Chapter 3.8 Exercise 39

Problem

The figure [not included here] shows a sector in a circle of radius r. The sector is the union of triangle T [formed by the two radii and the secant joining their ends] and segment S [the remaining area in the sector]. Suppose that the radius vector rotates counterclockwise with a constant angular velocity of omega radians per second. Show that the area of the sector changes at a constant rate, but the area of T and the area of S do not change at a constant rate.

Solution

First find expressions for area S, area T and area U := S+T:

• S = (r**2) (theta - sin theta) / 2
• T = (r**2) (sin theta) / 2
• U = (r**2) theta / 2

Then differentiate them:

• dS/dt = (r**2) (1 - cos theta) / 2
• dT/dt = (r**2) (cos theta) / 2
• dU/dt = (r**2) / 2

dU/dt is the only one of the three that is constant with respect to theta.