Question

Find the moment of inertia of this system about an axis that is perpendicular to the plane of the square and passes through one of the masses.

Answer 1

To find the moment of inertia of a system about an axis that is perpendicular to the plane of the square and passes through one of the masses, you can use the parallel axis theorem. The moment of inertia can be calculated by first finding the moment of inertia for the center of mass and then adding the moment of inertia due to the distance between the axis passing through the center of mass and the axis passing through one of the masses.

Step-by-step explanation:

The moment of inertia of a system about an axis that is perpendicular to the plane of a square and passes through one of the masses can be calculated using the parallel axis theorem.

1. First, find the moment of inertia for the center of mass using the formula ICM = ML2/12, where M is the total mass of the system and L is the distance between the masses.
2. Then, use the parallel axis theorem to find the moment of inertia about the point of rotation, given by I = ICM + Md2, where M is the mass of each individual mass, and d is the distance between the axis passing through the center of mass and the axis passing through one of the masses.

In the case of a system with two masses at the ends of a rod, each mass being a distance R away from the axis, the moment of inertia is I = 2mR2, where m is the mass of each individual mass.

Answer 2

We would need to know the geometry of the system. The formula for the moment of inertia about the axis the axis perpendicular to the plane for common geometries are
Solid cylinder: I = (1/2) MR^2
Hollow cylinder: I = MR^2
Solid Sphere: I = (1/12) ML^2
Square Plate: I = (2/3) MR^2 (where R is half the length of the diagonal)