Rotational Mechanics · Moment of Inertia

Rotate each rod structure about OCO′ and inspect the geometry

Each structure is made from uniform rods of mass m and length l. The dashed axis lies in the original plane of the structure. Watch the rods turn about OCO′ in the direction marked near O′, then compare the perpendicular distances that determine the moment of inertia.

P · two perpendicular rods

Dashed outline = original plane · solid rods = rotating structure

Choose a rod structure

Rotation speed 1.0×

P · Two perpendicular rods

Both rods make 45° with the axis. Each contributes (1/3)ml² sin²45°.

AxisO → C → O′

InertiaI = 1/3 ml²

Matched entryList-II (5)

Axis OCO′fixed in space and lying in the initial plane

Original structuredashed reference before rotation begins

Rotating rodssolid projection of the rigid assembly

How to read the rotation

01 / AXIS

OCO′ remains fixed

The dashed axis is in the original plane. Every rod sweeps through three-dimensional space around this line.

02 / DISTANCE

Only perpendicular distance matters

For each mass element, the contribution is set by its perpendicular distance from OCO′.

I = ∫ r_⊥² dm

03 / ROD

Use the rod angle

For a rod through the axis at one end, inclined by angle θ, the useful result is:

I = ⅓ ml² sin²θ

The correct match

two rods at 45°

List-II (5)

equilateral triangle

List-II (1)

square about diagonal

List-II (4)

two rods at 30°

List-II (2)

Therefore: P→5, Q→1, R→4, S→2.