A cube of wood supporting 200gm mass just floats in water. When the mass is removed, the cube ruses by 2cm. What is the size of the cube?

Explanation

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the Initial Condition
When the cube of wood is floating in water with a 200 g mass on it, it is in equilibrium. The weight of the cube plus the weight of the mass is balanced by the buoyant force acting on the cube.

### Step 2: Set Up the First Equation
Let \( L \) be the length of one side of the cube. The volume of the cube is \( L^3 \), and the weight of the cube is given by \( \text{Weight of cube} = \text{Density of wood} \times g \times L^3 \).

The buoyant force is equal to the weight of the water displaced by the submerged part of the cube. When the 200 g mass is on the cube, the equation for equilibrium can be written as:
\[ \text{Weight of cube} + 200 \, \text{g} = \text{Buoyant force} \]
This can be expressed as:
\[ D_{\text{wood}} \cdot g \cdot L^3 + 200 \cdot g = D_{\text{water}} \cdot g \cdot V_{\text{displaced}} \]
Where \( V_{\text{displaced}} = L^2 \cdot h \) (with \( h \) being the submerged height of the cube).

### Step 3: Set Up the Second Equation
When the mass is removed, the cube rises by 2 cm. The new height submerged is \( h - 2 \) cm. The new equilibrium condition can be expressed as:
\[ D_{\text{wood}} \cdot g \cdot L^3 = D_{\text{water}} \cdot g \cdot (L^2 \cdot (h - 2)) \]

### Step 4: Simplify the Equations
Assuming the density of water \( D_{\text{water}} = 1 \) g/cm³, we can simplify both equations.

For the first case (with mass):
\[ D_{\text{wood}} \cdot L^3 + 200 = L^2 \cdot h \]
For the second case (without mass):
\[ D_{\text{wood}} \cdot L^3 = L^2 \cdot (h - 2) \]

### Step 5: Solve the Equations
From the first equation:
\[ D_{\text{wood}} \cdot L^3 = L^2 \cdot h - 200 \]
Substituting this into the second equation:
\[ L^2 \cdot h - 200 = L^2 \cdot (h - 2) \]
Expanding and simplifying:
\[ L^2 \cdot h - 200 = L^2 \cdot h - 2L^2 \]
This leads to:
\[ 2L^2 = 200 \]
Thus:
\[ L^2 = 100 \quad \Rightarrow \quad L = 10 \, \text{cm} \]

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

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