A stream of a liquid of density p flowin horizontally with speed v rushes out of a tube of radius r and hits a verticla wall nearly normally. Assumi g that the liquid does not rebound from the wall, the force exerted o the wall by the impact of the liquid is given by

Options

A. `pirrhov`
B. `pirrhov^(2)`
C. `pir^(2)rhov`
D. `pir^(2)rhov^(2)`

Answer

To solve the problem of determining the force exerted on a vertical wall by a stream of liquid flowing horizontally, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the parameters**:
- Let the density of the liquid be \( \rho \).
- The speed of the liquid as it exits the tube is \( v \).
- The radius of the tube is \( r \).

2. **Calculate the cross-sectional area of the tube**:
- The cross-sectional area \( A \) of the tube can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]

3. **Determine the volume flow rate**:
- The volume flow rate \( Q \) (volume per unit time) of the liquid can be expressed as:
\[ Q = A \cdot v = \pi r^2 \cdot v \]

4. **Calculate the mass flow rate**:
- The mass flow rate \( \frac{dm}{dt} \) can be found using the relationship between mass, density, and volume:
\[ \frac{dm}{dt} = \rho \cdot Q = \rho \cdot (\pi r^2 v) = \pi r^2 \rho v \]

5. **Apply the principle of momentum change**:
- When the liquid hits the wall, it comes to a stop (assuming it does not rebound). The change in momentum per unit time (which is the force exerted on the wall) is given by:
\[ F = \frac{dP}{dt} = \frac{d(mv)}{dt} \]
- Since the velocity \( v \) is constant, we can simplify this to:
\[ F = v \cdot \frac{dm}{dt} \]

6. **Substitute the mass flow rate into the force equation**:
- Now substituting \( \frac{dm}{dt} \) into the force equation:
\[ F = v \cdot (\pi r^2 \rho v) = \pi r^2 \rho v^2 \]

7. **Final expression for the force**:
- Thus, the force exerted on the wall by the impact of the liquid is:
\[ F = \pi r^2 \rho v^2 \]

### Final Answer:
The force exerted on the wall by the impact of the liquid is given by:
\[ F = \pi r^2 \rho v^2 \]