Question

A hole is drilled in a metal sheet At $27^{\circ} C$, the diameter of hole is $5 cm$ When the sheet is heated to $177^{\circ} C$, the change in the diameter of hole is $d \times 10^{-3} cm$ The value of $d$ will be ___ if coefficient of linear expansion of the metal is $16 \times 10^{-5} { }^{\circ} C$

Answer 1

Correct Answer: 12

To solve the problem, we need to calculate the change in the diameter of the hole when the metal sheet is heated from $27^{\circ} C$ to $177^{\circ} C$. The formula for linear expansion is:

$\Delta L = L_0 \cdot \alpha \cdot \Delta T$

where:

• $\Delta L$ is the change in length (or diameter).
• $L_0$ is the original length (or diameter), which is $5 \mathrm{\ cm}$.
• $\alpha$ is the coefficient of linear expansion, which is $16 \times 10^{-5} \mathrm{\ {}^{\circ} C^{-1}}$.
• $\Delta T$ is the change in temperature, calculated as:

$\Delta T = 177^{\circ} C - 27^{\circ} C = 150^{\circ} C$

Substitute the known values into the formula:

$\Delta L = 5 \cdot 16 \times 10^{-5} \cdot 150$

Calculate:

$\Delta L = 5 \cdot 2400 \times 10^{-5}$

$\Delta L = 12000 \times 10^{-5}$

$\Delta L = 1.2 \times 10^{-2} \mathrm{\ cm}$

Since the change in diameter is given by $d \times 10^{-3} \mathrm{\ cm}$, let $d = 1.2 \times 10$:

$d = 12$

Therefore, the value of $d$ is $12$, which is within the expected range of $12$ to $12$.