Angular and Linear Velocity - 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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Question

A plane is traveling Northeast. If the eastern component of it's velocity is $300 \text{ mph}$ , how fast is the plane traveling?

Answer

Since the plane is traveling NE, we know that it is traveling at an angle of 45 degrees north of east. Therefore, we can represent this in a diagram:

We have all of the information we need to solve for the velocity of the plane. We have an angle and a side adjacent to that angle. Therefore, we can use the cosine function to solve for the velocity.

$cos(45^{\circ}) = \frac{300}{v}$

Rearranging for v, we get:

$v = \frac{300}{cos(45^{\circ})} = 424 \text{ mph}$

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Question

A car is traveling NW at a speed of mph. What are the magnitudes of each of its North and West components?

Answer

If the car is traveling NW, it means that the angle $\theta$ is degrees above the x-axis. Using this angle, and the fact that the magnitude of the velocity is mph, we can find the x and y components, or the west and north components using trigonometry.

$\text{West: } \cos(45)=\frac{x}{60}$

$\text{North: } \sin(45)=\frac{y}{60}$

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Question

An object is moving in a circular path. How will its linear velocity change if the diameter of the circular path was decreased by one half?

Answer

Write the formula for the linear velocity.

$v=\omega r$

The diameter is twice the radius.

If the diameter was decreased by one half, the radius will also decrease by one half.

Therefore:

$v=\omega \cdot \frac{1}{2} r$

$v=\frac{1}{2}( \omega r)$

The linear velocity will decrease by one half.

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Question

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.

Answer

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.

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

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Question

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

Answer

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.

The actual answer is $\frac{1.07cm}{second}$ .

To see why, note that the second hand spins around a total of 360 degrees or $2\pi$ .

$S=20.4cm\times \pi$ .

And how long does it take for the hand to go around? Linear speed of the clock second hand is

$\frac{20.4cm \times \pi}{60s}=$ $\frac{1.07cm}{second}$ (rounded answer).

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Question

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

Answer

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

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

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

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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.

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

Compare your answer with the correct one above

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.

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

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

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

Compare your answer with the correct one above

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 $2\pi$ .

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

Compare your answer with the correct one above

Question

A plane is traveling Northeast. If the eastern component of it's velocity is $300 \text{ mph}$ , how fast is the plane traveling?

Answer

Since the plane is traveling NE, we know that it is traveling at an angle of 45 degrees north of east. Therefore, we can represent this in a diagram:

We have all of the information we need to solve for the velocity of the plane. We have an angle and a side adjacent to that angle. Therefore, we can use the cosine function to solve for the velocity.

$cos(45^{\circ}) = \frac{300}{v}$

Rearranging for v, we get:

$v = \frac{300}{cos(45^{\circ})} = 424 \text{ mph}$

Compare your answer with the correct one above

Question

A car is traveling NW at a speed of mph. What are the magnitudes of each of its North and West components?

Answer

If the car is traveling NW, it means that the angle $\theta$ is degrees above the x-axis. Using this angle, and the fact that the magnitude of the velocity is mph, we can find the x and y components, or the west and north components using trigonometry.

$\text{West: } \cos(45)=\frac{x}{60}$

$\text{North: } \sin(45)=\frac{y}{60}$

Compare your answer with the correct one above

Question

An object is moving in a circular path. How will its linear velocity change if the diameter of the circular path was decreased by one half?

Answer

Write the formula for the linear velocity.

$v=\omega r$

The diameter is twice the radius.

If the diameter was decreased by one half, the radius will also decrease by one half.

Therefore:

$v=\omega \cdot \frac{1}{2} r$

$v=\frac{1}{2}( \omega r)$

The linear velocity will decrease by one half.

Compare your answer with the correct one above

Question

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.

Answer

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.

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

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