-410-shear-and-moment-diagrams)

Determine the force in members $AB$, $AC$, $BD$, $CD$, and $CE$ of the cantilever truss shown. If the loads were applied at $C$ and $E$ instead of at $B$, specify which members would have their internal force changed.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-411-cantilever-truss-method-joints)

Cantilever beam carrying a distributed load with intensity varying from $w_0$ at the free end to zero at the wall, as shown. Write shear and moment equations for the beam and draw shear and moment diagrams. Also, specify values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam, and neglect the mass of the beam in each problem.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-411-shear-and-moment-diagrams)

Compute the force in each member of the truss shown in. If the loads at $B$ and $D$ are shifted vertically downward to add to the loads at $C$ and $E$, would there be any change in the reactions? Which members, if any, would undergo a change in internal force?

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-412-right-triangular-truss-method-joints)

Write shear and moment equations for the beam and draw shear and moment diagrams. Also, specify values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam, and neglect the mass of the beam in each problem.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-412-shear-and-moment-diagrams)

Determine the force in each member of the crane shown.

$AC = 131 \textrm{ kN (T)}$ $AB = 120 \textrm{ kN (C)}$ $BC = 60 \textrm{ kN (C)}$ $BD = 104 \textrm{ kN (C)}$ $CD = 0$ $CE = 90 \textrm{ kN (T)}$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-413-crane-method-joints)

Write shear and moment equations for the beam and draw shear and moment diagrams. Also, specify values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam, and neglect the mass of the beam in each problem.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-413-shear-and-moment-diagrams)

Determine the force in members $AB$, $BD$, and $CD$ of the truss shown. Also solve for the force on members $FH$, $DF$, and $DG$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-414-truss-method-joints)

Cantilever beam carrying the load shown. Write shear and moment equations for the beam and draw shear and moment diagrams. Also, specify values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam, and neglect the mass of the beam in each problem.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-414-shear-and-moment-diagrams)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-415-shear-and-moment-diagrams)

Beam carrying uniformly varying load. Write shear and moment equations for the beam and draw shear and moment diagrams. Also, specify values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam, and neglect the mass of the beam in each problem.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-416-shear-and-moment-diagrams)

Using the method of sections, determine the force in members $BD$, $CD$, and $CE$ of the roof truss shown.

$F_{BD} = 160 \, \text{kN (C)}$ $F_{CD} = 200 \, \text{kN (C)}$ $F_{CE} = 320 \, \text{kN (T)}$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-417-roof-truss-method-sections)

A beam is carrying the triangular loading shown. Write shear and moment equations for the beam and draw shear and moment diagrams. Also, specify values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam, and neglect the mass of the beam in each problem.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-417-shear-and-moment-diagrams)

The Warren truss loaded as shownis supported by a roller at $C$ and a hinge at $G$. By the method of sections, compute the force in the members $BC$, $DF$, and $CE$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-418-warren-truss-method-sections)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-418-shear-and-moment-diagrams)

Use the method of sections to determine the force in members $BD$, $CD$, and $CE$ of the Warren truss described in Problem 408.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-419-warren-truss-method-sections)

Write shear and moment equations for the beam and draw shear and moment diagrams. Also, specify values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam, and neglect the mass of the beam in each problem.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-419-shear-and-moment-diagrams)

Determine the force in members $DF$, $DG$, and $EG$ of the Howe truss shown.

