A wooden cube just floats inside water when a 200 gm mass is placed on it. When the mass is removed, the cube is 2 cm above the water level. The size of the cube is

Options

A. 5 cm
B. 10 cm
C. 15 cm
D. 20 cm

Explanation

To solve the problem, we need to analyze the conditions under which the wooden cube floats and the implications of the mass placed on it.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A wooden cube floats in water when a 200 gm mass is placed on it.
- When the mass is removed, the cube is 2 cm above the water level.

2. **Establishing Variables**:
- Let the side length of the cube be \( x \) cm.
- The volume of the cube, \( V = x^3 \) cm³.
- The density of water, \( \rho_w = 1 \) gm/cm³ (for simplicity).

3. **Weight of the Cube**:
- The weight of the cube, \( W_{cube} = \text{mass} \times g = \rho_{cube} \cdot V \cdot g \).
- Since the cube is wooden, we can assume its density is less than that of water, allowing it to float.

4. **Condition When Mass is Added**:
- When the 200 gm mass is placed on the cube, the total weight is \( W_{total} = W_{cube} + 200 \) gm.
- The buoyant force equals the weight of the displaced water, which is equal to the volume submerged times the density of water times \( g \).
- Let \( V_{sub} \) be the volume submerged. Then, \( V_{sub} = A \cdot h \), where \( A \) is the area of the base of the cube and \( h \) is the height submerged.

5. **Buoyant Force Equation**:
- The buoyant force when the mass is added can be expressed as:
\[ W_{total} = V_{sub} \cdot \rho_w \cdot g \]
- Thus, we have:
\[ W_{cube} + 200 = V_{sub} \cdot \rho_w \]

6. **Condition When Mass is Removed**:
- When the mass is removed, the cube rises 2 cm above the water level.
- This means the new submerged height is \( h - 2 \) cm.
- The new buoyant force can be expressed as:
\[ W_{cube} = V_{sub}' \cdot \rho_w \]
- Where \( V_{sub}' = A \cdot (h - 2) \).

7. **Setting Up the Equations**:
- From the first condition:
\[ W_{cube} + 200 = A \cdot h \cdot \rho_w \]
- From the second condition:
\[ W_{cube} = A \cdot (h - 2) \cdot \rho_w \]

8. **Solving the Equations**:
- Substituting \( W_{cube} \) from the second equation into the first:
\[ A \cdot (h - 2) \cdot \rho_w + 200 = A \cdot h \cdot \rho_w \]
- Simplifying gives:
\[ 200 = A \cdot 2 \cdot \rho_w \]
- Since \( \rho_w = 1 \) gm/cm³, we have:
\[ 200 = 2A \implies A = 100 \text{ cm}^2 \]

9. **Finding the Side Length of the Cube**:
- Since the area \( A = x^2 \):
\[ x^2 = 100 \implies x = 10 \text{ cm} \]

### Final Answer:
The size of the cube is **10 cm**.