6-Problems.pdf

Problems

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Problems

Momentum Equation

6.1 Using the Internet or some other source as reference, define in your own words the meaning of

“inertial reference frame.”

6.2 The surface of the earth is not a true inertial reference frame because there is a centripetal

acceleration due to the earth's rotation. The earth rotates once every 24 hours and has a diameter of 8000

miles. What is the centripetal acceleration on the surface of the earth, and how does it compare to the

gravitational acceleration?

Answer:

a n = 0.112 ft/s 2 , a n /g = 0.0035

6.3 Newton's second law can be stated that the force is equal to the rate of change of momentum,

F = d ( mv )/ dt . Taking the derivative by parts yields F = m ( dv )/ dt + ν( dm ) / ( dt ) This does not correspond to

F = ma What is the source of the discrepancy?

Jets

6.4 Give five examples of jets and how they are used in practice.

6.5 A “balloon rocket” is a balloon suspended from a taut wire by a hollow tube (drinking straw) and string.

The nozzle is formed of a 1 cm–diameter tube, and an air jet exits the nozzle with a speed of 40 m/s and a

density of 1.2 kg/m 3 . Find the force F needed to hold the balloon stationary. Neglect friction.

6.6 The balloon rocket is held in place by a force F . The pressure inside the balloon is 8 inH 2 O, the nozzle

diameter is 1.0 cm, and the air density is 1.2 kg/m 3 . Find the exit velocity ν and the force F . Neglect friction

and assume the air flow is inviscid and irrotational.

PROBLEM 6.5, 6.6

Answer:

F = 0.31 N, ν = 57.6 m/s

6.7 A water jet of diameter 30 mm and speed ν = 20 m/s is filling a tank. The tank has a mass of 20 kg and

contains 20 liters of water at the instant shown. The water temperature is 15°C. Find the force acting on the

bottom of the tank and the force acting on the stop block. Neglect friction.

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PROBLEM 6.7, 6.8

6.8 A water jet of diameter 2 inches and speed ν = 50 ft/s is filling a tank. The tank has a mass of 25 lbm and

contains 5 gallons of water at the instant shown. The water temperature is 70°F. Find the minimum

coefficient of friction such that the force acting on the stop block is zero.

Answer:

= 0.218

6.9 A design contest features a submarine that will travel at a steady speed of V sub = 1 m/s in 15°C water. The

sub is powered by a water jet. This jet is created by drawing water from an inlet of diameter 25 mm, passing

this water through a pump and then accelerating the water through a nozzle of diameter 5 mm to a speed of

V jet . The hydrodynamic drag force ( F D ) can be calculated using

where the coefficient of drag is C D = 0.3 and the projected area is A p = 0.28 m 2 . Specify an acceptable value

of V jet .

PROBLEM 6.9

6.10 A horizontal water jet at 70°F impinges on a verticalperpendicular plate. The discharge is 2 cfs. If the

external force required to hold the plate in place is 200 lbf, what is the velocity of the water?

PROBLEMS 6.10, 6.11

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Answer:

ν 1 = 51.5 ft/s

6.11 A horizontal water jet at 70°F issues from a circular orifice in a large tank. The jet strikes a vertical plate

that is normal to the axis of the jet. A force of 600 lbf is needed to hold the plate in place against the action

of the jet. If the pressure in the tank is 25 psig at point A , what is the diameter of the jet just downstream of

the orifice?

6.12 An engineer, who is designing a water toy, is making preliminary calculations. A user of the product will

apply a force F F that moves a piston ( D = 80 mm) at a speed of V pistion = 300 mm/s. Water at 20°C jets out

of a converging nozzle of diameter d = 15 mm. To hold the toy stationary, the user applies a force F 2 to the

handle. Which force ( F 1 versus F 2 ) is larger? Explain your answer using concepts of the momentum

principle. Then calculate F 1 and F 2 . Neglect friction between the piston and the walls.

PROBLEM 6.12

Answer:

F 1 = 182 N, F 2 = 169 N

6.13 A firehose on a boat is producing a 3 in.–diameter water jet with a speed of V = 70 mph. The boat is held

stationary by a cable attached to a pier, and the water temperature is 50°F. Calculate the tension in the

cable.

6.14 A boat is held stationary by a cable attached to a pier. A firehose directs a spray of 5°C water at a speed of

V = 50 m/s. If the allowable load on the cable is 5 kN, calculate the mass flow rate of the water jet. What is

the corresponding diameter of the water jet?

PROBLEMS 6.13, 6.14

Answer:

= 200 kg/s, d = 7.14 cm

6.15 A tank of water (15°C) with a total weight of 200 N (water plus the container) is suspended by a vertical

cable. Pressurized air drives a water jet ( d = 12 mm) out the bottom of the tank such that the tension in the

vertical cable is 10 N. If H = 425 mm, find the required air pressure in units of atmospheres (gage).

Assume the flow of water is irrotational.

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PROBLEM 6.15

6.16 A jet of water (60°F) is discharging at a constant rate of 2.0 cfs from the upper tank. If the jet diameter at

section 1 is 4 in., what forces will be measured by scales A and B ? Assume the empty tank weighs 300 lbf,

the crosssectional area of the tank is 4 ft 2 , h = 1 ft, and H = 9 ft.

PROBLEM 6.16

Answer:

F A = 643 lbf, F B = 88.9 lbf

6.17 A conveyor belt discharges gravel into a barge as shown at a rate of 50 yd 3 /min. If the gravel weighs 120

lbf/ft 3 , what is the tension in the hawser that secures the barge to the dock?

PROBLEM 6.17

6.18 The hemicircular nozzle sprays a sheet of liquid through 180° of arc as shown. The velocity is V at the

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efflux section where the sheet thickness is t . Derive a formula for the external force F (in the y direction)

required to hold the nozzle system in place. This force should be a function of ρ, V , r , and t .

PROBLEM 6.18

Answer:

F y = 2ρν 2 tr

6.19 The expansion section of a rocket nozzle is often conical in shape; and because the flow diverges, the

thrust derived from the nozzle is less than it would be if the exit velocity were everywhere parallel to the

nozzle axis. By considering the flow through the spherical section suspended by the cone and assuming

that the exit pressure is equal to the atmospheric pressure, show that the thrust is given by

where is the mass flow through the nozzle, V e is the exit velocity, and α is the nozzle halfangle.

PROBLEM 6.19

Vanes

6.20 Determine the external reactions in the x and y directions needed to hold this fixed vane, which turns the

oil jet in a horizontal plane. Here V 1 is 18 m/s, V 2 = 17 m/s, and Q = 0.15 m 3 /s.

Answer:

6.21 Solve Prob. 6.20 for V 1 = 90 ft/s, V 2 = 85 ft/s, and Q = 2 cfs.

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