A balloon inflated with helium gas (density = `0.2 kg//m^(3)`) has a volume of `6 xx 10^(-3) m^(3)` . If the density of air is `1.3 kg//m^(3)`. What is the buoyant force exerted on the balloon ?

Options

A. `0.01 `N
B. `0.8 ` N
C. `0.08 N`
D. `1.3` N

Answer

To solve the problem of finding the buoyant force exerted on a helium-filled balloon, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Given Values**:
- Volume of the balloon, \( V_b = 6 \times 10^{-3} \, \text{m}^3 \)
- Density of air, \( \rho_{\text{air}} = 1.3 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)

2. **Understand the Concept of Buoyant Force**:
- The buoyant force (\( F_b \)) acting on an object submerged in a fluid is given by Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. The formula is:
\[ F_b = V \cdot \rho \cdot g \]
- Here, \( V \) is the volume of the fluid displaced (which is equal to the volume of the balloon), \( \rho \) is the density of the fluid (air in this case), and \( g \) is the acceleration due to gravity.

3. **Substitute the Values into the Formula**:
\[ F_b = V_b \cdot \rho_{\text{air}} \cdot g \]
\[ F_b = (6 \times 10^{-3} \, \text{m}^3) \cdot (1.3 \, \text{kg/m}^3) \cdot (9.8 \, \text{m/s}^2) \]

4. **Calculate the Buoyant Force**:
- First, calculate the product of the density of air and the acceleration due to gravity:
\[ \rho_{\text{air}} \cdot g = 1.3 \, \text{kg/m}^3 \cdot 9.8 \, \text{m/s}^2 = 12.74 \, \text{kg/(m}^2\text{s}^2\text{)} = 12.74 \, \text{N/m}^3 \]
- Now, multiply this result by the volume of the balloon:
\[ F_b = 6 \times 10^{-3} \, \text{m}^3 \cdot 12.74 \, \text{N/m}^3 = 0.07644 \, \text{N} \]

5. **Final Result**:
- The buoyant force exerted on the balloon is approximately:
\[ F_b \approx 0.07644 \, \text{N} \]

### Conclusion
The buoyant force exerted on the balloon is approximately **0.07644 N**, which can be rounded to **0.08 N** for practical purposes.