The angular acceleration of a wheel is given in by where t is in seconds.If the wheel starts from rest at when is the next time the wheel is at rest?

Question

The angular acceleration of a wheel is given in  by  where t is in seconds.If the wheel starts from rest at  when is the next time the wheel is at rest?

Quizplus · Accepted Answer

From the given equation, we have:

$\alpha = 0.8t - 0.03t^2$

Angular velocity, $\omega$ at time t is given by integrating $\alpha$ with respect to t:

$\omega = \int \alpha \,dt$
          
$\omega = 0.4t^2 - 0.01t^3 + C_1$

Initial condition: At t = 0, $\omega$ = 0

$0 = 0 + C_1$

$C_1 = 0$

Therefore,

$\omega = 0.4t^2 - 0.01t^3$

Linear velocity, v at time t is given by:

$v = r\omega = 0.05(0.4t^2 - 0.01t^3)$

Initial condition: At t = 0, v = 15

$15 = 0$

This initial condition is incorrect, which indicates that there is an error in the question.

Ignoring the incorrect initial condition, the wheel will be at rest when the linear velocity v is equal to zero. Therefore, we need to find the roots of the following equation:

$0 = 0.4t^2 - 0.01t^3$

$0.01t^2(40-t) = 0$

$t = 0$ or $t = 40$

Since the wheel starts from rest at t = 0, the next time it is at rest is when $t = 40$ is not a valid solution. Therefore, the wheel will be at rest when:

$0.4t^2 - 0.01t^3 = 0$

$0.01t^2(40-t) = 0$

$t = 0$ or $t = 40$

The wheel is at rest at t = 0, so we need to find the next root.

$40-t = 0$

$t = 40$

Therefore, the wheel is next at rest at t = 40 seconds, which is not one of the answer choices.

Since none of the answer choices are correct, this question cannot be answered as given.

The Angular Acceleration of a Wheel Is Given in