Understanding Cylinder Volume Increase Due to Thermal Expansion

When a cylinder is heated, its dimensions (radius and height) increase due to thermal expansion. This increase in dimensions leads to an increase in the overall volume of the cylinder. We can calculate this volume increase using the concept of volume expansion, which is related to the linear expansion coefficient.

Given Parameters for the Cylinder Volume Increase

We are given the following information:

• Initial radius of the cylinder, $r_0 = 1 \text{ cm}$
• Initial height of the cylinder, $h_0 = 4 \text{ cm}$
• Initial temperature, $T_i = 0^\circ\text{C}$
• Final temperature, $T_f = 100^\circ\text{C}$
• Coefficient of linear expansion, $\alpha = 4 \times 10^{-4} /^\circ\text{C}$

Calculating Initial Volume and Temperature Change

First, let's calculate the initial volume of the cylinder and the change in temperature.

• The initial volume of a cylinder is given by the formula $V_0 = \pi r_0^2 h_0$.
• Substituting the given values: $V_0 = \pi (1 \text{ cm})^2 (4 \text{ cm}) = \pi (1 \text{ cm}^2)(4 \text{ cm}) = 4\pi \text{ cm}^3$.
• The change in temperature is $\Delta T = T_f - T_i$.
• Substituting the given temperatures: $\Delta T = 100^\circ\text{C} - 0^\circ\text{C} = 100^\circ\text{C}$.

Relating Linear and Volume Expansion Coefficients

For most isotropic materials (materials that expand equally in all directions), the coefficient of volume expansion, $\gamma$, is approximately three times the coefficient of linear expansion, $\alpha$.

So, $\gamma = 3\alpha$.

Using the given value of $\alpha$: $\gamma = 3 \times (4 \times 10^{-4} /^\circ\text{C}) = 12 \times 10^{-4} /^\circ\text{C}$.

Calculating the Increase in Volume of the Cylinder

The increase in volume ($\Delta V$) due to thermal expansion is given by the formula:

$\Delta V = V_0 \gamma \Delta T$

Now, we substitute the values we calculated:

$\Delta V = (4\pi \text{ cm}^3) \times (12 \times 10^{-4} /^\circ\text{C}) \times (100^\circ\text{C})$

Let's perform the multiplication:

$\Delta V = 4\pi \times 12 \times 10^{-4} \times 100 \text{ cm}^3$

$\Delta V = (4 \times 12) \pi \times (10^{-4} \times 10^2) \text{ cm}^3$

$\Delta V = 48 \pi \times 10^{-2} \text{ cm}^3$

$\Delta V = 0.48\pi \text{ cm}^3$

This calculation shows the increase in the volume of the cylinder when heated from $0^\circ\text{C}$ to $100^\circ\text{C}$ with the given linear expansion coefficient. This result matches one of the provided options for the cylinder volume increase.