Solve Angular Velocity Problems - Pre-Calculus

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Question

If a ball is travelling in a circle of diameter $10 \text{m}$ with velocity $20\text{m}/ \text{s}$ , find the angular velocity of the ball.

Answer

Using the equation,

$\omega=\frac{v}{r}$ where

$\omega$ =angular velocity, =linear velocity, and =radius of the circle.

In this case the radius is 5 (half of the diameter) and linear velocity is 20 m/s.

$\omega=\frac{20}{5}=4$ .

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Question

Suppose a car tire rotates times a second. The tire has a diameter of inches. Find the angular velocity in radians per second.

Answer

Write the formula for angular velocity.

$\omega =2\pi f_$

The frequency of the tire is 8 revolutions per second. The radius is not used.

Substitute the frequency and solve.

$\omega =2\pi f= 2\pi(8)=16\pi$

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Question

A $20\ in$ diamter tire on a car makes revolutions per second. Find the angular speed of the car.

Answer

Recall that $angular\ velocity=\theta/t$ .

Since the tire revolves 9.3 times/second it would seem that the tire would rotate

$9.3 (2\pi) radians/second$ or $18.6 \pi radians/ second$ .

We use $2\pi$ to indicate that the tire is rolling 360 degrees or $2\pi$ radians each revolution (as it should).

Thus,

$18.6 \pi radians/ second$ is your final answer.

Note that radians is JUST a different way of writing degrees. The higher numbers in the answers above are all measures around the actual linear speed of the tire, not the angular speed.

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Question

What is the angular velocity of a spinning top if it travels $\pi$ radians in a third of a second?

Answer

Write the formula for average velocity.

$\omega=\frac{\theta}{t}$

The units of omega is radians per second.

Substitute the givens and solve for omega.

$\omega=\frac{\theta}{t}=\frac{\pi}{\frac{1}{3}}=\pi*3=3\pi$

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Question

A car wheel of radius 20 inches rotates at 8 revolutions per second on the highway. What is the angular speed of the tire?

Answer

Angular speed is the same as linear speed, but instead of distance per unit time we use degrees or radians. Any object traveling has both linear and angular speed (though objects only have angular speed when they are rotating).

$Angular\hspace{1mm}speed= \frac{\Theta }{t}$

Since our tire completes 8 revolutions per second we multiply by $2\pi$ since a full rotation (360°) equals $2\pi$ .

$Angular\hspace{1mm}speed= \frac{16\pi }{1 s}=16\pi\frac{rad}{s}$

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