Question

four thin rods each of mass m and length l are joined to make a square. Find moment of inertia of all the four rods about any side of the square.

Answer 1

Given: Four rods of mass m and length l are joined to form a square. (Refer the diagram given below).

To find: Moment of inertia about one side.

Solution:

• We have to find moment of inertia(I) about any one side of the square. We consider about side CD.
• Let moment of inertia of rod AB be I₁,rod BC be I₂, rod CD be I₃ and rod AD be I₄.
• so, I = I₁ + I₂ + I₃ + I₄
• The rod CD lies on the axis . So the distance between the rod and the axis is zero. Hence, its moment of inertia I₃ = 0.
• The rods AD and BC are perpendicular to the axis. They have the same mass and lengths. Hence, I₂ = I₄.
• The formula for moment of inertia of a rod about its end is,

        I = (ml²)/3

• Hence, I₂ = I₄ = (ml²)/3
• The rod AB is parallel to the axis. Hence we use the parallel axis theorem.
• We consider an axis parallel to CD passing through AB. As the axis is passing through AB its moment of inertia about the parallel axis will be zero. Also, the distance between the two axes is l.
• Hence, I₁ = 0 + ml²
• Finally, adding all the values we will get,

        I = ml² + (ml²)/3 + 0 + (ml²)/3

        I = (5ml²)/3

The answer is (5/3) ml².