Angular and CoM Velocities of a Solid Sphere

[m1k4chu]

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 v_cm 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, [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!

 

[Doc Al]

Looks good to me. (Of course there was no need to solve it twice, since v = ωR.)