Question

A flywheel turns through 26 rev as it slows from an angular speed of 2.9 rad/s to a stop. (a) Assuming a constant angular acceleration, find the time for it to come to rest. (b) What is its angular acceleration? (c) How much time is required for it to complete the first 13 of the 26 revolutions?

Answer 1

a) The wheel takes 112.7 s to comes to rest

b) The angular acceleration of the wheel is $-0.0257 rad/s^2$

c) The wheel takes 33.0 s to complete 13 revolutions

Explanation:

a)

The rotational motion of the wheel is at constant angular acceleration, so we can use the equivalent of the suvat equations for rotational motion. In particular, to find the time the wheel takes to come to a stop, we use the following:

$\theta = (\frac{\omega_f+\omega_i}{2})t$

where:

$\theta$ is the angular displacement

$\omega_i$ is the initial angular speed

$\omega_f$ is the final angular speed

t is the time

For the wheel in this problem:

$\omega_i = 2.9 rad/s$

$\omega_f = 0$

$\theta = 26 rev \cdot (2\pi)= 163.4 rad$

And solving for t, we find the time:

$t=\frac{2\theta}{\omega_i + \omega_f}=\frac{2(163.4)}{0+2.9}=112.7 s$

b)

We can find the angular acceleration by using the equation:

$\alpha = \frac{\omega_f - \omega_i}{t}$

where here we have:

$\omega_f = 0$ is the final angular speed

$\omega_i = 2.9 rad/s$ is the initial angular speed

t = 112.7 s is the time

And substituting,

$\alpha = \frac{0-2.9}{112.7}=-0.0257 rad/s^2$

And the negative sign is because the wheel is decelerating.

c)

In order to find the time taken to complete 13 revolutions, we use the following suvat equation:

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

where:

$\omega_i = 2.9 rad/s$ is the initial angular speed

$\theta = 13 rev \cdot (2\pi)=81.7 rad$ is the angular displacement

$\alpha=-0.0257 rad/s^2$ is the angular acceleration

Substituting into the equation:

$81.7 = 2.9 t +\frac{1}{2}(-0.0257)t^2\\0.0129t^2-2.9t+81.7 = 0$

which has two solutions:

t = 33.0 s

t = 191.8 s

And here we have to take the first solution (33.0 s), because it corresponds to the first time at which the angular displacement of the wheel is equal to 13 revolutions (the second solution is the time at which the angular displacement of the wheel returns to 13 revolutions, after the wheel has came to rest and it started to rotate in the opposite direction).

Learn more about rotational motions:

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