A cube of iron whose sides are of length L, is put into mercury. The weight of iron cube is W. The density of irons is `rho_(1)`, that of mercury is `rho_(M)`. The depth to which the cube sinks is given by the expression

Options

A. `WL^(2)rho_(I)`
B. `WL^(2)rho_(M)`
C. `W/(L^(2)rho_(I))`
D. `W/(L^(2)rho_(M)g)`

Answer

To solve the problem, we need to find the depth to which the iron cube sinks into the mercury. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have a cube of iron with side length \( L \) and weight \( W \). The cube is submerged in mercury, which has a density \( \rho_M \). We need to find the depth \( x \) to which the cube sinks in the mercury.

### Step 2: Identify the Forces Acting on the Cube
When the cube is submerged in mercury, two main forces act on it:
1. The weight of the cube \( W \) acting downwards.
2. The buoyant force \( F_B \) acting upwards.

### Step 3: Write the Expression for Buoyant Force
The buoyant force can be calculated using Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the submerged part of the object. The formula for buoyant force is:
\[ F_B = \rho_{M} \cdot V_{displaced} \cdot g \]
where:
- \( \rho_M \) is the density of mercury,
- \( V_{displaced} \) is the volume of mercury displaced,
- \( g \) is the acceleration due to gravity.

### Step 4: Calculate the Volume of the Displaced Mercury
The volume of the displaced mercury when the cube sinks to a depth \( x \) is given by:
\[ V_{displaced} = L^2 \cdot x \]
where \( L^2 \) is the area of the base of the cube.

### Step 5: Substitute the Volume into the Buoyant Force Equation
Substituting the volume of the displaced mercury into the buoyant force equation, we get:
\[ F_B = \rho_{M} \cdot (L^2 \cdot x) \cdot g \]

### Step 6: Set Up the Equilibrium Condition
For the cube to be in equilibrium (not sinking further), the buoyant force must equal the weight of the cube:
\[ F_B = W \]
Thus, we have:
\[ \rho_{M} \cdot (L^2 \cdot x) \cdot g = W \]

### Step 7: Solve for Depth \( x \)
Rearranging the equation to solve for \( x \):
\[ x = \frac{W}{\rho_{M} \cdot L^2 \cdot g} \]

### Final Expression
The depth to which the cube sinks in mercury is given by:
\[ x = \frac{W}{\rho_{M} \cdot L^2 \cdot g} \]