A round object has a rotational inertia of I = B M R 2 , where Bis a constant, M

Question

A round object has a rotational inertia of I = BMR2, where Bis a constant, M is its mass and R is its radius. It rolls a distance d down a plane inclined at an angle theta, without slipping.

Part A: If it starts from rest, what is its speed v at the distance d? Express the answer in terms of the given variables and any other constants necessary.

Part B: What fraction of the total kinetic energy is rotational? Express your answer in terms of B.

PLEASE SHOW ALL OF YOUR WORK! I NEED TO UNDERSTAND THIS!!

Explanation / Answer

by energy conservation

mgh = mgd*sin(theta) = 1/2mv^2+1/2IW(v/r)^2

I=BMr^2

so mgd*sin(theta) = 1/2mv^2(1+B)

v=sqrt[2gdsin(theta)/(1+B)]

b. total energy = 1/2mv^2(1+B)

rotational energy = 1/2Bmv^2

so rotational energy / total energy = B/(1+B)

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