Q. 97 - Precalculus Enhanced with Graphing Utilities

Question

An object is traveling around a circle with a radius of $5$centimeters. If in $20$seconds a central angle of $\frac{1}{3}$radian is swept out, what is the angular speed of the object? What is its linear speed? 

Step-by-Step Solution

Answer

The angular speed is $\frac{1}{60}radian/second$ and the linear speed is $\frac{1}{12}cm/second$

Step 1. Given information

Given, the travelling path of the object is a circle of radius $5cm$. and in $20$seconds a central angle of $\frac{1}{3}$ radian is swept out.

Therefore,  $\theta =\frac{1}{3}radians$ and $t=20seconds$ and $r=5cm$

Step 2. Used formulae

• The angular speed $\omega$ of this object is the angle $\theta$ swept out, divided by the elapsed time $t$; that is, $\omega =\frac{\theta }{t}$
  
• If an object moves around a circle of radius $r$ at a constant speed. If $s$ is the distance traveled in time $t$ around this circle of radius $r$ and the angle made is $\theta$, then the linear speed $v$ of the object is defined as $v=\frac{s}{t}$ and the arc length $s=r\theta$, that is $v=\frac{r\theta }{t}=r·\omega$

Step 3. Calculations

The arc length $s=5·\frac{1}{3}=\frac{5}{3}cm$

Substituting the values in the mentioned formulae, we get, 

Angular speed $\omega =\frac{{\displaystyle \frac{1}{3}}}{20}=\frac{1}{60}radians/second$

and the linear speed $v=5·\frac{1}{60}=\frac{1}{12}cm/second$