A non-uniform round object of mass M and radius R is found to have its total kinetic energy to be given by
When rolling down an incline.What is this object's moment of inertia?

Question

A non-uniform round object of mass M and radius R is found to have its total kinetic energy to be given by 
When rolling down an incline.What is this object's moment of inertia?

Quizplus · Accepted Answer

The total kinetic energy of a rolling object is given by:

KE = (1/2)mv^2 + (1/2)Iω^2

where m is the mass of the object, v is its translational velocity, I is its moment of inertia about the axis of rotation, and ω is its angular velocity.

When the object is rolling without slipping, we have the relationship:

v = Rω

where R is the radius of the object. Substituting this into the expression for KE, we get:

KE = (1/2)mR^2ω^2 + (1/2)Iω^2

Simplifying, we obtain:

KE = (1/2)(mR^2 + I)ω^2

Comparing this with the given expression for KE, we see that:

(1/2)(mR^2 + I) = 11ea7a51_6367_38a6_89bd_7d77c89913cc_TB5259_11

Solving for I, we get:

I = 2KE - mR^2 = 11ea7a51_6367_38a8_89bd_4383a91afcc0_TB5259_11

Therefore, the answer is (B).

A Non-Uniform Round Object of Mass M and Radius R