Trigonometry - Angle Applications

What is the formula for angular velocity?

$\omega = \frac{r}{l_A}$

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

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

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

Explanation

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.

What is the formula for angular velocity?

$\omega = \frac{r}{l_A}$

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

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

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

Explanation

Which of the following is the correct formula for finding arc length?

$l_A = 2\theta r^2$

$l_A = \theta r^2$

$l_A = \theta r$

$l_A = 2\theta r$

Explanation

The circumference of an entire circle is $2\pi r$ . When considering the length of an arc, the angle is less than $2\pi$ denoted by angle $\theta$ . So the formula for finding the length of an arc is replacing the angle of an entire circle, $2\pi$ , with the angle that forms the arc, $\theta$ . This gives us the formula $l_A = \theta r$ .

Which of the following is the correct formula for finding arc length?

$l_A = 2\theta r^2$

$l_A = \theta r^2$

$l_A = \theta r$

$l_A = 2\theta r$

Explanation

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?

1 second

8 seconds

20 seconds

24 seconds

Explanation

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.

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?

1 second

8 seconds

20 seconds

24 seconds

Explanation

$10 *2\pi = 20\pi$

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.

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

$\omega = 2\pi$ radians/seconds

$\omega = 15$ radians/seconds

$\omega = \frac{2\pi}{15}$ radians/seconds

$\omega = \frac{7\pi}{13}$ radians/seconds

Explanation

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

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

$\omega = 2\pi$ radians/seconds

$\omega = 15$ radians/seconds

$\omega = \frac{2\pi}{15}$ radians/seconds

$\omega = \frac{7\pi}{13}$ radians/seconds

Explanation

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

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

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

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?

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

$3\pi$

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

$\theta = \frac{8\pi}{9}$

Explanation

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 $\theta$ )

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

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

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?

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

$3\pi$

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

$\theta = \frac{8\pi}{9}$

Explanation

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 $\theta$ )

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

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

Angle Applications

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