A wooden cube of side 10 cm has mass 700g What part of it remains above the water surface while floating vertically on the water surface?

Answer

To solve the problem of determining what part of a wooden cube remains above the water surface while floating vertically, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the dimensions and mass of the cube**:
- The side length of the cube (s) = 10 cm
- The mass of the cube (m) = 700 g

2. **Calculate the volume of the cube**:
- The volume (V) of the cube can be calculated using the formula for the volume of a cube:
\[ V = s^3 = (10 \, \text{cm})^3 = 1000 \, \text{cm}^3 \]

3. **Determine the weight of the cube**:
- The weight (W) of the cube can be calculated using the formula:
\[ W = m \cdot g \]
- Since we are working in grams and centimeters, we can consider \( g \) to be 1 for simplicity in this context:
\[ W = 700 \, \text{g} \cdot 1 = 700 \, \text{g} \]

4. **Establish the relationship between buoyant force and weight**:
- According to Archimedes' principle, the buoyant force (F_b) acting on the cube must equal the weight of the cube for it to float:
\[ F_b = W \]

5. **Calculate the volume of water displaced**:
- The volume of water displaced (V_2) is equal to the volume of the cube submerged in water. If \( h \) is the height of the cube above the water, then the submerged height is \( 10 \, \text{cm} - h \):
\[ V_2 = s^2 \cdot (10 - h) = 10^2 \cdot (10 - h) = 100 \cdot (10 - h) \, \text{cm}^3 \]

6. **Set up the equation for buoyant force**:
- The buoyant force can also be expressed as:
\[ F_b = \rho_w \cdot g \cdot V_2 \]
- Given that the density of water (\( \rho_w \)) is 1 g/cm³, we have:
\[ F_b = 1 \cdot 1 \cdot V_2 = V_2 \]

7. **Equate buoyant force to weight**:
- Setting the buoyant force equal to the weight of the cube:
\[ 100 \cdot (10 - h) = 700 \]

8. **Solve for h**:
- Rearranging the equation:
\[ 10 - h = \frac{700}{100} = 7 \]
- Therefore:
\[ h = 10 - 7 = 3 \, \text{cm} \]

9. **Conclusion**:
- The part of the cube that remains above the water surface is \( h = 3 \, \text{cm} \).

### Final Answer:
3 cm of the wooden cube remains above the water surface while floating vertically.