A rubber ball floats on water with its 1/3 rd volume outside water. What is the density of rubber?

Answer

To find the density of rubber when a rubber ball floats on water with one-third of its volume above the surface, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Volume of the Ball**:
Let the total volume of the rubber ball be \( V \).

2. **Determine the Volume Above Water**:
Since one-third of the ball's volume is above the water, we can express this as:
\[ \text{Volume above water} = \frac{1}{3} V \]

3. **Calculate the Volume Immersed in Water**:
The volume of the ball that is submerged in water can be calculated by subtracting the volume above water from the total volume:
\[ \text{Volume immersed} = V - \frac{1}{3} V = \frac{2}{3} V \]

4. **Apply Archimedes' Principle**:
According to Archimedes' Principle, the weight of the water displaced by the submerged part of the ball is equal to the weight of the ball itself. This can be expressed as:
\[ \text{Weight of water displaced} = \text{Weight of the ball} \]

5. **Express the Weights**:
The weight of the water displaced can be calculated using the density of water (\( \rho_{water} \)) and the volume immersed:
\[ \text{Weight of water displaced} = \rho_{water} \times \text{Volume immersed} = \rho_{water} \times \frac{2}{3} V \]
The weight of the ball can be expressed using the density of rubber (\( \rho_{rubber} \)):
\[ \text{Weight of the ball} = \rho_{rubber} \times V \]

6. **Set Up the Equation**:
Setting the two expressions for weight equal gives us:
\[ \rho_{water} \times \frac{2}{3} V = \rho_{rubber} \times V \]

7. **Simplify the Equation**:
We can cancel \( V \) from both sides (assuming \( V \neq 0 \)):
\[ \rho_{water} \times \frac{2}{3} = \rho_{rubber} \]

8. **Substitute the Density of Water**:
The density of water is approximately \( 1000 \, \text{kg/m}^3 \). Substituting this into the equation gives:
\[ \rho_{rubber} = 1000 \times \frac{2}{3} \]

9. **Calculate the Density of Rubber**:
Performing the multiplication:
\[ \rho_{rubber} = \frac{2000}{3} \approx 666.67 \, \text{kg/m}^3 \]
We can round this to:
\[ \rho_{rubber} \approx 667 \, \text{kg/m}^3 \]

### Final Answer:
The density of rubber is approximately \( 667 \, \text{kg/m}^3 \).