# A pulley with rotational inertia

• soccer_09
In summary, at 4 seconds, the rotational acceleration of the pulley is 68 rad/s^2. Its rotational speed is 2 rad/s.
soccer_09

## Homework Statement

A pulley, with a rotational inertia of 2.0 10-3 kg·m2 about its axle and a radius of 20 cm, is acted on by a force applied tangentially at its rim. The force magnitude varies in time as F = 0.50t + 0.30t2, where F is in Newtons and t in seconds. The pulley is initially at rest.

(a) At t = 4.0 s what is its rotational acceleration?
(b) At t = 4.0 s what is its rotational speed?

## Homework Equations

Torque = I * alpha

Torque = |r||F|sin(theta)

## The Attempt at a Solution

I have only tried part a. What I did was since I know the radius = .020 m, F, and I, I rearranged the formulas like so:

alpha = Torque / I

Since it's tangent, sin(90) = 1 therefore Torque = r*F

F at 4s = 6.8 N and multiplying this by the radius .020 m gives a torque of .136 Nm

So now alpha = .136 Nm / .002 kgm^2 = 68 rad/s^2

I plugged that in on webassign and it was wrong so apparently I'm not doing something right. Any guidance is appreciated, thanks.

Your method for calculating the angular acceleration is correct. However 20 cm = 0.20 m not 0.02 m.

Yep, I had a friend tell me I converted incorrectly. Thanks :)

From there though, my alpha = 680 rad /s^2

The equation for rotational veloctiy is:

omega(t) = omega (initial) + alpha (t)

so with alpha = 680 rad/s^2 and t = 4s that should give me a rotational speed of 2720 rad/s

This, however, is incorrect as well. The initial omega = 0 rad/s because the pulley is initially at rest. What am I doing wrong here??

The equation omega(t) = omega (initial) + alpha (t) only works for constant angular acceleration. Here, force is F = 0.50t + 0.30t2, so alpha can't possibly be constant. You'll have to use integration to get the speed.

Thanks :) I integrated from 0 to 4 of F(t) and got 10.4. using the constants I and R, I was able to get the correct answer. Thanks again.

## What is rotational inertia?

Rotational inertia, also known as moment of inertia, is the measure of an object's resistance to changes in its rotational motion. It is dependent on the mass and distribution of the object's mass around its axis of rotation.

## How does a pulley with rotational inertia work?

A pulley with rotational inertia works by using the principle of conservation of energy. When a force is applied to the rope attached to the pulley, the pulley rotates and the object attached to the other end of the rope moves. The rotational inertia of the pulley helps to reduce the force required to lift the object.

## What are the advantages of using a pulley with rotational inertia?

Using a pulley with rotational inertia allows for a reduction in the amount of force needed to lift an object. It also helps to distribute the load evenly, making it easier to lift heavy objects. Additionally, a pulley with rotational inertia can change the direction of the force, making it useful in various applications.

## What factors affect the rotational inertia of a pulley?

The rotational inertia of a pulley is affected by its mass, radius, and distribution of mass. The larger the mass and radius of the pulley, the greater its rotational inertia will be. The distribution of mass also plays a role, as a pulley with more mass concentrated towards its center will have a lower rotational inertia compared to one with mass distributed towards its edges.

## How can a pulley with rotational inertia be used in real life?

Pulleys with rotational inertia are commonly used in various machines and devices, such as cranes, elevators, and exercise equipment. They are also used in simple machines like block and tackle systems, which make it easier to lift heavy objects. In addition, they can be found in many household items, such as window blinds and garage doors.

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