A wooden cube floats just inside the water, when a mass of x(in grams) is placed on it. If the mass is removed, the cube floats with a height `(x)/(100)` (cm) above the water surface. The length of the side of cube is (density of water is 1000 `kg//m^(3)`)

Options

A. a. 10 cm
B. b. 15 cm
C. c. 20 cm
D. d. 30 cm

Answer

To solve the problem, we need to determine the length of the side of the wooden cube based on the information provided about its buoyancy and the mass placed on it. Let's break it down step by step.

### Step 1: Understand the Problem
The wooden cube floats in water when a mass \( x \) (in grams) is placed on it. When the mass is removed, the cube floats with a height of \( \frac{x}{100} \) cm above the water surface. We need to find the length of the side of the cube.

### Step 2: Convert Units
First, we convert the height above the water surface from centimeters to meters:
\[ h = \frac{x}{100} \text{ cm} = \frac{x}{100} \times \frac{1}{100} \text{ m} = \frac{x}{10^4} \text{ m} \]
Also, convert the mass \( x \) from grams to kilograms:
\[ m = \frac{x}{1000} \text{ kg} \]

### Step 3: Apply Archimedes' Principle
According to Archimedes' principle, the buoyant force acting on the cube when the mass is removed is equal to the weight of the water displaced by the submerged part of the cube. The buoyant force can be expressed as:
\[ F_b = \text{Density of water} \times \text{Volume of water displaced} \times g \]
Where:
- Density of water \( \rho = 1000 \text{ kg/m}^3 \)
- Volume of water displaced \( V = L^2 \times h \) (where \( L \) is the side length of the cube)
- \( g \) is the acceleration due to gravity (approximately \( 10 \text{ m/s}^2 \))

### Step 4: Set Up the Equation
The weight of the mass \( m \) is given by:
\[ W = m \cdot g = \frac{x}{1000} \cdot 10 = \frac{x}{100} \text{ N} \]
The buoyant force when the mass is removed is:
\[ F_b = \rho \cdot V \cdot g = 1000 \cdot (L^2 \cdot \frac{x}{10^4}) \cdot 10 \]
Setting the weight equal to the buoyant force gives:
\[ \frac{x}{100} = 1000 \cdot (L^2 \cdot \frac{x}{10^4}) \cdot 10 \]

### Step 5: Simplify the Equation
Cancelling \( x \) from both sides (assuming \( x \neq 0 \)):
\[ \frac{1}{100} = 1000 \cdot L^2 \cdot \frac{10}{10^4} \]
This simplifies to:
\[ \frac{1}{100} = 1000 \cdot L^2 \cdot \frac{1}{1000} \]
\[ \frac{1}{100} = L^2 \]

### Step 6: Solve for \( L \)
Taking the square root of both sides:
\[ L = \frac{1}{10} \text{ m} = 0.1 \text{ m} \]
Converting to centimeters:
\[ L = 0.1 \times 100 = 10 \text{ cm} \]

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

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