A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass M = 7.01 kg and length L = 0.15 m that is pivoted freely about one end, with a solid sphere of the same mass, 7.01 kg, and a radius of L/2 centered about the free end of the rod.

a) What the moment of inertia of the pendulum about its pivot point?

b) What is the period of the pendulum for small oscillations?

c) Determine the length L that gives a period of T = 2.51 s.

To find the moment of inertia of the pendulum about its pivot point, we can use the parallel axis theorem. The parallel axis theorem states that the moment of inertia about an axis parallel to an axis through the center of mass is equal to the moment of inertia about the center of mass plus the product of the mass and the square of the distance between the two axes.

a) The moment of inertia of the rod about its center can be calculated using the formula for the moment of inertia of a thin rod rotating about an axis through its center:

I_rod = (1/12) * M * L^2

The moment of inertia of the sphere about its center can be calculated using the formula for the moment of inertia of a solid sphere rotating about an axis through its center:

I_sphere = (2/5) * M * (L/2)^2

Since the moment of inertia of the sphere is given about its center, we need to find the moment of inertia of the sphere about the pivot point. The distance between the pivot point and the center of the sphere is L, so we can use the parallel axis theorem:

I_sphere_pivot = I_sphere + M * L^2

Now we can find the total moment of inertia of the pendulum:

I_total = I_rod + I_sphere_pivot

Substituting the formulas for I_rod and I_sphere_pivot, we get:

I_total = (1/12) * M * L^2 + (2/5) * M * (L/2)^2 + M * L^2

Simplifying this expression will give the moment of inertia of the pendulum about its pivot point.

b) The period of a physical pendulum for small oscillations can be calculated using the formula:

T = 2π * sqrt(I_total / (M * g * L))

Where T is the period, I_total is the moment of inertia about the pivot point, M is the total mass, g is the acceleration due to gravity, and L is the length of the pendulum.

Substituting the values for I_total, M, and L in this formula will give the period of the pendulum.

c) To determine the length L that gives a period of T = 2.51 s, we rearrange the formula for the period of the pendulum to solve for L:

L = (T^2 * (M * g)) / (4π^2 * I_total)

Substituting the values for T, M, g, and I_total into this formula will give the length of the pendulum.