# Homework Help: Prove that is it SHM or not?

1. Jan 7, 2012

### vkash

1. The problem statement, all variables and given/known data

lets have a look on scene;
there is a concave of radius R.(like concave lens) there is a rubber ball of radius r(r<<R). It is placed in equilibrium position in the concave. Now it is displaced by small distance along concave and released,friction is sufficient enough for pure rolling of ball. What kind of motion will it did.

2. Relevant equations

all equations i know.

3. The attempt at a solution

let me say it's acceleration at any instant is a.net frictional force acting on the ball is f.
so our equation of motion comes out to be.
ma=mg*sin(θ)-f (here θ is the angle made by the center on the concave with line joining it with equilibrium position)
θ is too small so we can write sin(θ)=~θ
equation changes to
a=gθ-f/m
since it is pure rolling angular acceleration(α) = linear acceleration(a)/radius of ball(r)
so α=a/r
torque(T)=2/5mr2*α=2/5mr2a/r (taking around center of ball)
T=(frictional force)fr=2/5mr2a/r
from here we got the frictional=(2/5)ma
putting these values in first equation which was a=gθ-f/m
we got a+2/5a=gθ
simplifying it
a=5/7g θ
a=Ar=5/7gθ (A is angular acceleration with center of concave body with line joining it with mean position line)
A=5/7gθ/r
A directly proportional to θ.
hence this is equation of angular SHM.
this is my self generated question so can't tell what is it's answer in book. So it's u who will tell me either this answer is correct or incorrect?
Have i did any mistake in applying Torque equations?(i am not good in rotational dynamics)
If i am wrong somewhere then please tell me.I will thankful to u.
thanks!

Last edited: Jan 7, 2012
2. Jan 7, 2012

### SammyS

Staff Emeritus
All that the fiction accomplishes is that it causes the ball to roll, apparently without slipping. This will not dissipate any energy.

3. Jan 7, 2012

### vkash

If anything doesn't dissipate energy does it mean that it is SHM. I think no. You need to prove it to confirm.

4. Jan 7, 2012

### SammyS

Staff Emeritus
I was merely making a statement to clarify the role of friction in this situation.

I have not stated that it is SHM, nor have I stated that it's not SHM.

However, I will say here, that most small oscillation problems, like this one, are SHM.