To solve the problem step by step, we will follow these steps:
### Step 1: Identify the given data
- Cross-sectional area of the jet, \( A = 6 \, \text{cm}^2 = 6 \times 10^{-4} \, \text{m}^2 \)
- Velocity of the water jet, \( v = 12 \, \text{m/s} \)
- Angle of incidence with the normal, \( \theta = 60^\circ \)
### Step 2: Calculate the components of the velocity
The velocity of the water jet can be broken down into two components:
- The component perpendicular to the wall (y-component):
\[
v_y = v \sin(60^\circ) = 12 \sin(60^\circ) = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3} \, \text{m/s}
\]
- The component parallel to the wall (x-component):
\[
v_x = v \cos(60^\circ) = 12 \cos(60^\circ) = 12 \times \frac{1}{2} = 6 \, \text{m/s}
\]
### Step 3: Analyze the momentum change
Since the water jet rebounds elastically, the y-component of the velocity remains the same, while the x-component changes direction:
- Initial momentum in the x-direction: \( p_{x, \text{initial}} = m v_x = m (6) \)
- Final momentum in the x-direction (after rebounding): \( p_{x, \text{final}} = m (-v_x) = m (-6) \)
The change in momentum in the x-direction is:
\[
\Delta p_x = p_{x, \text{final}} - p_{x, \text{initial}} = m(-6) - m(6) = -12m
\]
### Step 4: Calculate the mass flow rate
The mass flow rate (\( \dot{m} \)) of the water jet can be calculated using the formula:
\[
\dot{m} = \rho A v
\]
Where:
- \( \rho \) (density of water) = \( 1000 \, \text{kg/m}^3 \)
- \( A = 6 \times 10^{-4} \, \text{m}^2 \)
- \( v = 12 \, \text{m/s} \)
Substituting the values:
\[
\dot{m} = 1000 \times (6 \times 10^{-4}) \times 12 = 7.2 \, \text{kg/s}
\]
### Step 5: Calculate the force acting on the wall
The force (\( F \)) acting on the wall is given by the rate of change of momentum:
\[
F = \frac{\Delta p_x}{\Delta t} = \dot{m} \cdot \Delta v_x
\]
Where \( \Delta v_x = 12 \, \text{m/s} \) (since the change in momentum is 12m for each second).
Thus:
\[
F = \dot{m} \cdot 12 = 7.2 \cdot 12 = 86.4 \, \text{N}
\]
### Final Answer
The force acting on the wall is \( 86.4 \, \text{N} \).
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