A glass sinker has a mass M in air, When weighed in a liquid at temperature `t_(1)`, the apparent mass is `M_(1)` and when weighed in the same liquid at temperature `t_(2)`, the apparent mass is `M_(2)`. If the coefficient of cubical expansion of the glass is `gamma_(g)`, then the real coefficient of expansion of the liquid is :

Options

A. `gamma_(g)+((M_(2)-M_(1))/(M-M_(2)))*(1)/((t_(2)-t_(1)))`
B. `gamma_(g)-((M_(2)-M_(1))/(M-M_(2)))*(1)/((t_(2)-t_(1)))`
C. `gamma_(g) - ((M-M_(2))/(M_(2)-M_(1)))*(1)/((t_(2)-t_(1)))`
D. `gamma_(g)+((M_(2)-M_(1))/(M_(2)+M_(1)))*(1)/((t_(2)-t_(1)))`

Explanation

To find the real coefficient of expansion of the liquid (denoted as \( \gamma_L \)), we can follow these steps:

### Step-by-Step Solution

1. **Understanding Apparent Mass in Liquid**:
- When the glass sinker is weighed in air, its mass is \( M \).
- When weighed in a liquid at temperature \( t_1 \), the apparent mass is \( M_1 \). The buoyant force acting on the sinker is given by Archimedes' principle.

2. **Buoyant Force Calculation**:
- The buoyant force \( F_b \) can be expressed as:
\[ F_b = \rho_L V g \]
where \( \rho_L \) is the density of the liquid, \( V \) is the volume of the sinker, and \( g \) is the acceleration due to gravity.

3. **Equating Apparent Mass**:
- The apparent weight when submerged is given by:
\[ M_1 g = M g - F_b \]
- Substituting the expression for buoyant force:
\[ M_1 g = M g - \rho_L V g \]
- Rearranging gives:
\[ M_1 = M - \rho_L V \]

4. **Volume Expansion of Glass**:
- The volume of the glass sinker at temperature \( t_1 \) is:
\[ V = V_0 (1 + \gamma_g (t_1 - t_0)) \]
- At temperature \( t_2 \), the volume becomes:
\[ V' = V_0 (1 + \gamma_g (t_2 - t_0)) \]

5. **Apparent Mass at Temperature \( t_2 \)**:
- When weighed at temperature \( t_2 \), the apparent mass is \( M_2 \):
\[ M_2 g = M g - \rho_L V' g \]
- Rearranging gives:
\[ M_2 = M - \rho_L V' \]

6. **Setting Up the Equations**:
- We have two equations:
\[ M_1 = M - \rho_L V_0 (1 + \gamma_g (t_1 - t_0)) \]
\[ M_2 = M - \rho_L V_0 (1 + \gamma_g (t_2 - t_0)) \]

7. **Finding the Density Change**:
- The change in density of the liquid with temperature can be expressed as:
\[ \rho_L' = \rho_L (1 - \gamma_L (t - t_0)) \]

8. **Relating \( M_1 \) and \( M_2 \)**:
- Substituting the expressions for \( M_1 \) and \( M_2 \) into the equations gives:
\[ M - M_1 = \rho_L V_0 (1 + \gamma_g (t_1 - t_0)) \]
\[ M - M_2 = \rho_L V_0 (1 + \gamma_g (t_2 - t_0)) \]

9. **Solving for \( \gamma_L \)**:
- By dividing the two equations, we can eliminate \( \rho_L V_0 \) and isolate \( \gamma_L \):
\[ \frac{M - M_2}{M - M_1} = \frac{1 + \gamma_g (t_2 - t_0)}{1 + \gamma_L (t_2 - t_0)} \]
- Rearranging gives us the expression for \( \gamma_L \):
\[ \gamma_L = \frac{(M - M_2)(1 + \gamma_g (t_1 - t_0)) - (M - M_1)(1 + \gamma_g (t_2 - t_0))}{(M - M_1)(t_2 - t_1)} \]

### Final Expression for \( \gamma_L \):
The real coefficient of expansion of the liquid is given by:
\[ \gamma_L = \frac{(M - M_2)(1 + \gamma_g (t_1 - t_0)) - (M - M_1)(1 + \gamma_g (t_2 - t_0))}{(M - M_1)(t_2 - t_1)} \]