An object is made of two equal parts by volume; one part has density ρ₀ and the other part has density 2ρ₀. What is the average density of the object?
- (a)3ρ₀
- (b)3⁄2 ρ₀
- (c)ρ₀
- (d)1⁄2 ρ₀
Correct — B, 3⁄2 ρ₀. Let each half have volume V. The two masses are then ρ₀V and 2ρ₀V, so the whole object has mass 3ρ₀V and volume 2V. Average density is total mass divided by total volume, that is 3ρ₀V ÷ 2V = 1·5 ρ₀. Because the two parts are equal in volume, the average density is simply the arithmetic mean of the two densities, (ρ₀ + 2ρ₀)/2.
- (a)3ρ₀ — 3ρ₀ is what you get by adding the two densities and forgetting to divide by the total volume 2V. It is also denser than the denser half, and no composite body can be denser than its densest component.
- (c)ρ₀ — ρ₀ is the density of the lighter half alone. Adding an equal volume of heavier material must pull the average above ρ₀, not leave it unchanged.
- (d)1⁄2 ρ₀ — 1⁄2 ρ₀ is lighter than either part. An average density always lies between the smallest and the largest component density, so here it must sit between ρ₀ and 2ρ₀.
Density is mass divided by volume, and for a body made of more than one material the average density is the TOTAL mass divided by the TOTAL volume. Densities cannot simply be added, and they can only be averaged directly when the volumes being combined are equal. Whatever the mixing rule, the result must lie between the lowest and the highest component density.
Work in masses, not densities. Assign each half a volume V, convert to mass with m = ρV, add the masses, then divide by 2V. The tempting error is to treat density like mass and add the two figures to get 3ρ₀ — a quick sanity check kills it, because the answer must fall strictly between ρ₀ and 2ρ₀, which leaves only 1·5 ρ₀ standing.
- Average density of a composite body is total mass ÷ total volume, never the sum of the densities.
- Mixing equal VOLUMES gives the arithmetic mean of the densities, (ρ₁ + ρ₂)/2.
- Mixing equal MASSES gives the harmonic mean, 2ρ₁ρ₂/(ρ₁ + ρ₂), which is always the smaller of the two averages.
- An average density must lie between the component densities — here between ρ₀ and 2ρ₀, so 1·5 ρ₀ passes the check while 3ρ₀ and 0·5 ρ₀ fail it.
Equal volumes mean the average density is the arithmetic mean of the two densities — 3⁄2 ρ₀, option (b).
- Adding densities instead of adding masses.
- Assuming equal masses and equal volumes give the same average — they do not.
- Skipping the sanity check that the answer must lie between the two component densities.
Either as a direct two-line calculation like this one, or as a harder variant that gives both the equal-volume and the equal-mass averages and asks for the individual densities.
Assertion (A): An iron ball floats on mercury but gets immersed in water. Reason (R): The specific gravity of iron is more than that of mercury.
- (a) Both A and R are individually true and R is the correct explanation of A
- (b) Both A and R are individually true but R is not a correct explanation of A
- (c) A is true but R is false
- (d) A is false but R is true
Answer(c) A is true but R is false
Same idea — an object's fate in a fluid is decided by comparing densities. This NDA item computes an average density from two parts; the UPSC item asks you to compare iron's density against mercury's and water's.
Two substances of densities ρ1 and ρ2 are mixed in equal volume and their relative density is 4. When they are mixed in equal masses, relative density is 3. The values of ρ1 and ρ2 respectively are
- (a) 6, 2
- (b) 3, 5
- (c) 12, 4
- (d) 9, 3
Answer(a) 6, 2
The same mixing rules, run backwards. This 2022 item asks for the equal-volume average directly; the 2019 item gives both the equal-volume and the equal-mass averages and asks for the component densities.
- practice — not a real PYQ
Two liquids of densities 2ρ and 4ρ are mixed in equal volumes. The density of the mixture is
- (a)2ρ
- (b)3ρ
- (c)6ρ
- (d)8ρ
Answer(b) 3ρ — equal volumes give the arithmetic mean of the densities, (2ρ + 4ρ)/2.
- practice — not a real PYQ
A solid body placed in a liquid will float when its average density is
- (a)greater than that of the liquid
- (b)less than that of the liquid
- (c)equal to that of the liquid
- (d)independent of that of the liquid
Answer(b) less than that of the liquid — the buoyant force then exceeds the body's weight when fully immersed.