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Home » A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm at 27.0 °C. What is the change in the diameter of the hole when the sheet is heated to 227 °C? Coefficient of linear expansion of copper = .
A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm at 27.0 °C. What is the change in the diameter of the hole when the sheet is heated to 227 °C? Coefficient of linear expansion of copper = 1.70 × 10^-5 K^-1.
CBSE | Class 11 | NCERT | Thermal Properties of Matter
A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm at 27.0 °C. What is the change in the diameter of the hole when the sheet is heated to 227 °C? Coefficient of linear expansion of copper = 1.70 × 10^-5 K^-1.
Solution:
Given in the question:
Diameter of hole ( D1) = 4.24 cm
Initial Temperature, T1 = 27.0 °C = 27+ 273 = 300 K
Final temperature, T2 =227 °C = 227+ 273 = 500K
Let the diameter of the hole at the final temperature be D2
Coefficient of linear expansion of copper, α= 1.70 × 10–5 K–1

Initial area of hole (Ao)= πr² =π ( 4.24/2)²

coefficient of superficial expansion (β) = 2× coefficient of linear expansion of copper (α)= 2 × 1.7 × 10-5 = 3.4 × 10-5 

Calculation:

Making use of the formula,
A = Ao( 1 + β∆T)

Evaluating the value of diameter,

A = π ( 4.24/2)² [1 + 3.4 × 10-5 × (500-300)]
= π ( 4.24/2)² [ 1 + 3.4 × 10-5 × 200]
= π ( 4.24/2)² [ 1 + 6.8 × 10-3 ]
= π ( 4.24/2)²  [ 1 + 0.0068]
=  π ( 4.24/2)² (1.0068)
πD2²/4 = π ( 4.24/2)² ( 1.0068 )
D2²= 17.97 (1.0068)

D2²=  18.092
= 4.253
D2 = 4.253 cm
Change in diameter (∆D) = D2 – D1
= 4.253 – 4.24
= 0.013 cm
Final Answer: Thus, the change in diameter is 0.013 cm
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