A wooden cube of side 10 cm and density `0.8" gm/cm"^(3)` is floating in water (density `=1" gm/cm"^(3)`). A mass of `(100x)` gm is placed over the cube so that cube is completely immersed without wetting the mass. Find value of x.

Answer

To solve the problem, we need to determine the value of \( x \) when a wooden cube is completely immersed in water after placing a mass of \( 100x \) grams on it.

### Step-by-Step Solution:

1. **Calculate the volume of the wooden cube:**
The side of the cube is given as \( 10 \, \text{cm} \).
\[ V = \text{side}^3 = 10^3 = 1000 \, \text{cm}^3 \]

2. **Calculate the mass of the wooden cube:**
The density of the wooden cube is given as \( 0.8 \, \text{g/cm}^3 \).
\[ M_w = \text{density} \times \text{volume} = 0.8 \, \text{g/cm}^3 \times 1000 \, \text{cm}^3 = 800 \, \text{g} \]

3. **Determine the buoyant force acting on the cube:**
The buoyant force \( F_b \) is equal to the weight of the water displaced by the cube. The density of water is \( 1 \, \text{g/cm}^3 \).
\[ F_b = \text{density of water} \times \text{volume of the cube} \times g = 1 \, \text{g/cm}^3 \times 1000 \, \text{cm}^3 \times g = 1000g \, \text{g} \cdot \text{cm/s}^2 \]

4. **Set up the equilibrium condition:**
When the cube is completely immersed, the total downward force (weight of the cube plus the weight of the mass placed on it) equals the buoyant force.
\[ M_w \cdot g + (100x) \cdot g = F_b \]
Substituting the values we have:
\[ 800g + 100xg = 1000g \]

5. **Simplify the equation:**
We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):
\[ 800 + 100x = 1000 \]

6. **Solve for \( x \):**
Rearranging the equation gives:
\[ 100x = 1000 - 800 \]
\[ 100x = 200 \]
\[ x = \frac{200}{100} = 2 \]

### Final Answer:
The value of \( x \) is \( 2 \).

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