# Angular Displacement and Velocity

• Bigdane
In summary, the person riding a bicycle has wheels with an initial angular velocity of +22.0 rad/s. When the brakes are applied, the bike comes to a uniform stop and the angular displacement of each wheel is +11.0 revolutions. To find the time it takes for the bike to come to rest, the equation 11=22t is used and the result is t=1/2 s. To find the angular acceleration of each wheel, the equation \omega^2 = \omega_o^2 + 2 \alpha(\theta - \theta_o) is used and the result is rad/s2.
Bigdane

## Homework Statement

A person is riding a bicycle, and its wheels have an angular velocity of +22.0 rad/s. Then, the brakes are applied and the bike is brought to a uniform stop. During braking, the angular displacement of each wheel is +11.0 revolutions.

(a) How much time does it take for the bike to come to rest?
s

(b) What is the angular acceleration of each wheel?

## Homework Equations

theta = (angular velocity)(time)

## The Attempt at a Solution

11=22t
11/22 = t=1/2

thats wrong i guess.
can you help me out?

Bigdane said:

## Homework Statement

A person is riding a bicycle, and its wheels have an angular velocity of +22.0 rad/s. Then, the brakes are applied and the bike is brought to a uniform stop. During braking, the angular displacement of each wheel is +11.0 revolutions.

(a) How much time does it take for the bike to come to rest?
s

(b) What is the angular acceleration of each wheel?

## Homework Equations

theta = (angular velocity)(time)

## The Attempt at a Solution

11=22t
11/22 = t=1/2

thats wrong i guess.
can you help me out?

Here's an interesting thing: The kinematic equations for rotational motion (be sure to use radians) have the same form as those for straight line motion:

$$X = X_o + V_o t + \frac{1}{2} a t^2$$

becomes

$$\theta = \theta_o + \omega_o t + \frac{1}{2} \alpha t^2$$

and so on. Therefore you can use many of the same ideas that you would use for straight line motion problems with simple rotational kinematics. For example, you could use

$$\omega^2 = \omega_o^2 + 2 \alpha(\theta - \theta_o)$$

to find the angular acceleration in your problem above. I'll leave it to you to figure out the other parts.

$$\omega$$ is the angular velocity; $$\alpha$$ is the angular acceleration in the above equations.

(a) To find the time it takes for the bike to come to rest, we can use the equation theta = (angular velocity)(time). Since the angular displacement of each wheel is +11.0 revolutions and the angular velocity is +22.0 rad/s, we can set up the following equation:

Solving for time, we get:

time = 11.0 revolutions / 22.0 rad/s = 0.5 seconds

Therefore, it takes 0.5 seconds for the bike to come to rest.

(b) To find the angular acceleration of each wheel, we can use the equation angular acceleration = (change in angular velocity) / (time). Since the bike is coming to a stop, the final angular velocity is 0 rad/s. So, the change in angular velocity is -22.0 rad/s. We already calculated the time in part (a) to be 0.5 seconds. Therefore, the angular acceleration is:

Note that the negative sign indicates that the angular acceleration is in the opposite direction of the angular velocity.

## 1. What is angular displacement?

Angular displacement is the measure of the change in position of an object around a fixed point or axis. It is typically measured in radians or degrees.

## 2. How is angular displacement different from linear displacement?

Angular displacement refers to the change in position of an object around an axis, while linear displacement refers to the change in position of an object in a straight line.

## 3. What is angular velocity?

Angular velocity is the rate of change of angular displacement over time. It is a vector quantity and is typically measured in radians per second.

## 4. How is angular velocity related to linear velocity?

Angular velocity and linear velocity are related by the radius of the circular path. The linear velocity of an object moving in a circular path is equal to the angular velocity multiplied by the radius of the circle.

## 5. How can angular displacement and velocity be calculated?

Angular displacement can be calculated by subtracting the initial angular position from the final angular position. Angular velocity can be calculated by dividing the change in angular displacement by the time it took to make that change.

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