$F_{DF} = $ 280 kN (C) $F_{DG} = $ 150 kN (C) $F_{EG} = $ 400 kN (T) [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-420-howe-truss-method-sections)

A total distributed load of 30 kips supported by a uniformly distributed reaction as shown. Write shear and moment equations for the beam and draw shear and moment diagrams. Also, specify values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam, and neglect the mass of the beam in each problem.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-420-shear-and-moment-diagrams)

Use the method of sections to compute for the force in members $DF$, $EF$, and $EG$ of the cantilever truss.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-421-cantilever-truss-method-sections)

Write the shear and moment equations as functions of the angle $\theta$ for the built-in arch shown.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-421-shear-and-moment-equations)

Refer to the truss described in Problem 412 and compute the force in members $BD$, $CD$, and $CE$ by the method of sections.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-422-right-triangular-truss-method-sections)

Write the shear and moment equations for the semicircular arch shown if (a) the load $P$ is vertical as shown, and (b) the load is applied horizontally to the left at the top of the arch.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-422-shear-and-moment-equations)

Use the method of sections to determine the force acting in members $DF$, $EF$, and $EG$ of the Howe truss described in Problem 409.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-423-howe-roof-truss-method-sections)

For the truss shown, determine the force in $BF$ by the method of joints and then check this result using the method of sections. Hint: To apply the method of sections, first obtain the value of $BE$ by inspection.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-424-solution-method-joints-and-method-sections)

In the Fink truss shown, the web members $BC$ and $EF$ are perpendicular to the inclined members at their midpoints. Use the method of sections to determine the force in members $DF$, $DE$, and $CE$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-425-fink-truss-method-sections)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-425-relation-between-load-shear-and)

Using the method of sections, compute the force in bars $FH$, $GH$, and $EK$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-426-fink-truss-method-sections)

Cantilever beam acted upon by a uniformly distributed load and a couple as shown. Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-426-relation-between-load-shear-and)

Determine the force in bars $BD$, $CD$, and $DE$ of the nacelle truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-427-interior-members-nacelle-truss-method-sections)

Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-427-relation-between-load-shear-and)

Use the method of sections to determine the force in members $DF$, $FG$, and $GI$ of the triangular Howe truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-428-howe-truss-method-sections)

Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-428-relation-between-load-shear-and)

For the cantilever truss shown, determine the forces in members $DF$, $FH$, $FI$, $GI$, and $FG$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-429-cantilever-truss-method-sections)

Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-429-relationship-between-load-shear)

The loads on the Parker truss are shown, determine the forces in members $BD$, $BE$, $CE$, and $DE$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-430-parker-truss-method-sections)

Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-430-relationship-between-load-shear)

Determine the force in the members $DF$, $DG$, and $EG$ for the Parker truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-431-members-third-panel-parker-truss)

Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-431-relationship-between-load-shear)

Use the method of sections to compute the force in members $AB$, $AD$, $BC$, and $BD$ of the truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-432-force-members-truss-method-sections)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-432-relationship-between-load-shear)

Compute the forces in bars $AB$, $AC$, $DF$, and $DE$ of the scissors truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-433-scissors-truss-method-sections)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-433-relationship-between-load-shear)

Compute the force in bars $GI$, $GH$, $EH$, and $HI$ for the scissors truss shown.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-434-scissors-truss-method-sections)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-434-relationship-between-load-shear)

For the transmission tower shown, determine the force in member $CJ$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-435-transmission-tower-method-sections)

Beam loaded and supported as shown. Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-435-relationship-between-load-shear)

Assume that counter diagonals act from $B$ to $E$ and from $E$ to $F$ in addition to the counter diagonals $CD$ and $DG$ shown in the figure. Assuming that these counter diagonals can support tension only, determine which diagonals are acting and the force in each.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/436-howe-truss-counter-braces)

Draw the shear and bending moment diagrams for the beam and loading shown, specifying values at all change of loading positions and at points of zero shear. Let $x$ be the distance measured from left end of the beam. Neglect the weight of the beam.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-436-relationship-between-load-shear)

The center panel of the truss shown contains two flexible cables. What load $P$ will cause a compressive force of 20 kN in BD? Then determine which tension diagonal $BE$ or $CD$ is acting and the force in it.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-437-truss-counter-diagonals)

A cantilever beam loaded as shown in the diagram. Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-437-relationship-between-load-shear)

The center diagonals of the truss can support tension only. Compute the force in each center diagonal and the force in $BC$, $DE$, and $FG$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-438-truss-redundant-members)

The beam loaded as shown in consists of two segments joined by a frictionless hinge at which the bending moment is zero. Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-438-relationship-between-load-shear)

A beam supported on three reactions as shown consists of two segments joined by frictionless hinge at which the bending moment is zero. Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-439-relationship-between-load-shear)

For the frame loaded as shown, determine the horizontal and vertical components of the pin pressure at $B$. Specify directions (up or down; left or right) of the force as it acts upon member $CD$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-440-frame-analysis-method-members)

The structure shown is hinged at $A$ and $C$. Find the horizontal and vertical components of the hinge force at $B$, $C$, and $A$.

