State and prove the parallel axis theorem.

III. Long Answer Questions · Q8

Q. State and prove the parallel axis theorem.

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Step 1. Statement of the theorem.

The moment of inertia of a rigid body about any axis equals the sum of (a) its moment of inertia about a PARALLEL axis passing through its own center of mass, and (b) the product of the body's total mass and the square of the perpendicular distance between the two axes.

Step 2. Setting up the proof.

Let $I_C$ be the (known) moment of inertia of the body about an axis $AB$ passing through its center of mass, and let $DE$ be a second axis, parallel to $AB$ , at perpendicular distance $d$ from it. Call $I$ the (unknown) moment of inertia about $DE$ , which is to be found.

Step 3. A representative point mass.

Consider a small point mass $m$ somewhere in the body, at perpendicular distance $x$ from the center-of-mass axis $AB$ (measuring $x$ as a signed distance, positive on one side of $AB$ and negative on the other). Since $DE$ is offset from $AB$ by $d$ , this same point mass is at distance $(x+d)$ from $DE$ . Its contribution to the moment of inertia about $DE$ is therefore

m(x+d)^2.

Step 4. Summing over the whole body.

Adding up this contribution over every point mass making up the body:

I=\sum m(x+d)^2=\sum m\left(x^2+2xd+d^2\right)=\sum mx^2+2d\sum mx+d^2\sum m.

Step 5. Identifying each of the three sums.

$\sum mx^2$ is, by definition, exactly $I_C$ — the moment of inertia about the center-of-mass axis $AB$ .
$\sum m$ is simply the total mass $M$ of the body. …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set ANNUAL5 marks
$Q. (a) State and prove the parallel axis theorem. OR$
(b) Explain how overtones are produced in a closed organ pipe.
›Reveal solutionSolution
The parallel axis theorem, I = I_cm + Md^2, relates the moment of inertia about any axis to the moment of inertia about a parallel axis through the centre of mass, proved by expanding the defining integral and showing the cross term vanishes.

Statement: The moment of inertia (I) of a rigid body about any axis is equal to the sum of its moment of inertia (I_cm) about a parallel axis passing through its centre of mass, and the product of the mass (M) of the body and the square of the perpendicular distance (d) between the two parallel axes:

I = I_cm + Md^2

Proof:

Consider a rigid body of mass M, and let AB be an axis through its centre of mass C, with moment of inertia I_cm about AB. Let A'B' be another axis, parallel to AB, at a perpendicular distance d from it, about which we want to find the moment of inertia I.

Set up coordinates with the centre of mass C at the origin, and the axis AB along the z-direction through the origin. The parallel axis A'B' is then a line parallel to AB but displaced by distance d in the x-direction.

Consider a small mass element dm of the body, located at perpendicular distance r from axis AB (through the centre of mass), with coordinates (x, y) in the plane perpendicular to AB (so r^2 = x^2 + y^2).

The perpendicular distance of this same mass element from the new axis A'B' (displaced by d along x) is r', where, by the geometry of the displaced axis,

r'^2 = (x-d)^2 + y^2 = x^2 - 2xd + d^2 + y^2 = r^2 - 2xd + d^2

The moment of inertia about the new axis A'B' is:

I = integral of r'^2 dm = integral of (r^2 - 2xd + d^2) dm

I = integral of r^2 dm - 2d*(integral of x dm) + d^2*(integral of dm)

Now examine each term:

integral of r^2 dm = I_cm, the moment of inertia about the axis through the centre of mass (by definition). …
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Q.State and prove the parallel axis theorem.

Concept understanding — Parallel Axis Theorem

Q.State and prove the parallel axis theorem.

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