# Increase moment of inertia, slipping down an incline

1. Aug 18, 2013

### IBAStudent

Hi, I'm trying to find out what effect an increase in moment of inertia of a cylinder would have on slipping when the cylinder rolls down an incline.
Starting with: ma=mgsin(theta)-f, where f is the friction force, which is less than or equal to µmgcos(theta), normal force times µ.
We already know that an increase in moment of inertia without increasing the radius will decrease the linear acceleration, and therefore, for instance a hoop, will roll down the incline slower than a cylinder of equal mass and radius.
Therefore ma will decrease, but mgsin(theta) is constant, so f has to increase, right? An increase in moment of inertia requires a higher friction force.
Does this mean that if we were to let a solid cylinder and a hoop race down a ramp of angle (theta), the hoop would start to slip first if we were to increase the angle (theta)? If this is true, wouldn't it be possible for the hoop to win the race, if at a specific angle, the hoop started to slip and the cylinder didn't, because less friction is needed to sustain its rolling motion?
Could anyone please confirm my train of thoughs? Thanks :)

2. Aug 20, 2013

### bgq

You have to work it out in details, here are the guidelines:

1) First note that increasing the moment of inertia without increasing the radius means to increase the mass.

2) Apply Ʃτ = Iα.

3) Apply ƩF = ma.

4) Knowing that m could be expressed in terms of I, and α could be expressed in terms of a; the above equations form a system of two equations of two unknowns: I and a.

5) Solve the system to find a in terms of I.

6) Then analyze the obtained equation.

3. Aug 21, 2013

### IBAStudent

I'm sorry for not being clear enough: The way I though of increasing the moment of inertia wasn't by keeping the radius constant and increasing the mass, but change the mass distribution so that the mass is concentrated away from the center of rotation.