- #1

Lengalicious

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## Homework Statement

Consider a solid sphere of radius r to be placed at the bottom of a spherical bowl radius R, after the ball is given a push it oscillates about the bottom. By using the Lagrangian approach find the period of oscillation.

## Homework Equations

## The Attempt at a Solution

Ok so this is as far as i get, not sure if what I am doing is correct:

General coords (∅,x,y)

T = 1/2*m*(xdot

^{2}+ydot

^{2}) + 1/2*I*∅dot

^{2}

T = 1/2*m*xdot

^{2}+1/2*m*ydot

^{2}+(mr

^{2}∅

^{2})/5 (Since Inertia of Solid Sphere is = 2/5*m*r

^{2})

V = mgy

L = 1/2*m*xdot

^{2}+1/2*m*ydot

^{2}+(mr

^{2}∅

^{2})/5 - mgy

∂L/∂∅dot = 2/5*m*r

^{2}*∅dot

∂L/∂∅ = 0

∂L/∂xdot = m*xdot

∂L/∂x = 0

∂L/∂ydot = m*ydot

∂L/∂y = -mg

d(2/5*m*r

^{2}*∅dot)/dt = 0

d(m*xdot)/dt = 0

d(m*ydot)/dt = -mg

Up to here I am not sure whether I've been doing the right thing, and also, not quite sure how this helps me find the period of oscillation?

EDIT: Sigh, I know my equations of motion are wrong because i should be getting something similar to the DE for an SHO. Could really use some help.

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