PHYSM-1. A horizontal water jet strikes a plate wi

1. A horizontal water jet strikes a plate with mass flow rate

\boldsymbol{m}_{l}

and velocity

V_{l}

as shown in Figure 1. Upon striking the plate, the jet is split into two streams, mass flow rate

m_{2}

flowing upwards and

m_{3}

flowing downwards, with velocity

V_{2}

and

V_{3}

respectively. The exit diameter of the jet nozzle is

30 \mathrm{~mm}

and its corresponding velocity is

8 \mathrm{~m} / \mathrm{s}

(a) When

\theta

is

90^{\circ}

, what is the force required to hold the plate stationary? (5 marks) (b) When

\theta

is

90^{\circ}

and the plate is moving horizontally towards the jet at

3 \mathrm{~m} / \mathrm{s}

, what is the power required? (7 marks) (c) Derive an expression for

m_{2}

and

m_{3}

in terms of

m_{l}

and angle

\theta

. (10 marks) (d) Determine mass flow rate

m_{2}

and

m_{3}

when

\theta

is

60^{\circ}

. (3 marks) Figure 1

Solution

a

Apply conservation of linear momentum for the horizontal direction:

\sum p_{\text{initial}} = \sum p_{\text{final}}

b

Calculate the initial momentum:

p_{\text{initial}} = m_{l}V_{l}

c

Calculate the final momentum:

p_{\text{final}} = m_{2}V_{2} \cos(\theta) - m_{3}V_{3} \cos(\theta)

d

Since the plate is stationary, the net horizontal force required is the rate of change of momentum:

F = \frac{\Delta p}{\Delta t} = m_{l}V_{l} - (m_{2}V_{2} \cos(\theta) - m_{3}V_{3} \cos(\theta))

e

For

\theta = 90^{\circ}

, the cosines become zero, so the force is:

F = m_{l}V_{l}

Answer

F = m_{l}V_{l}

Key Concept

Conservation of Linear Momentum: The total linear momentum of a closed system remains constant if no external forces are acting on it.

Explanation

The force required to hold the plate stationary is equal to the rate of change of momentum of the water jet, which is the initial momentum since the final horizontal momentum is zero when

\theta = 90^{\circ}

.

Solution

a

Calculate the relative velocity of the water jet to the moving plate:

V_{\text{rel}} = V_{l} + 3 \text{ m/s}

b

The power required is the force times the velocity of the plate:

P = F \cdot V_{\text{plate}}

c

Use the force from part (a) and the velocity of the plate:

P = m_{l}V_{l} \cdot 3 \text{ m/s}

Answer

P = m_{l}V_{l} \cdot 3 \text{ m/s}

Key Concept

Power: Power is the rate at which work is done or energy is transferred. It is equal to the force times the velocity of the object in the direction of the force.

Explanation

The power required to move the plate horizontally towards the jet is the product of the force required to hold the plate stationary and the velocity of the plate.

Solution

a

Apply conservation of mass:

m_{l} = m_{2} + m_{3}

b

Apply conservation of linear momentum in the vertical direction:

0 = m_{2}V_{2} \sin(\theta) - m_{3}V_{3} \sin(\theta)

c

Solve the system of equations to express

m_{2}

and

m_{3}

in terms of

m_{l}

and

\theta

:

m_{2} = \frac{m_{l}V_{3} \sin(\theta)}{V_{2} \sin(\theta) + V_{3} \sin(\theta)}

and

m_{3} = \frac{m_{l}V_{2} \sin(\theta)}{V_{2} \sin(\theta) + V_{3} \sin(\theta)}

Answer

m_{2} = \frac{m_{l}V_{3} \sin(\theta)}{V_{2} \sin(\theta) + V_{3} \sin(\theta)}

and

m_{3} = \frac{m_{l}V_{2} \sin(\theta)}{V_{2} \sin(\theta) + V_{3} \sin(\theta)}

Key Concept

Conservation of Mass and Momentum: In a fluid system, the mass flow rate must be conserved, and the momentum must be conserved in each direction.

Explanation

By applying the conservation of mass and momentum, we can derive expressions for

m_{2}

and

m_{3}

in terms of

m_{l}

and

\theta

.

Solution

a

Use the expressions derived in part (c) for

m_{2}

and

m_{3}

b

Substitute

\theta = 60^{\circ}

into the expressions:

m_{2} = \frac{m_{l}V_{3} \sin(60^{\circ})}{V_{2} \sin(60^{\circ}) + V_{3} \sin(60^{\circ})}

and

m_{3} = \frac{m_{l}V_{2} \sin(60^{\circ})}{V_{2} \sin(60^{\circ}) + V_{3} \sin(60^{\circ})}

c

Calculate

m_{2}

and

m_{3}

using the given values for

V_{2}

and

V_{3}

Answer

The mass flow rates

m_{2}

and

m_{3}

when

\theta = 60^{\circ}

are calculated using the expressions from part (c) with the given velocities.

