The parallel axis theorem relates Icm, the moment of inertia of an object about an axis passing through its center of mass, to Ip, the moment of inertia of the same object about a parallel axis passing through point p. The mathematical statement of the theorem is Ip=Icm+Md2, where d is the perpendicular distance from the center of mass to the axis that passes through point p, and M is the mass of the object. Part A Suppose a uniform slender rod has length L and mass m. The moment of inertia of the rod about about an axis that is perpendicular to the rod and that passes through its center of mass is given by Icm=112mL2. Find Iend, the moment of inertia of the rod with respect to a parallel axis through one end of the rod. Express Iend in terms of m and L. Use fractions rather than decimal numbers in your answer. Part B Now consider a cube of mass m with edges of length a. The moment of inertia Icm of the cube about an axis through its center of mass and perpendicular to one of its faces is given by Icm=16ma2. Find Iedge, the moment of inertia about an axis p through one of the edges of the cube Express Iedge in terms of m and a. Use fractions rather than decimal numbers in your answer.

A) The moment of inertia of the rod with respect to a parallel axis through one end of the rod is; I_end = ¹/₃mL²

B) The moment of inertia about an axis p through one of the edges of the cube is; I_edge = ²/₃ma²

        We are given the formula for the moment of inertia of the same object about a parallel axis passing through point p using parallel axis theorem as;

I_p = I_cm + Md²

where;

d is the perpendicular distance from the center of mass to the axis that passes through point p

M is the mass of the object

I_cm is the moment of inertia of an object about an axis passing through its center of mass

A) We are told that;

Length = L

Mass = m

I_cm = ¹/₁₂mL²

We want to find I_end, the moment of inertia of the rod with respect to a parallel axis through one end of the rod.

This means that d = L/2. Thus;

I_end =  ¹/₁₂mL² + m(L/2)²

I_end =  ¹/₁₂mL² + ¹/₄mL²

I_end = ¹/₃mL²

B) We want to find I_edge, the moment of inertia about an axis p through one of the edges of the cube.

We are told that;

I_cm = ¹/₆ma²

Edge length is a. Thus;

d = a/√2

Thus;

I_edge = ¹/₆ma² + m(a/√2)²

I_edge = ¹/₆ma² +  ¹/₂ma²

I_edge = ²/₃ma²

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