Ball rolling in a hemispherical bowl (model oscillation)

[eon714]

Homework Statement

A uniform solid sphere of radius r is placed on the inside surface of a hemispherical bowl with radius R. The sphere is released from rest at an angle θ to the vertical and rolls without slipping. Model the motion of the sphere assuming no energy is lost, so it oscillates indefinitely. (the motion will be similar to a pendulum).

Homework Equations

F=ma
τ = Iα
I = (2/5)mr^2
N = normal force
α = angular acceleration
Theta is a small angle (approximations are acceptable)

The Attempt at a Solution

I attempted to model this similarly to a pendulum, but i just can figure out the torque. This is what i have so far :

Ncosθ=(2/5)mr^2(α)= (2/5)mr^2(d^2Θ/dt^2)

not sure where to go from here, this is just the rotational motion of the ball i believe, but we need to account for the angular motion of the ball as it traverses the bowl.
Note: assume theta is a small angle as well.

Thanks in advance for the help !

 

[Yuqing]

Normally, these problems are more easily modeled with conservation of energy. Try writing the energy equation for this system and see what you can do with it.

 

[eon714]

well you are right about that but that doesn't necessarily help me get the position of the ball at any time. I need to somehow arrive at a function that looks something like this :

Θ(t) = Acos(wt+Φ) ; in order to describe the ball's motion when it is let go. The energy only gives me velocity.

 

[Yuqing]

I need to head out for a bit now, but if you write the energy conservation equation and then differentiate it, you should be able to extract a differential equation that can be used.

 

[eon714]

ok thanks for your help ill give that a try and post whatever i get...

 

[eon714]

ok so i may have gotten somewhere on this problem: for this i didnt use energy analysis but i used Newtons 2nd for torque and linear motion.

For linear motion:

Fnet=ma
mgsinΘ - Friction=ma

Now to account for the ball rotating :
τ=Iα=(2/5)(m)(r^2)(a/r)=(2/5)mra
(Friction)(r)=(2/5)mra
Friction=(2/5)(ma)

now to capture the motion of entire ball as it traverses the bowl we plug in friction into the linear equation.

mgsinΘ - Friction=ma
mgsinθ-2/5ma=ma (masses cancel) (also for small angle aproximation sinθ=θ)
gθ=7/5(a)=7/5(d^2s/dt^2) ... (s=θR)
5/7(gθ)=(d^2θ/dt^2)(R)
d^2θ/dt^2 = -(5/7)(g/R)(θ) ... Solve the Diffy q and come too ...

θ(t)=θmaxcos([(5/7)(g/R)]t+Φ) ;;;; where ω=(5/7)(g/R) ; angular frequency, and Φ is the phase difference.

Does this look correct if anyone could verify this for me that would be amazing, I've been trying to figure this one out for quite sometime.
The assistance is greatly appreciated!