# Angular and CoM Velocities of a Solid Sphere

## Homework Statement

A solid sphere of mass M and radius R is rolling,without slipping, down a curved rail. The sphere is initially at rest at a height of h1. Find the angular velocity ω2 and the center of mass velocity of the sphere vcm at the end of the rail of height h2. You may assume that no vibration and heat are generated as the sphere rolls along the rail.

## Homework Equations

solid sphere, $I = \frac{2}{5}mR^2$

## The Attempt at a Solution

I'm not sure if I began with the correct equation.

$KE = \frac{1}{2} m v^2 + \frac{1}{2} I ω^2$
$= \frac{1}{2} m(ωR)^2 + \frac{1}{2} (\frac{2}{5}mR^2) ω^2$
$mg(h_1 -h_2) = \frac{7}{10}m ω^2 R^2$
$ω = √(\frac{10}{7}g(h_1-h_2)) / R$

$KE = \frac{1}{2} m v^2 + \frac{1}{2} I ω^2$
$= \frac{1}{2} m v^2 + \frac{1}{2} (\frac{2}{5}mR^2) (\frac{v}{R})^2$
$mg(h_1 -h_2) = \frac{7}{10}mv^2$
$v = √(\frac{10}{7}g(h_1-h_2))$