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?
3ρ/4
9ρ/16
7ρ/8
5ρ/8
Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC NDA-1 – 2024
Mass of solid sphere (M_sphere) = Density * Volume = ρ * (4/3)π(R/2)³ = ρ * (4/3)π(R³/8) = (1/6)πR³ρ.
Volume of the shell material (V_shell) = Volume of sphere with radius R – Volume of sphere with radius R/2 = (4/3)πR³ – (4/3)π(R/2)³ = (4/3)πR³ – (1/6)πR³ = (7/6)πR³.
Mass of the shell (M_shell) = Density * Volume = (ρ/2) * (7/6)πR³ = (7/12)πR³ρ.
Total mass (M_total) = M_sphere + M_shell = (1/6)πR³ρ + (7/12)πR³ρ = (2/12)πR³ρ + (7/12)πR³ρ = (9/12)πR³ρ = (3/4)πR³ρ.