An engine pumps a liquid of density 'd' continuously through a pipe of area of cross-section A. If the speed with which the liquid passes through a pipe is v, then force exerted on liquid.

Options

A. `Adv^(3)//2`
B. `( 1//2)adv`
C. `Adv^(2)//2`
D. `Adv^(2)`

Answer

To solve the problem of finding the force exerted on a liquid being pumped through a pipe, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Density of the liquid = \( d \)
- Area of cross-section of the pipe = \( A \)
- Speed of the liquid = \( v \)

2. **Calculate the Volume Flow Rate**:
- The volume of liquid flowing through the pipe per second can be calculated using the formula:
\[ \text{Volume flow rate} = \text{Area} \times \text{Velocity} = A \times v \]

3. **Calculate the Mass Flow Rate**:
- The mass flow rate (\( m \)) can be calculated by multiplying the volume flow rate by the density of the liquid:
\[ m = \text{Volume flow rate} \times \text{Density} = (A \times v) \times d = A \times v \times d \]

4. **Determine the Kinetic Energy per Second**:
- The kinetic energy (\( KE \)) of the liquid flowing through the pipe can be expressed as:
\[ KE = \frac{1}{2} m v^2 \]
- Substituting the mass flow rate into the kinetic energy equation gives:
\[ KE = \frac{1}{2} (A \times v \times d) v^2 \]

5. **Simplify the Expression**:
- This simplifies to:
\[ KE = \frac{1}{2} A d v^3 \]

6. **Conclusion**:
- The force exerted on the liquid can be related to the change in momentum or the kinetic energy. In this case, the force exerted on the liquid is given by:
\[ F = \frac{1}{2} A d v^3 \]

### Final Answer:
The force exerted on the liquid is:
\[ F = \frac{1}{2} A d v^3 \]