Angular Velocity and Radian Measurement in Physics Problems

In physics, you can combine angular velocity with radian measurement to calculate the distance and speed of a moving object. For example, you can calculate the angular speed of a car as it travels along a curve.

Here are some practice questions that you can try.

Practice questions

1. The moon rotates once on its axis in about 27 days. If the moon's radius is 1,734 kilometers, how far in kilometers does an astronaut standing on the moon's surface move in 1 day?

2. You drive along the freeway at 60 miles per hour. You go through a curve with a radius of one quarter mile. What is your angular speed in radians per hour in this curve?

Answers

The following are the answers to the practice questions:

1. 

   The angular speed of the moon is

   

   For this problem,

   

   (one complete revolution) and

   

   so the angular speed is

   

   To find the angle through which the astronaut rotates in one day, solve the equation above for

   

   (the subscript ast means that this is the angle through which the astronaut rotates in one day, not the angle through which the moon rotates in 27 days). The result is

   

   where

   

   is the time during which the astronaut rotates. Insert the known quantities to find

   

   To find the distance that the astronaut travels, use the equation relating arc length to arc radius and the angle subtended by the arc:

   

   Use r = 1,734 kilometers to find

   

2. 

   The angular speed is related to the tangential velocity and the radius of curvature by

   

   Solve this for

   

   and insert the known quantities of

   

   and

   

   The result is