Key Concept

Conservation of Mass and Momentum: In a fluid system, the mass flow rate must be conserved, and the momentum must be conserved in each direction.

Explanation

By substituting

\theta = 60^{\circ}

into the expressions derived in part (c), we can determine the mass flow rates

m_{2}

and

m_{3}

.

1 (a) A steady stream of water jet discharges from the nozzle with a uniform velocity of 10

\mathrm{m} / \mathrm{s}

at angle

60^{\circ}

as shown in Figure 1. The jet strikes a horizontal plate

2 \mathrm{~m}

above the nozzle exit. Upon impact, part of the flow is guided towards the left while the remainder towards the right, and the plate is suspended in the air. The weight of the plate is

25 \mathrm{~N}

. Assume frictionless flow along the plate and weight of water to be neglected. (i) Write down the momentum equation for

\mathrm{x}

and

\mathrm{y}

direction. (ii) Determine the diameter of the nozzle (at the exit). (iii) Determine the mass flow rate

\mathrm{A}

and

\mathrm{B}

. (12 marks) Figure 1

Solution

a

Write down the momentum equation for the x direction:

\sum F_x = \dot{m} \Delta V_x

where

\dot{m}

is the mass flow rate and

\Delta V_x

is the change in velocity in the x direction

b

Write down the momentum equation for the y direction:

\sum F_y = \dot{m} \Delta V_y - W

where

W

is the weight of the plate

c

To find the diameter of the nozzle, use the continuity equation

A_1 V_1 = A_2 V_2

and the fact that the velocity and area are related through the mass flow rate

\dot{m} = \rho A V

where

\rho

is the density of water

d

Determine the mass flow rate A and B by using the momentum equations in the x and y directions and solving for

\dot{m}_A

and

\dot{m}_B

Answer

The diameter of the nozzle and the mass flow rates A and B cannot be determined without additional information such as the density of water and the velocity of the jet at the exit.

Key Concept

Dynamics deals with the study of forces and torques and their effect on motion, particularly Newton's Laws of Motion and the conservation of momentum. In fluid mechanics, the continuity equation and the momentum equation are essential for analyzing fluid flow.

Explanation

The momentum equations in the x and y directions are used to analyze the forces due to the change in momentum of the fluid as it strikes the plate. The continuity equation relates the area and velocity of the jet at different points. Without the density of water or the velocity at the exit, the diameter and mass flow rates cannot be calculated.

1 (a) A steady stream of water jet discharges from the nozzle with a uniform velocity of 10

\mathrm{m} / \mathrm{s}

at angle

60^{\circ}

as shown in Figure 1. The jet strikes a horizontal plate

2 \mathrm{~m}

above the nozzle exit. Upon impact, part of the flow is guided towards the left while the remainder towards the right, and the plate is suspended in the air. The weight of the plate is

25 \mathrm{~N}

. Assume frictionless flow along the plate and weight of water to be neglected. (i) Write down the momentum equation for

\mathrm{x}

and

\mathrm{y}

direction. (ii) Determine the diameter of the nozzle (at the exit). (iii) Determine the mass flow rate

\mathrm{A}

and

\mathrm{B}

. (12 marks) Figure 1

Solution

a

Write down the momentum equation for the x and y direction:

b

For the x-direction, apply conservation of momentum:

\rho Q v_{\text{in}} \cos(\theta) = \rho A v_A \cos(\alpha) - \rho B v_B \cos(\beta)

c

For the y-direction, the momentum equation is:

\rho Q v_{\text{in}} \sin(\theta) = F_{\text{plate}} + \rho A v_A \sin(\alpha) + \rho B v_B \sin(\beta)

d

Determine the diameter of the nozzle: Use the continuity equation

Q = A_{\text{nozzle}} v_{\text{in}}

and solve for the area to find the diameter

e

Determine the mass flow rate A and B: Use the momentum equations from steps b and c to solve for A and B

Answer

The momentum equations for the x and y directions are needed to solve for the mass flow rates A and B. The diameter of the nozzle can be found using the continuity equation.

Key Concept

Conservation of Momentum: In the absence of external forces, the momentum of a system remains constant. The momentum equation in fluid dynamics is given by

\rho Q v = \rho A v_A - \rho B v_B

where

\rho

is the density,

Q

is the volume flow rate, and

v

,

v_A

, and

v_B

are the velocities of the jet, flow A, and flow B, respectively.

Explanation

The momentum equation is used to relate the incoming flow to the outgoing flows A and B, taking into account the angle of deflection and the forces acting on the plate.

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