Angular Displacement and Velocity

[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?
rad/s2

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?

 

[AEM]

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?
rad/s2

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.

 

[nlightner]

(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:

11.0 revolutions = (22.0 rad/s)(time)

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:

angular acceleration = (-22.0 rad/s) / (0.5 s) = -44.0 rad/s^2

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