Pre-Calculus - Angular and Linear Velocity

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.

$4 \text{ Hz}$

$\frac {1}{4} \text{ Hz}$

$100 \text{ Hz}$

$5 \text{ m}/\text{s}$

Explanation

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.

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

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.

$4 \text{ Hz}$

$\frac {1}{4} \text{ Hz}$

$100 \text{ Hz}$

$5 \text{ m}/\text{s}$

Explanation

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.

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

A clock has a second hand measuring 12cm. What is the linear speed of the tip of the second hand?

$\frac{1.26cm}{s}$

$\frac{6.12cm}{s}$

$\frac{2.16cm}{s}$

$\frac{1.62cm}{s}$

None of these/

Explanation

Linear speed is equal to the arc length traversed divided by the time. We use s to denote arc length.

$s=2\pi r=2\pi*12cm=24\pi cm$

$LS=\frac{arc length}{t}=\frac{s}{t}= \frac{24\pi cm}{60s}=1.26\frac{cm}{s}$

A clock has a second hand measuring 12cm. What is the linear speed of the tip of the second hand?

$\frac{1.26cm}{s}$

$\frac{6.12cm}{s}$

$\frac{2.16cm}{s}$

$\frac{1.62cm}{s}$

None of these/

Explanation

Linear speed is equal to the arc length traversed divided by the time. We use s to denote arc length.

$s=2\pi r=2\pi*12cm=24\pi cm$

$LS=\frac{arc length}{t}=\frac{s}{t}= \frac{24\pi cm}{60s}=1.26\frac{cm}{s}$

Find the linear velocity in meters per second of an object if it took hours to travel a distance of kilometers. Round to the nearest integer.

Explanation

Write the linear velocity formula.

$v=\frac{x}{t}$

Since the answer is required in meters per second, two conversions will need to be made.

Convert 30 kilometers to meters.

$30 : \textup{kilometers}\left(\frac{1000:\textup{meters}}{1 \textup{ kilometer}}\right)=30000 :\textup{meters}$

Convert 2 hours to seconds.

$2:\textup{hours}\left(\frac{60\textup{ minutes}}{1\textup{ hour}}\right)\left(\frac{60 \textup{ seconds}}{1 \textup{ minute}}\right)=7200:\textup{seconds}$

Apply the formula.

$v=\frac{x}{t}=\frac{30000\textup{ meters}}{7200\textup{ seconds}}=4.1\bar{6}=4$

Find the linear velocity in meters per second of an object if it took hours to travel a distance of kilometers. Round to the nearest integer.

Explanation

Write the linear velocity formula.

$v=\frac{x}{t}$

Since the answer is required in meters per second, two conversions will need to be made.

Convert 30 kilometers to meters.

$30 : \textup{kilometers}\left(\frac{1000:\textup{meters}}{1 \textup{ kilometer}}\right)=30000 :\textup{meters}$

Convert 2 hours to seconds.

$2:\textup{hours}\left(\frac{60\textup{ minutes}}{1\textup{ hour}}\right)\left(\frac{60 \textup{ seconds}}{1 \textup{ minute}}\right)=7200:\textup{seconds}$

Apply the formula.

$v=\frac{x}{t}=\frac{30000\textup{ meters}}{7200\textup{ seconds}}=4.1\bar{6}=4$

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

$18.6 \pi radians/ second$

$300.2 \pi radians/ second$

$475.6 \pi radians/ second$

$599 \pi radians/ second$

$22 \pi radians/ second$

Explanation

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.

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

$18.6 \pi radians/ second$

$300.2 \pi radians/ second$

$475.6 \pi radians/ second$

$599 \pi radians/ second$

$22 \pi radians/ second$

Explanation

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.

The second hand of a clock is long. Find the linear speed of the end of this second hand.

$\frac{1.07cm}{second}$

$\frac{3.2cm}{second}$

$\frac{2cm}{second}$

$\frac{2.1cm}{second}$

None of the other answers.

Explanation

Linear velocity is defined as distance over a period of time. For instance if a person ran 1 mile or approximately 1600 meters in 7 minutes, the they would have covered about 230 meters per minute. Let's assume this person was running around a track. We could also measure their speed from a central angle and represent their speed as the amount of degrees (or radians) they ran around per unit time as well. This is considered angular speed. A perfect example of both are the hands on a clock. There is a relationship between arc length which we designate s, the radius r, and the angle $\Theta$ (theta). The relationship is $s=r\times\theta$ . So the length of the arc (s) is equal to the radius of the circle the arc is on and what section of the pie it covers ( this is akin to how many degrees our track runner ran "through" or around) $\theta$ . Thus, if we wanted the linear speed around a circle we could say or $angular speed=\theta/t$ . Key measurements that you will need to know are how many degrees in a circle of which there are 360 or $2\pi$ . Before you follow the step by step solution below, go back and see if you can use this new information to arrive at the correct answer.