# Rotational Inertia of Pulley, Integration Help

• rmunoz
In summary, the pulley with a rotational inertia of 1.5 10-3 kg·m2 and radius of 21 cm is acted on by a force that varies with time (F = 0.50t + 0.30t^{2}). At t = 10.0 s, its angular acceleration is 4900 rad/s2 and its angular velocity is 140 rad/s. To determine the angular velocity, an integral must be taken of the angular acceleration, with a constant of integration of 0 since the pulley was initially at rest.

## Homework Statement

A pulley, with a rotational inertia of 1.5 10-3 kg·m2 about its axle and a radius of 21 cm, is acted on by a force applied tangentially at its rim. The force magnitude varies in time as F = 0.50t + 0.30t$$^{2}$$, where F is in Newtons and t in seconds. The pulley is initially at rest.

(a) At t = 10.0 s what is its angular acceleration?

(b) At t = 10.0 s what is its angular velocity?

## Homework Equations

Tnet=I$$\alpha$$

f=.50t + .30t$$^{2}$$

## The Attempt at a Solution

I allready got the acceleration for the pulley. It turned out to be 4900 rad/sec. But now i assume because the amount of force is reliant on the time (in other words, is accelerating at a non-constant rate), some integration will be needed. The problem for me is, i have only the vaguest idea of how to actually integrate. Would anyone mind helping me with this problem if it does in fact require integration, by showing me the exact steps? And if that is not the case, perhaps point me in the right direction?

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rmunoz said:

## Homework Statement

A pulley, with a rotational inertia of 1.5 10-3 kg·m2 about its axle and a radius of 21 cm, is acted on by a force applied tangentially at its rim. The force magnitude varies in time as F = 0.50t + 0.30t$$^{2}$$, where F is in Newtons and t in seconds. The pulley is initially at rest.

(a) At t = 10.0 s what is its angular acceleration?

(b) At t = 10.0 s what is its angular velocity?

## Homework Equations

Tnet=I$$\alpha$$

f=.50t + .30t$$^{2}$$

## The Attempt at a Solution

I allready got the acceleration for the pulley. It turned out to be 4900 rad/sec. But now i assume because the amount of force is reliant on the time (in other words, is accelerating at a non-constant rate), some integration will be needed. The problem for me is, i have only the vaguest idea of how to actually integrate. Would anyone mind helping me with this problem if it does in fact require integration, by showing me the exact steps? And if that is not the case, perhaps point me in the right direction?

You are correct. You will need to do an integral to determine the angular velocity.

You have already evaluated the a(t) at 10 sec to determine a|10 from

|a(t)| = |r|*|F(t)|/I = 140*|.5*t + .3*t2|

Since a(t) = dω/dt

then ω|10 = ∫a(t)*dt

or

ω|10 = 140*∫(.5t+.3t2)*dt = 140*(.25*t2 +.1*t3 + c) evaluated from 0 to 10.

Since you are told it was at rest at t=0, then the constant of integration c = 0.

Awesome, thank you for the walkthrough, that was incredibly helpful