A wooden cube first floats inside water when a 200 g mass is placed on it. When the mass is removed the cube is 2 cm above water level. The side of cube is

Options

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

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We have a wooden cube that floats in water. When a 200 g mass is placed on it, it floats at a certain level. When the mass is removed, the cube rises 2 cm above the water level. We need to find the side length of the cube.

### Step 2: Define Variables
Let:
- \( m \) = mass of the cube (in grams)
- \( V \) = volume of the cube (in cm³)
- \( \rho_w \) = density of water = 1 g/cm³
- \( g \) = acceleration due to gravity (we can ignore it since it will cancel out)

### Step 3: Analyze the Floating Condition with Mass
When the 200 g mass is placed on the cube, the total downward force is:
\[ F_{\text{down}} = mg + 200 \, \text{g} \]

The buoyant force \( F_b \) acting on the cube when it is submerged is given by:
\[ F_b = V \cdot \rho_w \cdot g \]
Since \( \rho_w = 1 \, \text{g/cm}^3 \), we can simplify:
\[ F_b = V \cdot g \]

Setting the forces equal for the floating condition:
\[ mg + 200 = V \cdot g \]
Cancelling \( g \) from both sides:
\[ m + 200 = V \]

### Step 4: Analyze the Floating Condition without Mass
When the mass is removed, the cube rises 2 cm above the water level. This means the height of the cube submerged in water is \( x - 2 \) cm, where \( x \) is the side length of the cube.

The buoyant force now is:
\[ F_b = V_{\text{submerged}} \cdot \rho_w \cdot g = (A \cdot (x - 2)) \cdot g \]
Where \( A \) is the area of the base of the cube:
\[ V_{\text{submerged}} = A \cdot (x - 2) \]

Setting the forces equal again:
\[ mg = A \cdot (x - 2) \cdot g \]
Cancelling \( g \):
\[ m = A \cdot (x - 2) \]

### Step 5: Relate the Two Conditions
From the first condition:
\[ m = V - 200 \]
From the second condition:
\[ m = A \cdot (x - 2) \]

Since the volume \( V \) of the cube is \( A \cdot x \):
\[ A \cdot x - 200 = A \cdot (x - 2) \]

### Step 6: Simplify the Equation
Expanding the equation:
\[ A \cdot x - 200 = A \cdot x - 2A \]
Rearranging gives:
\[ 200 = 2A \]
Thus:
\[ A = 100 \, \text{cm}^2 \]

### Step 7: Find the Side Length
Since \( A = x^2 \):
\[ x^2 = 100 \]
Taking the square root:
\[ x = 10 \, \text{cm} \]

### Conclusion
The side length of the cube is \( 10 \, \text{cm} \).