Mechanics Homework 20-4 Name_______________

Problem 20-4 First, watch this Khan Academy Video where Sal discusses how to integrate using polar coordinates. 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 \(\tilde{y}\) as a function of \(r\) and \(\theta\).

image

\(A = \pi r^2\), \(\bar{y} = \frac{4 R}{3 \pi}\).