What is the moment of inertia of the object starting from rest if it has a final velocity of 5.9 m/s? Express the moment of inertia as a multiple of MR2, where M is the mass of the object and R is its radius.

The moment of inertia is I = 0.126*R^2*M

Explanation:

We can calculate the moment of inertia of an object that starts from rest and has a final velocity using the energy conservation equation, as follows:

Ek1 + Ep1 = Ek2 + Ep2, where

Ek1 = kinetic energy of the object before to roll down

Ep1 = potential energy of the object

Ek2 = kinetic energy when the object comes down

Ep2 = potential energy of the object at the bottom

We have the follow:

Ek1 = 0

Ep1 = M*g*h

Ek2 = ((I*w) / 2) + ((M*v^2) / 2)

Ep2 = 0

Replacing values:

0 + M*g*h = ((I*w) / 2) + ((M*v^2) / 2) + 0

where:

M = mass of the object

g = gravitational acceleration

I = moment of the inertia

w = angular velocity = v/R

h = height

M*g*h = ((1/2) * I * (v^2/R^2)) + ((M*v^2) / 2)

M*9.8*2 = (I * (5.9^2) / (2*R^2)) + ((5.9^2 * M) / 2)

19.6 * M = ((17.4*I) / R^2) + 17.4*M

Clearing I, we have:

I = 0.126*R^2*M