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?
- (a)3ρ/4
- (b)9ρ/16
- (c)7ρ/8
- (d)5ρ/8
Correct — B, 9ρ/16. Split the ball into the inner solid sphere of radius R/2 and the outer shell from R/2 to R. Since volume scales as radius cubed, the inner sphere is (1/2)^3 = 1/8 of the total volume and the shell is the remaining 7/8. The average density is the volume-weighted average of the two densities, (1/8)ρ + (7/8)(ρ/2) = 2ρ/16 + 7ρ/16 = 9ρ/16. Equivalently, total mass ÷ total volume gives the same 9ρ/16.
- (a)3ρ/4 — 3ρ/4 is the plain average of the two densities, (ρ + ρ/2)/2, which ignores that the denser core fills only 1/8 of the volume. Densities must be weighted by the volume each occupies, giving 9ρ/16, not 3ρ/4.
- (c)7ρ/8 — 7ρ/8 mixes up the volume fractions — for example pairing the 7/8 shell fraction with the full density ρ. Weighting each density by its correct fraction (1/8 for the core, 7/8 for the shell) gives 9ρ/16.
- (d)5ρ/8 — 5ρ/8 (= 10ρ/16) is a final-step arithmetic slip — it adds the shell's 7ρ/16 to a mis-converted core term (3ρ/16 instead of the correct ρ/8 = 2ρ/16). The correct sum is 2ρ/16 + 7ρ/16 = 9ρ/16.
The average (mean) density of a composite body is its total mass divided by its total volume. Because mass is additive, this equals the average of the part densities weighted by the fraction of volume each part occupies — not the simple average of the densities themselves.
First find the volume fractions. Halving the radius cuts the volume to (1/2)^3 = 1/8, so the inner sphere is 1/8 of the ball and the shell is 7/8. Then take the volume-weighted average of ρ (core) and ρ/2 (shell). A useful check is that because the denser core is only a small fraction of the ball, the answer must lie only a little above the shell density ρ/2 — and 9ρ/16 ≈ 0.56ρ does just that.
- The volume of a sphere is proportional to the cube of its radius, so halving the radius gives one-eighth the volume.
- The inner sphere of radius R/2 is 1/8 of the total volume; the shell from R/2 to R is the remaining 7/8.
- Average density equals total mass ÷ total volume, which is the volume-weighted average of the part densities.
- (1/8)ρ + (7/8)(ρ/2) = 2ρ/16 + 7ρ/16 = 9ρ/16.
- Because the dense core is only 1/8 of the ball, 9ρ/16 ≈ 0.56ρ sits just above the shell's ρ/2.
- Taking the plain average of the two densities instead of weighting by volume.
- Forgetting that halving the radius cuts volume to one-eighth, not one-half.
- Swapping the 1/8 and 7/8 fractions between the core and the shell.
NDA/UPSC give a layered sphere with different densities and ask for the average density — find each volume fraction (radius cubed) and take the volume-weighted average.
No directly related past PYQ was found.
- practice — not a real PYQ
If the radius of a sphere is halved, its volume becomes
- (a)one-half
- (b)one-quarter
- (c)one-eighth
- (d)one-sixteenth
Answer(c) one-eighth — volume scales as the cube of the radius, so (1/2)^3 = 1/8.
- practice — not a real PYQ
Equal volumes of two liquids of densities ρ and 2ρ are mixed. The density of the mixture is
- (a)ρ
- (b)1.5ρ
- (c)2ρ
- (d)3ρ
Answer(b) 1.5ρ — with equal volumes the average is (ρ + 2ρ)/2 = 1.5ρ.