A block of volume `V` and density `sigma_(b)` is placed in liquid of density `sigma_(1)(sigma_(1) gt sigma_(b))`, then block is moved upward upto a height `h` and it is still in liquid. The increase in gravitational energy of the block is :

Options

A. `sigma_(b) Vgh`
B. `(sigma_(b) + sigma_(l)) Vgh`
C. `(sigma_(b) - sigma_(l)) Vgh`
D. None of these

Answer

To find the increase in gravitational energy of a block of volume \( V \) and density \( \sigma_b \) when it is moved upward to a height \( h \) in a liquid of density \( \sigma_1 \) (where \( \sigma_1 > \sigma_b \)), we can follow these steps:

### Step 1: Understand the Problem
The block is submerged in a liquid and is moved upward. We need to calculate the increase in gravitational potential energy of the block as it is raised to a height \( h \).

### Step 2: Calculate the Mass of the Block
The mass \( m \) of the block can be calculated using the formula:
\[ m = \text{density} \times \text{volume} = \sigma_b \times V \]

### Step 3: Write the Formula for Gravitational Potential Energy
The gravitational potential energy \( U \) of an object at height \( h \) is given by:
\[ U = m \cdot g \cdot h \]
where \( g \) is the acceleration due to gravity.

### Step 4: Substitute the Mass into the Potential Energy Formula
Substituting the expression for mass from Step 2 into the potential energy formula gives:
\[ U = (\sigma_b \cdot V) \cdot g \cdot h \]

### Step 5: Simplify the Expression
Thus, the increase in gravitational potential energy when the block is moved upward by height \( h \) is:
\[ \Delta U = \sigma_b \cdot V \cdot g \cdot h \]

### Conclusion
The increase in gravitational energy of the block is:
\[ \Delta U = \sigma_b \cdot V \cdot g \cdot h \]