A body of volume V and density `rho` is kept completely immersed in a liquid of density `rho_(L)`. If g is the acceleration due to gravity, write expressions for the following
(i) the weight of the body.
(ii) the upthrust on the body.
(iii) the apparent weight of the body in liquid,
(iv) the loss in weight of the body.

Answer

To solve the problem, we will derive the required expressions step by step.

### Step 1: Weight of the Body
The weight of the body (W_B) can be calculated using the formula:
\[ W_B = \text{mass} \times g \]
The mass of the body can be expressed in terms of its density (\(\rho\)) and volume (V):
\[ \text{mass} = \rho \times V \]
Thus, the weight of the body is:
\[ W_B = \rho \times V \times g \]

### Step 2: Upthrust on the Body
The upthrust (or buoyant force, F_B) acting on the body when it is immersed in a fluid is equal to the weight of the fluid displaced by the body. The weight of the displaced fluid can be calculated as:
\[ F_B = \text{density of fluid} \times \text{volume displaced} \times g \]
Since the body is completely immersed, the volume displaced is equal to the volume of the body (V):
\[ F_B = \rho_L \times V \times g \]
where \(\rho_L\) is the density of the liquid.

### Step 3: Apparent Weight of the Body in Liquid
The apparent weight (W_A) of the body when it is submerged in the liquid is the actual weight of the body minus the upthrust:
\[ W_A = W_B - F_B \]
Substituting the expressions from Steps 1 and 2:
\[ W_A = (\rho \times V \times g) - (\rho_L \times V \times g) \]
Factoring out common terms:
\[ W_A = (\rho - \rho_L) \times V \times g \]

### Step 4: Loss in Weight of the Body
The loss in weight (L) of the body when it is immersed in the fluid is equal to the upthrust (buoyant force):
\[ L = F_B \]
Thus, substituting the expression for upthrust from Step 2:
\[ L = \rho_L \times V \times g \]

### Summary of Expressions
1. Weight of the body:
\[ W_B = \rho \times V \times g \]
2. Upthrust on the body:
\[ F_B = \rho_L \times V \times g \]
3. Apparent weight of the body in liquid:
\[ W_A = (\rho - \rho_L) \times V \times g \]
4. Loss in weight of the body:
\[ L = \rho_L \times V \times g \]