Physics Q32 – JEE-MAIN 2025

Home Physics Rotational MotionQuestion #446

Moment of inertia of a rod of mass 'M' and length 'L' about an axis passing through its center and normal to its length is '

α

'. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is :

Clue 1 of 3: Understand the initial condition

The moment of inertia of the original rod is given as $α$ .

🥷

Ninja Strategy•Step-by-Step Scaling

Calculate the moment of inertia for a single cut piece first, considering its reduced mass and length, then sum the contributions of both pieces to form the cross.

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● Video Walkthrough

Step 1: Initial Moment of Inertia✦ Active

The moment of inertia of the original rod of mass $M$ and length $L$ about an axis passing through its center and normal to its length is given as $α$ .

α = \frac{M L^{2}}{12}

Step 2: Moment of Inertia of Each Part○ Expand

The rod is cut into two equal parts. Each part has mass $M^{'} = M / 2$ and length $L^{'} = L / 2$ . For each part, the axis of rotation passes through its center and is normal to its length (which is also the center of the cross).

I_{p a r t} = \frac{M^{'} (L^{'})^{2}}{12} = \frac{(M / 2) (L / 2)^{2}}{12} = \frac{(M / 2) (L^{2} / 4)}{12} = \frac{M L^{2}}{96}

💡 Teacher's Secret Hint

Remember to correctly apply the new mass and length for each segment.

Step 3: Total Moment of Inertia of the Cross○ Expand

The cross shape is formed by joining the two parts symmetrically at their centers. The total moment of inertia of the cross about the central axis (normal to the plane) is the sum of the moments of inertia of the two parts.

I_{c r o s s} = I_{p a r t 1} + I_{p a r t 2} = \frac{M L^{2}}{96} + \frac{M L^{2}}{96} = 2 \times \frac{M L^{2}}{96} = \frac{M L^{2}}{48}

Now, substitute the value of $α$ from Step 1 into this expression.

I_{c r o s s} = \frac{1}{4} (\frac{M L^{2}}{12}) = \frac{α}{4}

💡 Teacher's Secret Hint

Ensure you sum the moments of inertia for both parts, as they both contribute to the total moment of inertia of the cross.

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Physics Question 32 – JEE-MAIN 2025

Clue 1 of 3: Understand the initial condition

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