$A_V = $ 20 kN $A_H = $ 17.5 kN $B_V = $ 10 kN $B_H = $ 17.5 kN $C_V= $ 10 kN $C_H = $ 17.5 kN [More info.](http://mathalino.com/reviewer/engineering-mechanics/441-analysis-frame-method-members)

Each member of the frame shown weighs 50 lb per ft. Compute the horizontal and vertical components of the pin pressure at $C$, $D$, and $F$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-442-analysis-frame-method-members)

A beam carrying a uniformly varying load. Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-442-relationship-between-load-shear)

The frame shown is hinged to rigid supports at $A$ and $E$. Find the components of the hinge forces $A$ and $E$ and the forces in members $BC$ and $BD$.

$A_H = $ 12 kN $E_H = $ 12 kN $A_V = B_V = $ 6 kN $E_V = $ 6 kN $F_{BC} = $ 10 kN tension $F_{BD} = $ 20 kN compression [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-443-analysis-frame-method-members)

Beam carrying the triangular loads shown. Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-443-relationship-between-load-shear)

The frame shown is supported by a hinge at $A$ and a roller at $E$. Compute the horizontal and vertical components of the hinge forces at $B$ and $C$ as they act upon member $AC$.

$B_H = $ 4 kN to the right $B_V = $ 12 kN upward $C_H = $ 4 kN to the left $C_V = $ 9.6 kN downward [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-444-analysis-frame-method-members)

Beam loaded as shown. Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-444-relationship-between-load-shear)

The frame shown is supported by a hinge at $E$ and a roller at $D$. Compute the horizontal and vertical components of the hinge force at $C$ as it acts upon $BD$.

$C_V = $ 7 kN upward $C_H = $ 28 kN to the right [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-445-analysis-x-frame-method-members)

Beam carrying the loads show. Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-445-relationship-between-load-shear)

A three-hinged arch is composed of two trusses hinged together at $D$. Compute the components of the reaction at $A$ and find the forces acting in bars $AB$ and $AC$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-446-compound-truss-turned-three-hinged-arch)

Beam loaded and supported as shown. Without writing shear and moment equations, draw the shear and moment diagrams for the beam shown. Give numerical values at all change of loading positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-446-relationship-between-load-shear)

The truss are joined as shown to form a three-hinged arch. Determine the horizontal and vertical components of the hinge force at $B$ and then determine the type and magnitude of force in bars $BD$ and $BE$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-447-compound-truss-formed-three-hinged-arch)

Draw moment and load diagrams corresponding to the given shear diagram. Specify values at all change of load positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-447-relationship-between-load-shear)

A beam carrying the loads show is composed of three segments. It is supported by four vertical reactions and joined by two frictionless hinges. Determine the values of the reactions.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-448-beam-three-segments-joined-internal-hinges)

Draw moment and load diagrams corresponding to the given shear diagram. Specify values at all change of load positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-448-relationship-between-load-shear)

The bridge shown consists of two end sections, each weighing 20 tons with center of gravity at $G$, hinged to a uniform center span weighing 12 tons. Compute the reactions at $A$, $B$, $E$, and $F$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-443-reactions-supports-bridge)

Draw moment and load diagrams corresponding to the given shear diagram. Specify values at all change of load positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-449-relationship-between-load-shear)

The billboard $BC$ weighing 1000 lb is subjected to a wind pressure of 300 lb/ft. Neglecting the weights of the support-members, determine the component of the hinge forces at $A$ and $F$.

