A body of volume V and density d is completely immersed in a liquid of density p. Then the apparent weight of the body will be:

Options

A. Vdg
B. `V rho g`
C. `V(rho -d)g`
D. `V(d-rho)g`

Explanation

To find the apparent weight of a body that is completely immersed in a liquid, we can follow these steps:

### Step 1: Understand the Forces Acting on the Body
When a body is immersed in a fluid, two main forces act on it:
1. The weight of the body (downward force).
2. The buoyant force exerted by the fluid (upward force).

### Step 2: Calculate the Weight of the Body
The weight of the body can be calculated using the formula:
\[ \text{Weight of the body} = \text{Density of the body} \times \text{Volume of the body} \times g \]
Where:
- Density of the body = \(d\)
- Volume of the body = \(V\)
- \(g\) = acceleration due to gravity

Thus, the weight of the body is:
\[ W = d \times V \times g \]

### Step 3: Calculate the Buoyant Force
The buoyant force can be calculated using Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the body. The formula for buoyant force is:
\[ \text{Buoyant Force} = \text{Density of the liquid} \times \text{Volume of the displaced liquid} \times g \]
Since the body is completely immersed, the volume of the displaced liquid is equal to the volume of the body \(V\). Therefore, the buoyant force is:
\[ F_B = p \times V \times g \]
Where:
- Density of the liquid = \(p\)

### Step 4: Calculate the Apparent Weight of the Body
The apparent weight of the body is the actual weight of the body minus the buoyant force acting on it. Thus, the formula for apparent weight is:
\[ \text{Apparent Weight} = \text{Weight of the body} - \text{Buoyant Force} \]
Substituting the values we calculated:
\[ \text{Apparent Weight} = (d \times V \times g) - (p \times V \times g) \]

### Step 5: Simplify the Expression
We can factor out \(V \times g\) from the equation:
\[ \text{Apparent Weight} = V \times g \times (d - p) \]

### Final Answer
Thus, the apparent weight of the body when completely immersed in the liquid is:
\[ \text{Apparent Weight} = V \times (d - p) \times g \]

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