You are provided with a hollow iron ball A of volume `5cm^(3)` and mass 12 g and a solid iron ball B of mass 12 g. Both are plced on the surface of water contained in a largetub. A. Find upthrust on each ball. B. Which ball will sink? Give reason for your answer. (Density of iron `=8.0gm^(-3)`)

Answer

### Step-by-Step Solution

**Given:**
- Volume of hollow iron ball A, \( V_A = 5 \, \text{cm}^3 \)
- Mass of hollow iron ball A, \( m_A = 12 \, \text{g} \)
- Mass of solid iron ball B, \( m_B = 12 \, \text{g} \)
- Density of iron, \( \rho_{\text{iron}} = 8 \, \text{g/cm}^3 \)
- Density of water, \( \rho_{\text{water}} = 1 \, \text{g/cm}^3 \) (approximately)

**Part A: Finding Upthrust on Each Ball**

1. **Calculate Upthrust on Ball A:**
- The upthrust (or buoyant force) on an object submerged in a fluid is given by Archimedes' principle:
\[ F_{upthrust} = V \times \rho_{\text{fluid}} \times g \]
- For ball A:
\[ F_{upthrust, A} = V_A \times \rho_{\text{water}} \times g = 5 \, \text{cm}^3 \times 1 \, \text{g/cm}^3 \times g = 5g \, \text{gf} \]
- Thus, the upthrust on ball A is \( 5g \, \text{gf} \).

2. **Calculate Volume of Ball B:**
- The volume of solid ball B can be calculated using its mass and density:
\[ V_B = \frac{m_B}{\rho_{\text{iron}}} = \frac{12 \, \text{g}}{8 \, \text{g/cm}^3} = 1.5 \, \text{cm}^3 \]

3. **Calculate Upthrust on Ball B:**
- For ball B:
\[ F_{upthrust, B} = V_B \times \rho_{\text{water}} \times g = 1.5 \, \text{cm}^3 \times 1 \, \text{g/cm}^3 \times g = 1.5g \, \text{gf} \]
- Thus, the upthrust on ball B is \( 1.5g \, \text{gf} \).

**Part B: Which Ball Will Sink?**

1. **Determine the Weight of Each Ball:**
- The weight of both balls can be calculated as:
\[ W_A = m_A \times g = 12 \, \text{g} \times g = 12g \, \text{gf} \]
\[ W_B = m_B \times g = 12 \, \text{g} \times g = 12g \, \text{gf} \]

2. **Compare Upthrust with Weight:**
- For ball A:
- Upthrust: \( 5g \, \text{gf} \)
- Weight: \( 12g \, \text{gf} \)
- Since \( 5g < 12g \), ball A will sink.
- For ball B:
- Upthrust: \( 1.5g \, \text{gf} \)
- Weight: \( 12g \, \text{gf} \)
- Since \( 1.5g < 12g \), ball B will also sink.

**Conclusion:**
Both balls A and B will sink because the upthrust (buoyant force) on each ball is less than their respective weights.