When the temperature of a metal wire is increased from 0ᵒC to 10ᵒC, its length increases by 0.02%. The percentage change in its mass density will be closest to :

Solution

Physics sir · 4 steps

Mass is fixed, so density falls as volume grows, and volume strain is three times length strain.

1. Mass is fixed. Heating does not change the mass of the wire. Since ρ = m/V, the density falls by the same fraction that the volume rises.
2. Volume from length. A solid expands equally in all three directions, so every length grows by the same fraction. Volume scales as length cubed, which gives γ = 3α.

   (Δ ρ)/ρ = -(Δ V)/V = -3 (Δ L)/L

3. Put in numbers. (Δ L)/L = 0.02%, so (Δ V)/V = 3 × 0.02% = 0.06%. The density therefore decreases by 0.06%.
4. So. The percentage change in mass density is about 0.06, option (b). The minus sign only tells you the density goes down.

The idea to keep. For small heating of a solid, volume changes by 3 times the fractional length change, and density changes by the same amount with the opposite sign.

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