A hole is drilled in a metal sheet. At 27° C, the diameter of hole is 5 cm. When the sheet is heated to 177° C, the change in the diameter of hole is d×10⁻³ cm. The value of d will be __________ if coefficient of linear expansion of the metal is 1.6×10⁻⁵/°C.

Solution

Physics sir · 3 steps

The hole grows exactly as if it were a disc of the same metal, so use linear expansion on the diameter.

1. Hole expands too. When the sheet is heated, every length on it scales up by the same factor, including the diameter of the hole. So the hole gets bigger, just as a solid disc of that metal would.

   Δ d = d₀ α Δ T

2. Put in numbers. Here d₀ = 5 cm, α = 1.6 × 10⁻⁵ °C⁻¹ and Δ T = 177 - 27 = 150 °C. So Δ d = 5 × 1.6 × 10⁻⁵ × 150 = 1.2 × 10⁻² cm.
3. So. Δ d = 12 × 10⁻³ cm, so d = 12.

The common slip: 24. Using the areal coefficient 2α for the diameter gives 24. A diameter is a length, so only α applies.

The idea to keep. A hole in a heated plate expands like the material it replaced. For a diameter, a length, use α. Use 2α only for an area and 3α only for a volume.

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