A piece of metal floats on mercury. The coefficient of volume expansion of metal and mercury are `gamma_1 and gamma_2`, respectively. if the temperature of both mercury and metal are increased by an amount `Delta T`, by what factor does the frection of the volume of the metal submerged in mercury changes ?

Explanation

To solve the problem, we need to analyze how the fraction of the volume of the metal submerged in mercury changes when both the metal and mercury are heated by a temperature change \( \Delta T \).

### Step-by-Step Solution:

1. **Understand the Initial Condition**:
- Let the density of the metal be \( \rho_s \) and the density of mercury be \( \rho_l \).
- The fraction of the volume of the metal submerged in mercury can be expressed as:
\[ f_0 = \frac{\rho_s}{\rho_l} \]

2. **Apply the Volume Expansion Formula**:
- When the temperature increases by \( \Delta T \), the new densities of the metal and mercury can be expressed using their coefficients of volume expansion \( \gamma_1 \) and \( \gamma_2 \):
\[ \rho_s' = \frac{\rho_s}{1 + \gamma_1 \Delta T} \]
\[ \rho_l' = \frac{\rho_l}{1 + \gamma_2 \Delta T} \]

3. **Calculate the New Fraction**:
- The new fraction of the volume of the metal submerged in mercury after the temperature increase becomes:
\[ f = \frac{\rho_s'}{\rho_l'} = \frac{\frac{\rho_s}{1 + \gamma_1 \Delta T}}{\frac{\rho_l}{1 + \gamma_2 \Delta T}} = \frac{\rho_s}{\rho_l} \cdot \frac{1 + \gamma_2 \Delta T}{1 + \gamma_1 \Delta T} \]
- Substituting \( f_0 \) into the equation gives:
\[ f = f_0 \cdot \frac{1 + \gamma_2 \Delta T}{1 + \gamma_1 \Delta T} \]

4. **Simplify the Expression**:
- For small values of \( \gamma_1 \Delta T \) and \( \gamma_2 \Delta T \) (which is often the case in thermal expansion problems), we can use the approximation:
\[ \frac{1 + x}{1 + y} \approx 1 + (x - y) \quad \text{for small } x \text{ and } y \]
- Applying this approximation:
\[ f \approx f_0 \cdot \left(1 + (\gamma_2 - \gamma_1) \Delta T\right) \]

5. **Final Result**:
- Thus, the fraction of the volume of the metal submerged in mercury changes by a factor of:
\[ f \approx f_0 \cdot \left(1 + (\gamma_2 - \gamma_1) \Delta T\right) \]