$A_H$ = 4500 lb to the left $A_V$ = 2250 lb downward $F_H$ = 1500 lb to the right $F_V$ = 3250 lb upward [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-450-reactions-hinge-support-frame-holding)

Draw moment and load diagrams corresponding to the given shear diagram. Specify values at all change of load positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-450-relationship-between-load-shear)

The frame shown is hinged at $E$ and roller supported at $A$. Determine the horizontal and vertical components of the hinge forces at $B$, $C$, and $D$. Neglect the weights of the members.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-451-components-hinge-forces-frame)

Draw moment and load diagrams corresponding to the given shear diagram. Specify values at all change of load positions and at all points of zero shear.

[More info.](http://mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-451-relationship-between-load-shear)

For the frame shown, determine the horizontal and vertical components of the hinge force at $B$ as it acts upon member $AC$.

$B_V = $ 150 kN upward $B_H = $ 160 kN to the left [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-452-weight-supported-cable-which-runs-over-frictionless)

For the frame shown, determine the resultant hinge forces at $B$, $C$, and $E$.

$R_B$ = 252.98 kN $R_C$ = 80 kN $R_E$ = 505.96 kN [More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-453-analysis-frame-weightless-cylinder)

Determine the horizontal and vertical components of the hinge force at $A$ for the structure shown. Neglect the weights of the members and of the pulleys.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/problem-454-system-booms-and-pulleys)

Determine the centroid of the shaded area shown, which is bounded by the $x$-axis, the line $x = a$ and the parabola $y^2=kx$

$A = \frac{2}{3}ab$ $\bar{x} = \frac{3}{5}a$ $\bar{y} = \frac{3}{8}b$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/705-centroid-parabolic-segment-integration)

Use integration to prove that the centroid of the quarter circle whose radius is $r$ is \\(\bar{x} = \bar{y} = \frac{4r}{3\pi} \\).

[More info.](http://mathalino.com/reviewer/engineering-mechanics/706-centroiud-quarter-circle-integration)

Determine the centroid of the quadrant of the ellipse shown. The equation of the ellipse is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/707-centroid-quarter-ellipse-integration)

Compute the area of the spandrel bounded by the $x$ -axis, the line $x = b$, and the curve $y = kx^n$ where $n ≥ 0$. What is the location of its centroid from the line $x = b$? Determine also the $y$ coordinate of the centroid.

$A = \dfrac{bh}{n + 1}$ $\bar{x} = \dfrac{b}{n + 2}$ $\bar{y} = \dfrac{n + 1}{4n + 2}h$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/708-centroid-and-area-spandrel-integration)

Locate the centroid of the area bounded by the $x$-axis and the sine curve $y = a \sin \dfrac{\pi x}{L}$ from $x = 0$ to $x = L$.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/709-centroid-area-bounded-one-arc-sine-curve-and-x-axis)

The dimensions of the T-section of a cast-iron beam are shown in the diagram. How far is the centroid of the area above the base?

$y = 3.07\'\'$ above the base [More info.](http://mathalino.com/reviewer/engineering-mechanics/714-inverted-t-section-centroid-composite-figure)

Determine the coordinates of the centroid of the area shown with respect to the given axes.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/715-semicircle-and-triangle-centroid-composite-figure)

A slender homogeneous wire of uniform cross section is bent into the shape shown. Determine the coordinates of the centroid.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/716-semicircular-arc-and-lines-centroid-composite-figure)

Locate the centroid of the bent wire shown in. The wire is homogeneous and of uniform cross-section.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/717-symmetricall-arcs-and-line-centroid-composite-line)

Locate the centroid of the shaded area shown.

