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Q56 (NDA-I/2024) Science & Technology › Basic Science (Physics, Chemistry, Biology) Official Key

A spherical shell of outer radius R and inner radius R/2 contains a solid sphere of radius R/2 (see figure). The density of the material of the solid sphere is ρ and that of the shell is ρ/2. What is the average mass density of the larger sphere thus formed?

Result
Your answer: —  ·  Correct: B
Explanation

The average mass density is calculated by dividing the total mass of the composite sphere by its total volume. Let R be the outer radius.

  • Inner Solid Sphere: Radius is R/2 and density is ρ. Mass (M₁) = Volume × Density = (4/3)π(R/2)³ × ρ = (1/6)πR³ρ.
  • Outer Spherical Shell: Inner radius is R/2, outer radius is R, and density is ρ/2. Volume = (4/3)π[R³ - (R/2)³] = (4/3)π(7R³/8) = (7/6)πR³. Mass (M₂) = Volume × Density = (7/6)πR³ × (ρ/2) = (7/12)πR³ρ.
  • Total Mass: M₁ + M₂ = (2/12 + 7/12)πR³ρ = (9/12)πR³ρ = (3/4)πR³ρ.
  • Total Volume: V = (4/3)πR³.
  • Average Density: (Total Mass) / (Total Volume) = [(3/4)πR³ρ] / [(4/3)πR³] = (3/4) × (3/4)ρ = 9ρ/16.

Hence, Option B is correct.

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