A jet of water of diameter 150 mm strikes a flat plate with a velocity of 12 m/s. The plate is moving with a velocity of 6 m/s in the direction of the jet. If the jet strikes the plate such that the normal to the plate makes an angle of 30° to the axis of the jet, find : (i) The normal force exerted on the plate, (ii) power, and (iii) efficiency of the jet.

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Force of a Jet on a Moving, Inclined Plate

Problem Statement

A jet of water of diameter 150 mm strikes a flat plate with a velocity of 12 m/s. The plate is moving with a velocity of 6 m/s in the direction of the jet. If the jet strikes the plate such that the normal to the plate makes an angle of 30° to the axis of the jet, find : (i) The normal force exerted on the plate, (ii) power, and (iii) efficiency of the jet.

Given Data & Constants

  • Diameter of jet, \(d = 150 \, \text{mm} = 0.15 \, \text{m}\)
  • Velocity of jet, \(V = 12 \, \text{m/s}\)
  • Velocity of plate, \(u = 6 \, \text{m/s}\)
  • Angle between jet and normal of plate, \(\theta = 30^\circ\)
  • Density of water, \(\rho = 1000 \, \text{kg/m}^3\)

Solution

(i) The Normal Force Exerted on the Plate (\(F_n\))

The force normal to the plate is based on the rate of change of momentum in the normal direction. The mass striking the plate depends on the relative velocity component normal to the plate, \( (V-u)\cos\theta \).

$$ \text{Area of jet, } A = \frac{\pi}{4} d^2 = \frac{\pi}{4} (0.15)^2 \approx 0.01767 \, \text{m}^2 $$ $$ F_n = \rho A (V - u)^2 \cos(\theta) $$ $$ F_n = 1000 \times 0.01767 \times (12 - 6)^2 \times \cos(30^\circ) $$ $$ F_n = 1000 \times 0.01767 \times 36 \times 0.866 $$ $$ F_n \approx 550.9 \, \text{N} $$

(ii) Power

Power is the work done per second, which is the force component in the direction of motion multiplied by the plate's velocity.

$$ \text{Force in direction of jet, } F_x = F_n \cos(\theta) $$ $$ F_x = 550.9 \times \cos(30^\circ) \approx 477.1 \, \text{N} $$ $$ \text{Power} = F_x \times u $$ $$ \text{Power} = 477.1 \, \text{N} \times 6 \, \text{m/s} \approx 2862.6 \, \text{W} $$

(iii) Efficiency of the Jet

Efficiency is the ratio of the useful power delivered to the plate to the initial power available in the jet.

$$ \text{Power of Jet} = \frac{1}{2} \rho A V^3 = \frac{1}{2} \times 1000 \times 0.01767 \times (12)^3 \approx 15268 \, \text{W} $$ $$ \eta = \frac{\text{Power}}{\text{Power of Jet}} $$ $$ \eta = \frac{2862.6}{15268} \approx 0.1875 $$
Final Results:

(i) Normal force exerted on the plate: \( \approx 550.9 \, \text{N} \)

(ii) Power: \( \approx 2862.6 \, \text{W} \) (or 2.86 kW)

(iii) Efficiency of the jet: \( \approx 18.75\% \)

Explanation of the Inclined Plate

  • Normal Force: When the plate is inclined, only the component of the relative velocity that is perpendicular (normal) to the plate is destroyed. This component is \((V-u)\cos\theta\). The force is calculated based on this change in momentum.
  • Power: The plate is moving horizontally. Therefore, to calculate the power, we only care about the force component that acts in the horizontal direction. This is found by resolving the normal force back into the direction of the jet's motion (\(F_x = F_n \cos\theta\)).
  • Efficiency: The efficiency is significantly lower than in the normal-impact case because a large component of the jet's momentum is not used to push the plate forward; it is instead deflected along the plate's surface.

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