Angular Velocity - Trigonometry

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Question

What is the formula for angular velocity?

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Answer

Angular velocity is a measure of the time it takes to travel over a certain arc that is formed by a central angle. $\omega$ is usually used to represent the angular velocity, $\theta$ is the measure of the central angle, and is the time it takes to travel from point A to B. This formula looks similar to the measure of an arc, $\theta = $\frac{r}{l_A}$$ , but we are considering the velocity in terms of the measure of the angle over the amount of time it took to cover that distance.

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Question

True or False: The angular velocity is a measure of an angle of rotation over time.

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Answer

consider the figure below. Think of line A being connected to a hinge where $\theta$ is. We are measuring the amount of time it takes for line A to rotate and align with line B. This is a measure of angle rotation over time.

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Question

What is the angular velocity of an angle whose measure is $\frac{5\pi}{4}$ when it took a point 6 seconds to rotate through the angle.

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Answer

We have enough information to plug into the angular velocity formula to solve this problem. $\theta = $\frac{5\pi}{4}$$ and seconds.

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

$\omega = ($\frac{5\pi}{4}$)($\frac{1}{6}$)$

$\omega = $\frac{5\pi}{24}$$

So our answer is $\omega = $\frac{5\pi}{24}$$ radians/seconds

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Question

What is the angular velocity of an angle whose measure is $\frac{\pi}{3}$ when it took a point 10 seconds to rotate through the angle.

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Answer

We have enough information to plug into the angular velocity formula to solve this problem. $\theta = $\frac{\pi}{3}$$ and seconds.

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

$\omega = ($\frac{\pi}{3}$)($\frac{1}{10}$)$

$\omega = $\frac{\pi}{30}$$

And so our answer is $\omega = $\frac{\pi}{30}$$ radians/seconds.

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Question

If you are walking around a circular track and your angular velocity through the angle $\theta = $\frac{\pi}{6}$$ is $\omega = $\frac{\pi}{12}$$ , how long would it take you to complete a full rotation?

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Answer

This is simply solving for an unknown. Here our unknown is time, . So we know that it takes you have an angular velocity of $\omega = $\frac{\pi}{12}$$ through the angle $\theta = $\frac{\pi}{6}$$ . Assuming that your angular velocity is consistent, we will be able to solve for . First we must manipulate our original angular velocity formula to solve for .

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

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

Now we will plug in our known value for $\omega$ and the angle of a full rotation. The angle of a full rotation is simply that of a full circle, $2\pi$ .

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

$t = $\frac{2\pi}{\frac{\pi}${12}}$

$t = 2\pi ($\frac{12}{\pi}$)$

So it takes you 24 seconds to complete a full rotation

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Question

You are on a carousel and have completed one full rotation in 15 seconds. What is your angular velocity?

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Answer

Even though we are not directly given an angle measurement, we are told the time it takes to complete one full rotation. One rotation is the same as the angle of an entire circle, $2\pi$ . Now we have enough information to plug into the angular velocity formula to solve this problem.

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

$\omega = $\frac{2\pi}{15}$$

So the angular velocity is $\omega = $\frac{2\pi}{15}$$ radians/seconds

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Question

A wheel is making one full rotation with an angular velocity of $\frac{5\pi }{2}$ radians/seconds. How long will it take the wheel to make 10 full rotations?

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Answer

We know that the angular velocity to complete a full rotation (rotate through an angle of $2\pi$ ) is $\frac{5\pi}{2}$ radians/seconds. The angle of 10 full rotations will be 10 rotations through the angle $2\pi$ , so we multiply these two quantities.

$10 *2\pi = 20\pi$

And this will be our $\theta$ . Now we must manipulate our angular velocity formula to solve for .

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

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

Now we can plug in our $\theta$ for the 10 rotations and our angular velocity.

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

$t = $\frac{20\pi}{\frac{5\pi}${2}}$

$t = 20\pi $\frac{2}{5\pi}$$

And so it will take 8 seconds to make 10 rotations.

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Question

If it takes you 3 seconds to rotate through an angle with an angular velocity of $\frac{8\pi}{9}$ , what is the measure of the angle you are rotating through?

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Answer

We are able to use the angular velocity formula and solve for the unknown $\theta$ .

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

$\omega t = \theta$

$$\frac{8\pi}{9}$ (3) = \theta$ (making it so we are solving for $\theta$ )

$$\frac{24\pi}{9}$ = \theta$

$$\frac{8\pi}{3}$ = \theta$

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Question

What is the formula for angular velocity?

Tap to reveal answer

Answer

Angular velocity is a measure of the time it takes to travel over a certain arc that is formed by a central angle. $\omega$ is usually used to represent the angular velocity, $\theta$ is the measure of the central angle, and is the time it takes to travel from point A to B. This formula looks similar to the measure of an arc, $\theta = $\frac{r}{l_A}$$ , but we are considering the velocity in terms of the measure of the angle over the amount of time it took to cover that distance.

← Didn't Know|Knew It →

Question

True or False: The angular velocity is a measure of an angle of rotation over time.

Tap to reveal answer

Answer

consider the figure below. Think of line A being connected to a hinge where $\theta$ is. We are measuring the amount of time it takes for line A to rotate and align with line B. This is a measure of angle rotation over time.

← Didn't Know|Knew It →

Question

What is the angular velocity of an angle whose measure is $\frac{5\pi}{4}$ when it took a point 6 seconds to rotate through the angle.

Tap to reveal answer

Answer

We have enough information to plug into the angular velocity formula to solve this problem. $\theta = $\frac{5\pi}{4}$$ and seconds.

