Thermal Expansion

To understand the problem, we need to calculate the increase in volume of a metal box when it is heated by a small temperature . The box is in the shape of a cube with side length 'a' at the initial temperature 'T'. The coefficient of linear expansion of the metal sheet is given as .

First, we need to know how thermal expansion works in a three-dimensional object like a cube:

• The linear expansion in solids is given by the formula: , where is the original length, and is the coefficient of linear expansion.
• For a cube, each side will expand, and its new side length will be:

Now, the original volume of the cube is . The volume of the cube after heating, , will be:

Expanding this expression using binomial theorem, we get:

• Assuming is small, we approximate using binomial expansion up to the first order:
  

Next, substituting back to find the new volume:

The increase in volume is given by:

This simplifies to:

Thus, the correct answer is , which corresponds to the first option.

To determine the temperature at which a gold ring expands enough to fit on a wooden bangle, we use the formula for linear expansion:

L = L_0 \cdot (1 + \alpha_L \cdot \Delta T)

Where:

• is the final length (or diameter in this case) after expansion.
• is the initial length (or initial diameter).
• is the coefficient of linear thermal expansion.
• is the change in temperature.

Given:

• Initial diameter of the gold ring,
• Final diameter needed,
• Initial room temperature,

We need to find the temperature such that the expanded diameter equals .

First, we calculate the required change in temperature :

Using the formula, set and solve for :

6.241 = 6.230 \times (1 + 1.4 \times 10^{-5} \times \Delta T)

Simplifying, we find:

1 + 1.4 \times 10^{-5} \times \Delta T = \frac{6.241}{6.230}

1.4 \times 10^{-5} \times \Delta T = \frac{6.241}{6.230} - 1

\Delta T = \frac{6.241 - 6.230}{6.230 \times 1.4 \times 10^{-5}}

\Delta T \approx \frac{0.011}{6.230 \times 1.4 \times 10^{-5}} \approx 98.7 \, \text{C}

Therefore, the final temperature is:

T = T_0 + \Delta T = 27 + 98.7 = 125.7^\circ \text{C}

Thus, the temperature to which the gold ring should be heated is , making the correct answer 125.7°C.

This calculation shows the change in diameter needed to fit the bangle and verifies the application of the linear expansion formula.