# Moment of Inertia and Torque

A wheel free to rotate about its axis that is not frictionless is initially at rest. A constant external torque of +40 N·m is applied to the wheel for 24 s, giving the wheel an angular velocity of +610 rev/min. The external torque is then removed, and the wheel comes to rest 120 s later. (Include the sign in your answers.)

(a) Find the moment of inertia of the wheel.
kg·m2
(b) Find the frictional torque, which is assumed to be constant.
N·m

This whole topic of angular/moment of inertia and torque taking its toll on me. Please bear with me, the more I read in textbook the more confused I get.

Relevant Equations:
Inertia = sum of mi * ri^2
Inertia for continuous objects = integral of r^2 dm

I would use the equation for inertia for continuous objects correct? since the textbook used the other one when given points.
The problem doesn't give us any masses/radius -- How would I start this?

For part B:
relevant equations:
Torque = r x F
Tnet = I alpha

What should I use for the Moment of Inertia for the wheel? It doesn't state it,
some exercise from the book says consider a wheel of a bicycle to be a hoop I = MR^2

Im not sure if this is correct,
I did for translation equation
Fnet = Ma
Ext F + friction = Ma

For rotation
Tnet = I alpha
RF - Rf = 1/2 M R^2 (a/R)
= F-f = 1/2 Ma

2F = 3/2 Ma
a = 4F/3M

From equation 1
F+f = Ma
f = F/3

Is this even remotely correct?

## The Attempt at a Solution

Hey there,

For part a) I don't think you need any more information. In this question you wouldn't find the moment of inertia by summing or integrating.

A word of warning: You haven't been the given the angular velocity, so to start with I would change your rev/min to rad/sec.

See this site: http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html

You need to use the rotational equivalent of v=v-0+a*t, along with Newtons second law rotational equivalent.

Hope this gets you going for the first part

It says it gave the wheel an angular velocity of +610 rev/min
I got 63.87905062 approx 64 rad/sec.

63.87905062 rad/sec / 24s = 2.661627109 approx 2.7 rad/s^2 is the acceleration
Tnet = I alpha
so Tnet/alpha = I
40 Nm /2.661627109 rad/s^2 = 15.02840110 approx 15 is the Inertia?

Is this correct?

Last edited:
Could someone check if what I did for part A/B is correct?

Yes your answer is the same as the one I got. For part b I'd use the same equations except this time you know the moment of inertia and initial and final angular velocities but want to find the frictional torque. You can use the same equations since they say that the frictional torque is constant.