A hole is drilled in a metal sheet. At $27^{\circ} \mathrm{C}$, the diameter of hole is 5 cm . When the sheet is heated to $177^{\circ} \mathrm{C}$, the change in the diameter of hole is $\mathrm{d} \times 10^{-3} \mathrm{~cm}$. The value of d will be \_\_\_\_ if coefficient of linear expansion of the metal is $1.6 \times 10^{-5} /{ }^{\circ} \mathrm{C}$.

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Heat & Thermodynamics

A hole is drilled in a metal sheet. At

2 7^{\circ} C

, the diameter of hole is 5 cm . When the sheet is heated to

17 7^{\circ} C

, the change in the diameter of hole is

d \times 1 0^{- 3} cm

. The value of d will be ____ if coefficient of linear expansion of the metal is

1.6 \times 1 0^{- 5} /^{\circ} C

Answer: 12

Solution

A hole in a heated plate widens exactly as the surrounding metal, so its diameter grows like any linear dimension:

Δ d = d_{0} α Δ T

. With

d_{0} = 5 cm

α = 1.6 \times 1 0^{- 5} /^{\circ} C

and

Δ T = 177 - 27 = 150^{\circ} C

, the change is

5 \times 1.6 \times 1 0^{- 5} \times 150 = 12 \times 1 0^{- 3} cm

. The value of

d

12

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Integer answer

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