$\bar{x} = 5\'\'$ $\bar{y} = 6\'\'$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/718-square-and-triangles-centroid-composite-area)

Determine the centroid of the lines that form the boundary of the shaded area.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/719-closed-straight-lines-centroid-composite-lines)

The centroid of the shaded area is required to lie on the $y$ axis. Determine the distance $b$ that will fulfill this requirement.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/720-two-triangles-centroid-composite-area)

To what value should the 6-in. width of the flange be changed so that the centroid of the area is 2.5 in. above the base?

$f = 10$ in [More info.](http://mathalino.com/reviewer/engineering-mechanics/721-increasing-width-flange-relocate-centroid-inverted-t-beam)

Use the composite area method to locate the centroid of the shaded area shown, which was created by cutting a semicircle of diameter $r$ from a quarter circle of radius $r$.

$\bar{x} = \dfrac{2r}{\pi} = 0.6366r$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/722-semicircle-and-quarter-circle-centroid-composite-area)

Locate the centroid of the shaded area.

$\bar{x} = 39.71$ mm $\bar{y} = 30.39$ mm [More info.](http://mathalino.com/reviewer/engineering-mechanics/723-rectangle-quarter-circle-and-triangle-centroid-composite-area)

Find the coordinates of the centroid of the shaded area shown.

$\bar{x} = 7.736$ in $\bar{y} = 5.075$ in [More info.](http://mathalino.com/reviewer/engineering-mechanics/724-rectangle-semicircle-quarter-circle-and-triangle-centroid-composi)

Locate the centroid of the shaded area enclosed by the curve $y^2 = a x$ and the straight line. Hint: Observe that the curve $y^2 = a x$ relative to the $y$-axis is of the form $y = kx^2 $ with respect to the $x$-axis.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/726-area-enclosed-parabola-and-straigh-line-centroid-composite-area)

A rectangle is 3 in. by 6 in. Determine the polar moment of inertia and the radius of gyration with respect to a polar axis through one corner.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/816-polar-moment-inertia-and-radius-gyration-one-corner-rectangle)

Determine the moment of inertia and radius of gyration with respect to a polar centroidal axis of the cross section of a hollow tube whose outside diameter is 6 in. and inside diameter is 4 in.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/817-hollow-tube-moment-inertia-and-radius-gyration)

A hollow square cross section consists of an 8 in. by 8 in. square from which is subtracted a concentrically placed square 4 in. by 4 in. Find the polar moment of inertia and the polar radius of gyration with respect to a z axis passing through one of the outside corners.

[More info.](http://mathalino.com/reviewer/engineering-mechanics/818-hollow-square-section-moment-inertia-and-radius-gyration)

Determine the moment of inertia of the T-section shown with respect to its centroidal $X_0$ axis.

$\bar{I} = 290.67 \, \text{ in.}^4$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/819-inverted-t-section-moment-inertia)

Determine the moment of inertia of the area shown with respect to its centroidal axes.

$\bar{I}_x = 855.3 \, \text{ in.}^4$ $\bar{I}_y = 163 \, \text{ in.}^4$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/820-unsymmetrical-i-section-moment-inertia)

Find the moment of inertia about the indicated $x$-axis for the shaded area shown.

$I_x = 906.14 ~ \text{in}^4$ [More info.](http://mathalino.com/reviewer/engineering-mechanics/821-rectangle-minus-semi-circle-moment-inertia)

Given the parabola $3x^2 + 40y – 4800 = 0$ What is the area bounded by the parabola and the X-axis? a. 6 200 unit ² b. 8 300 unit ² c. 5 600 unit ² d. 6 400 unit ² What is the moment of inertia, about the X-axis, of the area bounded by the parabola and the $x$-axis? a. 15 045 000 unit ⁴ b. 18 362 000 unit ⁴ c. 11 100 000 unit ⁴ d. 21 065 000 unit ⁴ What is the radius of gyration, about the $x$-axis, of the area bounded by the parabola and the $x$-axis? a. 57.4 units b. 63.5 units c. 47.5 units d. 75.6 units

[More info.](http://mathalino.com/reviewer/engineering-mechanics/001-area-moment-inertia-and-radius-gyration-parabolic-section)