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## 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 h

_{1}. Find the angular velocity ω

_{2}and the center of mass velocity of the sphere v

_{cm}at the end of the rail of height h

_{2}. You may assume that no vibration and heat are generated as the sphere rolls along the rail.

## Homework Equations

solid sphere, [itex]I = \frac{2}{5}mR^2 [/itex]

## The Attempt at a Solution

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

[itex]KE = \frac{1}{2} m v^2 + \frac{1}{2} I ω^2[/itex]

[itex]= \frac{1}{2} m(ωR)^2 + \frac{1}{2} (\frac{2}{5}mR^2) ω^2[/itex]

[itex]mg(h_1 -h_2) = \frac{7}{10}m ω^2 R^2[/itex]

[itex]ω = √(\frac{10}{7}g(h_1-h_2)) / R[/itex]

[itex]KE = \frac{1}{2} m v^2 + \frac{1}{2} I ω^2[/itex]

[itex] = \frac{1}{2} m v^2 + \frac{1}{2} (\frac{2}{5}mR^2) (\frac{v}{R})^2[/itex]

[itex]mg(h_1 -h_2) = \frac{7}{10}mv^2[/itex]

[itex]v = √(\frac{10}{7}g(h_1-h_2))[/itex]

Thanks in advance!