A pendulum consists of a uniform disk with radius r = 10 cm and mass 480 g attached to a uniform rod with length L = 500 mm and mass 175 g.

(a) Calculate the rotational inertia of the pendulum about the pivot.

(b) What is the distance between the pivot and the center of mass of the pendulum?

(c) Calculate the period of oscillation.

(i have absolutely no idea how to do this) thanks..

On a, use the parallel axis theorem.
on b, I assume it is 60 cm. Use pendulum formulaa on c.

I will be happy to critique your work.

Sure! I can guide you through the steps to solve each part of the problem.

(a) To calculate the rotational inertia of the pendulum about the pivot, you can use the parallel axis theorem. This theorem states that the rotational inertia of a body about an axis parallel to and a distance d away from the axis passing through its center of mass can be found by adding the rotational inertia of the body about its center of mass and the product of its mass and the square of the distance d. In this case, we have two components: the disk and the rod.

For the disk:
The rotational inertia of a uniform disk about its central axis is given by the formula I_disk = (1/2) * m * r^2, where m is the mass of the disk and r is its radius. Plugging in the values, we have:
I_disk = (1/2) * 480 g * (10 cm)^2.

For the rod:
The rotational inertia of a uniform rod pivoted at one end and rotating about an axis perpendicular to its length and passing through its other end is given by the formula I_rod = (1/3) * m * L^2, where m is the mass of the rod and L is its length. Plugging in the values, we have:
I_rod = (1/3) * 175 g * (50 cm)^2.

To find the total rotational inertia of the pendulum, simply add the rotational inertia of the disk and the rod together: I_total = I_disk + I_rod.

(b) To find the distance between the pivot and the center of mass of the pendulum, you can use the concept of center of mass. The center of mass of a system is the point at which its mass can be considered to be concentrated, and it can be found by taking the weighted average of the positions of its individual components.

In this case, the center of mass will be the position where the disk and the rod are balanced. Since the disk has a larger mass, we can assume the center of mass is closer to the disk. Let's call the distance between the pivot and the center of mass as d_cm.

To calculate d_cm, you can use the formula: d_cm = (m_disk * d_disk + m_rod * d_rod) / (m_disk + m_rod), where m_disk and m_rod are the masses of the disk and the rod respectively, and d_disk and d_rod represent the distances between their respective centers of mass and the pivot.

(c) To calculate the period of oscillation, you can use the formula for the period of a simple pendulum, which is given by the formula T = 2π * sqrt(L/g), where L is the length of the pendulum and g is the acceleration due to gravity.

Plugging in the value of L, you will get T in seconds.

I hope this explanation helps you in solving the problem. Let me know if you have any further questions or if you would like me to check your work!