Q.A uniform square plate lies in the $x$-$y$ plane with the $z$-axis passing through its centre, perpendicular to its face. A small piece $Q$ of irregular shape, originally located away from the centre (in the region above the $x$-axis), is cut out of the plate and re-glued exactly at the centre of the plate, leaving an empty hole at its original off-centre location. What happens to the moment of inertia of the plate about the $z$-axis?

Rotational Inertia Comparison: From Intuition to Precision

Imagine pushing a shopping cart that's nearly empty, then pushing the same cart loaded with bricks. The loaded cart is harder to get moving — it resists changes to its motion more. That resistance is inertia, and it depends only on how much mass is there.

Now imagine spinning a bicycle wheel. If you hold the axle and try to tilt the spinning wheel, it fights you. But here's the twist: a lightweight wheel that's large in diameter can be harder to spin or stop than a heavy wheel that's small in diameter, even if the heavy wheel has more mass. Why? Because rotational inertia depends not just on how much mass, but on where that mass is placed relative to the axis of rotation.

The Core Idea: Mass × Distance²

The precise statement is this:

For a collection of point masses, rotational inertia $I$ is the sum of each mass $m_i$ multiplied by the square of its perpendicular distance $r_i$ from the axis of rotation.

That square is crucial. Doubling the distance from the axis quadruples the rotational inertia. A mass far from the axis contributes much more to rotational inertia than the same mass close to the axis.

Why Comparison Matters

When you compare two objects, you're asking: Which is harder to spin? The answer depends on both mass and shape.

Example 1: A ring vs. a disk of the same mass and radius

• A ring has all its mass at the outer edge ($r = R$ for all mass). Its rotational inertia is $I_{\text{ring}} = MR^2$.
• A solid disk has mass spread evenly from center to edge. Its rotational inertia is $I_{\text{disk}} = \frac{1}{2}MR^2$.

The ring has twice the rotational inertia of the disk. Same mass, same radius — but the ring is harder to spin because its mass is concentrated farther from the axis.

Example 2: A long rod vs. a short rod of the same mass

• A rod spun about its center: $I = \frac{1}{12}ML^2$.
• A rod spun about one end: $I = \frac{1}{3}ML^2$.

The same rod, same mass — but spinning it about the end is four times harder than spinning it about the center. The mass is, on average, farther from the axis.

The Intuition Behind the Square

Why distance squared? Think of a spinning object. A mass far from the axis has to travel a longer path in the same time — it has a higher linear speed for the same angular speed. To change that speed (to accelerate or decelerate the rotation), you need to apply a force over that longer distance. The square comes from the geometry: the work required scales with distance, and the lever-arm effect also scales with distance. The two factors multiply.

A Quick Comparison Table