Moment of Inertia of a Cross-shaped object

The moment of inertia is a measure of an object's resistance to rotational motion. It is calculated using the object's mass, shape, and the axis of rotation. In this problem, we are asked to find the moment of inertia of a cross-shaped object about an axis passing through the point where the two rods are joined and perpendicular to the plane of the cross.

Uniform Thin Rods

A uniform thin rod is an object with a uniform mass distribution along its length. It is characterized by its mass, length, and moment of inertia. The moment of inertia of a thin rod about an axis perpendicular to its length and passing through its center of mass is given by the formula I = (1/12)ML^2, where M is the mass of the rod and L is its length.

Cross-shaped Object

To find the moment of inertia of the cross-shaped object, we can treat it as two identical thin rods of mass M and length L joined together at their centers.

- Divide the cross into two identical thin rods
- Calculate the moment of inertia of each rod
- Use the parallel axis theorem to find the moment of inertia of the cross-shaped object

Divide the Cross into Two Identical Thin Rods

To calculate the moment of inertia of the cross-shaped object, we can divide it into two identical thin rods. Each rod has a mass of M and a length of L.

Calculate the Moment of Inertia of Each Rod

The moment of inertia of each rod is given by the formula I = (1/12)ML^2. Since the two rods are identical, their moments of inertia are equal.

Use the Parallel Axis Theorem to Find the Moment of Inertia of the Cross-shaped Object

The parallel axis theorem states that the moment of inertia of an object about an axis parallel to its center of mass is equal to the moment of inertia about a parallel axis passing through the center of mass plus the product of the object's total mass and the square of the distance between the two axes.

In this problem, the axis of rotation passes through the point where the two rods are joined and is perpendicular to the plane of the cross. The distance between the center of mass of each rod and the axis of rotation is L/2.

Using the parallel axis theorem, the moment of inertia of the cross-shaped object about the given axis is:

I = 2(1/12)ML^2 + 2M(L/2)^2
= (1/6)ML^2 + (1/2)ML^2
= (1/6 + 1/2)ML^2
= (2/3)ML^2
= Ml²/6

Therefore, the moment of inertia of the cross-shaped object about the given axis is Ml²/6.

MI of a rod is Ml^2/12. ( for axis perpendicular to plane of rod,nd passing th' mid point). As both rods r joint at middle point bcs cross is formed, MI = Ml^2/12+ Ml^2/12= Ml^2/6