Set 8 Problem number 2

An object is moving on a circle whose radius is 3 meters.

If the angle of the radial line from the center of the circle to the object is changing at `pi radians/second, then how long does the object to complete one transit around the circle, and how fast is the object moving?

Solution

Since there are 2 `pi radians in a revolution, it takes 2 seconds to complete one revolution at `pi radians per second.

If the radius is 3 meters, then the circumference of the circle is 2 `pi ( 3 meters) = 18.8496 meters.
If the object moves this far in 2 seconds, its speed must be 18.8496 /2 meters per second = 9.4248 meters per second.

Alternatively, if the radius is 3 meters, then 1 radian corresponds to 3 meters on the circle.

In that case, `pi radians corresponds to 3 ( `pi ) meters along the arc of the circle, and `pi radians per second corresponds to 3 `pi meters per second = 9.4248 meters per second.

Generalized Solution

In general if we are moving at `omega radians/second then since 2 `pi radians constitutes a revolution we require 2 `pi / `omega seconds to complete a revolution.

This time is called the period of the motion.
If we are moving at `omega radians/second on a circle of radius r, then since each radian of angular motion corresponds to a distance r on the circle, we travel distance `omega * r every second.
Our speed is therefore v = `omega * r.

Explanation in terms of Figure(s), Extension

The figure below indicates motion through an angle of `pi radians, corresponding to 1 second's motion.

Clearly two seconds will be required to complete the revolution.

The most common Greek symbols used in describing rotational motion, and some of the equations using these symbols, are summarized on the two tables below. You should make careful note of these symbols for reference throughout this problem set.