A solid uniform sphere and a uniform spherical shell, both having the same mass

Question

A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle Theta above the horizontal. Both spheres start from rest at the same vertical height h .

a. How fast is each sphere moving when it reaches the bottom of the hill?
V (sub: solid)

b. V(sub:hollow)

c. Which sphere will reach the bottom first, the hollow one or the solid one?

Explanation / Answer

The equation of motion is: (Mom. of Inertia) d^O/dt^2 = mgasinA where A = angle of slope, a = radius of object and ) is the angle turned through by the sphere. d^O/dt^2 = mgasinA/(Mom. of Inertia) so the angular acceleration is inversely proportional to the moment of inertia of the solid. sphere = 2ma^2/5 + ma^2 = 7(ma^2)/5 about point of contact with plane hollow sphere = 2ma^2/3 + ma^2 = 5(ma^2)/3 So, the hollow sphere has the smaller moment of inertia => greater angular acceleration so it will reach the bottom of the slope first. Loss in PE = Gain in KE mgh = (1/2)I w^2 where I = Moment of inertia of object. w^2 = 2mgh/I which will be larger for the smaller value of I (the hollow sphere). Just put the values in to get the answers.

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