What Is the Linear Acceleration of a Sphere Rolling Down an Incline?

[rleung3]

Hey,

I have been thinking about this problem for a while to no avail. Any input would be greatly appreciated. Thanks!

Homework Statement

A solid sphere of mass M and radius R rolls without slipping down a 30 degree incline (see figure attached).

What is the linear acceleration, a, of the sphere down the incline?

Homework Equations

I=(2/5)MR^2

torque = (Ia)/R = rxF

The Attempt at a Solution

I only know of two ways to solve for this: energy methods or torque methods. To do torque, I would need to know the value of the static friction that is keeping the ball from slipping, but I don't have that information. For energy methods, I don't have a height from where the ball began to roll, so I don't see how I can use that.

Here is my attempt at torque:

torque = (-Ia)/R = -fR where f=force of static friction

=> Ia = fR^2 => substitute I=(2/5)MR^2 => (2/5)(MR^2)a = fR^2 => f=(2/5)Ma => a=(5f)/(2M)

I don't know where to go from here. The answer is (5/14)g. I must be missing some type of other alternative to solving this problem, but I can't think of what. Thanks so much! I appreciate it.

Ryan

 

[marlon]

You need three essential parts :
F' is friction


1) Mgsin(30) - F' = Ma (a along incline and the positive direction is along the incline as well)

2) F' R = I $$\alpha$$

3) $$\alpha$$ = a/R

Put the F' from 1) into 2) to get rid of it. Then use 3) to solve for a

marlon

 

[AlephZero]

If you want to use an energy method, you can assume it rolls an arbitrary distance starting from rest, find the final linear and angular velocities, then find the acceleration using the formulas for motion with constant acceleration.

 

[rleung3]

Ohh! Thank you. Yea, I am definitely rusty at this..has been a while.