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

$\omega = ($\frac{5\pi}{4}$)($\frac{1}{6}$)$

$\omega = $\frac{5\pi}{24}$$

So our answer is $\omega = $\frac{5\pi}{24}$$ radians/seconds

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Question

What is the angular velocity of an angle whose measure is $\frac{\pi}{3}$ when it took a point 10 seconds to rotate through the angle.

Tap to reveal answer

Answer

We have enough information to plug into the angular velocity formula to solve this problem. $\theta = $\frac{\pi}{3}$$ and seconds.

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

$\omega = ($\frac{\pi}{3}$)($\frac{1}{10}$)$

$\omega = $\frac{\pi}{30}$$

And so our answer is $\omega = $\frac{\pi}{30}$$ radians/seconds.

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Question

If you are walking around a circular track and your angular velocity through the angle $\theta = $\frac{\pi}{6}$$ is $\omega = $\frac{\pi}{12}$$ , how long would it take you to complete a full rotation?

Tap to reveal answer

Answer

This is simply solving for an unknown. Here our unknown is time, . So we know that it takes you have an angular velocity of $\omega = $\frac{\pi}{12}$$ through the angle $\theta = $\frac{\pi}{6}$$ . Assuming that your angular velocity is consistent, we will be able to solve for . First we must manipulate our original angular velocity formula to solve for .

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

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

Now we will plug in our known value for $\omega$ and the angle of a full rotation. The angle of a full rotation is simply that of a full circle, $2\pi$ .

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

$t = $\frac{2\pi}{\frac{\pi}${12}}$

$t = 2\pi ($\frac{12}{\pi}$)$

So it takes you 24 seconds to complete a full rotation

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Question

You are on a carousel and have completed one full rotation in 15 seconds. What is your angular velocity?

Tap to reveal answer

Answer

Even though we are not directly given an angle measurement, we are told the time it takes to complete one full rotation. One rotation is the same as the angle of an entire circle, $2\pi$ . Now we have enough information to plug into the angular velocity formula to solve this problem.

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

$\omega = $\frac{2\pi}{15}$$

So the angular velocity is $\omega = $\frac{2\pi}{15}$$ radians/seconds

← Didn't Know|Knew It →

Question

A wheel is making one full rotation with an angular velocity of $\frac{5\pi }{2}$ radians/seconds. How long will it take the wheel to make 10 full rotations?

Tap to reveal answer

Answer

We know that the angular velocity to complete a full rotation (rotate through an angle of $2\pi$ ) is $\frac{5\pi}{2}$ radians/seconds. The angle of 10 full rotations will be 10 rotations through the angle $2\pi$ , so we multiply these two quantities.

$10 *2\pi = 20\pi$

And this will be our $\theta$ . Now we must manipulate our angular velocity formula to solve for .

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

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

Now we can plug in our $\theta$ for the 10 rotations and our angular velocity.

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

$t = $\frac{20\pi}{\frac{5\pi}${2}}$

$t = 20\pi $\frac{2}{5\pi}$$

And so it will take 8 seconds to make 10 rotations.

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Question

If it takes you 3 seconds to rotate through an angle with an angular velocity of $\frac{8\pi}{9}$ , what is the measure of the angle you are rotating through?

Tap to reveal answer

Answer

We are able to use the angular velocity formula and solve for the unknown $\theta$ .

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

$\omega t = \theta$

$$\frac{8\pi}{9}$ (3) = \theta$ (making it so we are solving for $\theta$ )

$$\frac{24\pi}{9}$ = \theta$

$$\frac{8\pi}{3}$ = \theta$

← Didn't Know|Knew It →

Question

What is the formula for angular velocity?

Tap to reveal answer

Answer

Angular velocity is a measure of the time it takes to travel over a certain arc that is formed by a central angle. $\omega$ is usually used to represent the angular velocity, $\theta$ is the measure of the central angle, and is the time it takes to travel from point A to B. This formula looks similar to the measure of an arc, $\theta = $\frac{r}{l_A}$$ , but we are considering the velocity in terms of the measure of the angle over the amount of time it took to cover that distance.

← Didn't Know|Knew It →

Question

True or False: The angular velocity is a measure of an angle of rotation over time.

Tap to reveal answer

Answer

consider the figure below. Think of line A being connected to a hinge where $\theta$ is. We are measuring the amount of time it takes for line A to rotate and align with line B. This is a measure of angle rotation over time.

← Didn't Know|Knew It →

Question

What is the angular velocity of an angle whose measure is $\frac{5\pi}{4}$ when it took a point 6 seconds to rotate through the angle.

Tap to reveal answer

Answer

We have enough information to plug into the angular velocity formula to solve this problem. $\theta = $\frac{5\pi}{4}$$ and seconds.

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

$\omega = ($\frac{5\pi}{4}$)($\frac{1}{6}$)$

$\omega = $\frac{5\pi}{24}$$

So our answer is $\omega = $\frac{5\pi}{24}$$ radians/seconds

← Didn't Know|Knew It →

Question

What is the angular velocity of an angle whose measure is $\frac{\pi}{3}$ when it took a point 10 seconds to rotate through the angle.

Tap to reveal answer

Answer

We have enough information to plug into the angular velocity formula to solve this problem. $\theta = $\frac{\pi}{3}$$ and seconds.

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

$\omega = ($\frac{\pi}{3}$)($\frac{1}{10}$)$

$\omega = $\frac{\pi}{30}$$

And so our answer is $\omega = $\frac{\pi}{30}$$ radians/seconds.

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