The solid cylinder of length 80 cm and mass M has a radius of 20 cm. Calculate the density of the material used if the moment of inertia of the cylinder about an axis CD parallel to AB as shown in figure is 2.7 kg m².

Solution

Physics sir · 4 steps

Shift the axis with the parallel axis theorem, get the mass, then divide by the cylinder's volume.

1. Read the figure. AB is the cylinder's own axis, through its centre of mass. CD is parallel to AB at a distance d = L/2 = 0.4 m. The radius is r = 0.2 m.

   I_(CD) = 1/2Mr² + M(L/2)²

2. Find the mass. 1/2r² = 0.02 m² and (L/2)² = 0.16 m², so I_(CD) = 0.18M. Setting 0.18M = 2.7 gives M = 15 kg.
3. Find the volume. V = π r² L = π × 0.04 × 0.8 = 0.032π ≈ 0.1005 m³.
4. So. ρ = M/V = 15/0.1005 ≈ 149 kg m⁻³ = 1.49 × 10² kg m⁻³, which is option (d).

The idea to keep. The parallel axis theorem only works from an axis through the centre of mass: I = I_(cm) + Md². For a solid cylinder about its own axis, I_(cm) = 1/2Mr².

Answer from the NTA final answer key. Solution by PrepWiser.