Question

Part 1: A beaker of mass 1.2 kg containing 2.5 kg of water rests on a scale. A 3.9 kg block of a metallic alloy of density 4500 kg/m^3 is suspended from a spring scale and is submerged in the water of density 1000 kg/m^3 as shown in the figure. What does the hanging scale read? The acceleration of gravity is 9.8 m/s^2. Answer in units of N.

Part 2: What does the lower scale read? Answer in units of N.

Answer 1

To find the readings on the hanging scale and the lower scale given the scenario, we will follow these steps:

### Part 1: Hanging Scale Reading

1. **Calculate the Volume of the Metallic Block:**
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}} \implies \text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]
For the metallic block:
\[
\text{Volume} = \frac{3.9 \, \text{kg}}{4500 \, \text{kg/m}^3} = 0.00086667 \, \text{m}^3
\]

2. **Calculate the Buoyant Force Acting on the Block:**
The buoyant force can be calculated using Archimedes' principle:
\[
\text{Buoyant Force} = \text{Volume} \times \text{Density of Water} \times g
\]
\[
\text{Buoyant Force} = 0.00086667 \, \text{m}^3 \times 1000 \, \text{kg/m}^3 \times 9.8 \, \text{m/s}^2 = 8.5 \, \text{N} \text{ (approximately)}
\]

3. **Calculate the Weight of the Metallic Block:**
\[
\text{Weight} = \text{Mass} \times g = 3.9 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 38.22 \, \text{N}
\]

4. **Calculate the Reading on the Hanging Scale:**
The reading on the hanging scale (T) is the weight of the block minus the buoyant force:
\[
T = \text{Weight} - \text{Buoyant Force} = 38.22 \, \text{N} - 8.5 \, \text{N} = 29.72 \, \text{N}
\]

### Part 1 Answer:
The hanging scale reads approximately **29.72 N**.

---

### Part 2: Lower Scale Reading

1. **Total Weight Supported by the Lower Scale:**
The total force on the lower scale includes the weight of the beaker, the weight of the water, and the effective weight of the submerged block (which includes the buoyant force):
\[
\text{Total Weight} = \text{Weight of Beaker} + \text{Weight of Water} + (\text{Weight of Block} - \text{Buoyant Force})
\]

2. **Calculate the Weight of the Beaker and Water:**
- Mass of beaker = 1.2 kg
- Mass of water = 2.5 kg
\[
\text{Weight of Beaker} = 1.2 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 11.76 \, \text{N}
\]
\[
\text{Weight of Water} = 2.5 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 24.5 \, \text{N}
\]

3. **Calculate the Effective Weight of the Block:**
\[
\text{Effective Weight of Block} = \text{Weight of Block} - \text{Buoyant Force} = 38.22 \, \text{N} - 8.5 \, \text{N} = 29.72 \, \text{N}
\]

4. **Total Weight on the Lower Scale:**
\[
\text{Total Weight} = 11.76 \, \text{N} + 24.5 \, \text{N} + 29.72 \, \text{N} = 66.98 \, \text{N}
\]

### Part 2 Answer:
The lower scale reads approximately **66.98 N**.