A brass sphere weighs 100 gm. Wt in air. It is suspended by a thread in a liquid of specific gravity `=0.8`. If the specific energy gravity of brass is 8, the tension in the thread in newtons is

Options

A. 0.0882
B. 8.82
C. 0.882
D. 0.00882

Explanation

To solve the problem, we need to determine the tension in the thread when a brass sphere is suspended in a liquid with a specific gravity of 0.8. Here's a step-by-step solution:

### Step 1: Understand the Forces Acting on the Sphere
When the brass sphere is suspended in the liquid, there are three main forces acting on it:
1. The weight of the sphere (W) acting downwards.
2. The buoyant force (F_b) acting upwards.
3. The tension (T) in the thread acting upwards.

### Step 2: Calculate the Weight of the Sphere
The weight of the sphere (W) can be calculated using the formula:
\[ W = m \cdot g \]
Where:
- \( m = 100 \, \text{g} = 0.1 \, \text{kg} \) (convert grams to kilograms)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)

So,
\[ W = 0.1 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 0.98 \, \text{N} \]

### Step 3: Calculate the Volume of the Sphere
The specific gravity of brass is given as 8. This means:
\[ \text{Specific Gravity} = \frac{\text{Density of brass}}{\text{Density of water}} \]
Assuming the density of water is \( 1000 \, \text{kg/m}^3 \):
\[ \text{Density of brass} = 8 \cdot 1000 \, \text{kg/m}^3 = 8000 \, \text{kg/m}^3 \]

The volume of the sphere (V) can be calculated using the mass and density:
\[ V = \frac{m}{\text{Density of brass}} = \frac{0.1 \, \text{kg}}{8000 \, \text{kg/m}^3} = 1.25 \times 10^{-5} \, \text{m}^3 \]

### Step 4: Calculate the Buoyant Force
The buoyant force (F_b) can be calculated using Archimedes' principle:
\[ F_b = V \cdot \text{Density of liquid} \cdot g \]
The density of the liquid can be calculated from its specific gravity:
\[ \text{Density of liquid} = 0.8 \cdot 1000 \, \text{kg/m}^3 = 800 \, \text{kg/m}^3 \]

Now, substituting the values:
\[ F_b = 1.25 \times 10^{-5} \, \text{m}^3 \cdot 800 \, \text{kg/m}^3 \cdot 9.8 \, \text{m/s}^2 \]
\[ F_b = 1.25 \times 10^{-5} \cdot 800 \cdot 9.8 = 0.098 \, \text{N} \]

### Step 5: Apply the Equation of Motion
Using the equilibrium of forces:
\[ W = T + F_b \]
Rearranging gives us:
\[ T = W - F_b \]
Substituting the values:
\[ T = 0.98 \, \text{N} - 0.098 \, \text{N} \]
\[ T = 0.882 \, \text{N} \]

### Final Answer
The tension in the thread is:
\[ T = 0.882 \, \text{N} \]