Fluid Mechanics in SI Units

R. C. Hibbeler, Kai Beng Yap

Chapter 6

Fluid Momentum - all with Video Answers

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Chapter Questions

Problem 1

Determine the linear momentum of a mass of fluid in a $0.2-\mathrm{m}$ length of pipe if the velocity profile for the fluid is a paraboloid as shown. Compare this result with the linear momentum of the fluid using the average velocity of flow. Take $\rho=800 \mathrm{~kg} / \mathrm{m}^{3}$.

Prabhat Tyagi

Prabhat Tyagi

Numerade Educator

Problem 2

Flow through the circular pipe is turbulent, and the velocity profile can be modeled using Prandtl's one-seventh power law, $v=V_{\max }(1-r / R)^{1 / 7} .$ If $\rho$ is the density, show that the momentum of the fluid per unit time passing through the pipe is $(49 / 72) \pi R^{2} \rho V_{\max }^{2}$. Then show that $V_{\max }=(60 / 49) V,$ where $V$ is the average velocity of the flow. Also, show that the momentum per unit time is $(50 / 49) \pi R^{2} \rho V^{2}$

James Kiss

Numerade Educator

Problem 3

An Oil of density, $\rho_{o}=900 \mathrm{~kg} / \mathrm{m}^{3}$ flows at $0.08 \mathrm{~m}^{3} / \mathrm{s}$
through the transition at section $A B$. If the pressure at the transition $C$ is $10 \mathrm{kPa}$, determine the resultant horizontal shear force acting along the seam $A B$ that holds the cap to the larger pipe.

James Kiss

Numerade Educator

Problem 4

Water flowing through a hose with a velocity of 6 $\mathrm{m} / \mathrm{s}$, impacts on a wall. Determine the force exerted on the wall by the water flow. Assume the water does not splash back off the wall.

Ankur S

Numerade Educator

Problem 5

A small marine ascidian called a styela fixes itself on the sea floor and then allows moving water to pass through it in order to feed. If the opening at $A$ has a diameter of $2 \mathrm{~mm},$ and at the exit $B$ the diameter is $1.5 \mathrm{~mm}$, determine the horizontal force needed to keep this organism attached to the rock at $C$ when the water is moving at $0.2 \mathrm{~m} / \mathrm{s}$ into the opening at $A$. Take $\rho=1050 \mathrm{~kg} / \mathrm{m}^{3}$.

Penny Riley

Numerade Educator

Problem 6

A pipe of $80 \mathrm{~mm}$ diameter is bent at $45^{\circ} .$ Oil flows through the pipe with a velocity of $6 \mathrm{~m} / \mathrm{s}$. If the pressure in the pipe at $A$ and $B$ is $50 \mathrm{kPa}$, determine the $x$ and $y$ components of force the flow exerts on the elbow. The flow occurs in the horizontal plane. Take $\rho_{o}=900 \mathrm{~kg} / \mathrm{m} 3$.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 7

The apparatus or "jet pump" used in an industrial plant is constructed by placing the tube within the pipe. Determine the increase in pressure $\left(p_{B}-p_{A}\right)$ that occurs between the back $A$ and front $B$ of the pipe if the velocity of the flow within the 200 -mm-diameter pipe is $2 \mathrm{~m} / \mathrm{s}$, and the velocity of the flow through the 20 -mm-diameter tube is $40 \mathrm{~m} / \mathrm{s}$. The fluid is ethyl alcohol having a density of $\rho_{e a}=790 \mathrm{~kg} / \mathrm{m}^{3} .$ Assume the pressure at each cross section of the pipe is uniform.

Penny Riley

Numerade Educator

Problem 8

A jet of water flows through a nozzle of diameter 80 $\mathrm{mm}$ and strikes on a fixed vane with the velocity of $5 \mathrm{~m} / \mathrm{s}$. If the jet after the strike on the fixed vane deflects in the direction as shown, determine the normal force the jet exerts on the vane.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 9

A jet of water flows through a nozzle of diameter 80 $\mathrm{mm}$ and strikes on a fixed vane with the velocity of $5 \mathrm{~m} / \mathrm{s}$ If the jet, after the strike on the fixed vane deflects in the direction as shown, determine the volume flow towards $A$ and towards $B$ if the tangential component of the force that the water exerts on the vane is zero.

James Kiss

Numerade Educator

Problem 10

The nozzle has a diameter of $40 \mathrm{~mm}$. If it discharges water with a velocity of $20 \mathrm{~m} / \mathrm{s}$ against the fixed blade, determine the horizontal force exerted by the water on the blade. The blade divides the water evenly at an angle of 5

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 11

The nozzle has a diameter of $40 \mathrm{~mm}$. If it discharges water with a velocity of $20 \mathrm{~m} / \mathrm{s}$ against the fixed blade, determine the horizontal force exerted by the water on the blade as a function of the blade angle
\theta. Plot this force (vertical axis) versus $\theta$ for $0^{\circ} \leq \theta \leq 75^{\circ}$. Give values for increments of $\Delta \theta=15^{\circ}$. The blade divides the water evenly.

James Kiss

Numerade Educator

Problem 12

The hemispherical bowl of mass $m$ is held in equilibrium by the vertical jet of water discharged through a nozzle of diameter $d$. If the volumetric flow is $Q,$ determine the height $h$ at which the bowl is suspended. The water density is $\rho_{w}$.

James Kiss

Numerade Educator

Problem 13

The 500 -g hemispherical bowl is held in equilibrium by the vertical jet of water discharged through the 10-mm-diameter nozzle. Determine the height $h$ of the bowl as a function of the volumetric flow $Q$ of the water through the nozzle. Plot the height $h$ (vertical axis) versus $Q$ for $0.5\left(10^{-3}\right) \mathrm{m}^{3} / \mathrm{s} \leq Q \leq 1\left(10^{-3}\right) \mathrm{m}^{3} / \mathrm{s}$. Give values for
increments of $\Delta Q=0.1\left(10^{-3}\right) \mathrm{m}^{3} / \mathrm{s}$

James Kiss

Numerade Educator

Problem 14

Water flows through the 200 -mm-diameter pipe at $4 \mathrm{~m} / \mathrm{s}$. If it exits into the atmosphere through the nozzle, determine the resultant force the bolts must develop at the connection $A B$ to hold the nozzle onto the pipe.

James Kiss

Numerade Educator

Problem 15

Water enters $A$ with a velocity of $8 \mathrm{~m} / \mathrm{s}$ and pressure of $70 \mathrm{kPa}$. If the velocity at $C$ is $9 \mathrm{~m} / \mathrm{s}$, determine the horizontal and vertical components of the resultant force that must act on the transition to hold it in place. Neglect the size of the transition.

James Kiss

Numerade Educator

Problem 16

An industrial waste liquid having a density of $1.3 \mathrm{Mg} / \mathrm{m}^{3}$ is to be loaded into a barge. If the average velocity of flow out of the 120 -mm-diameter pipe is $V_{A}=4 \mathrm{~m} / \mathrm{s}$, determine the force in the tie rope needed to hold the barge stationary.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 17

An industrial waste liquid having a density of $1.3 \mathrm{Mg} / \mathrm{m}^{3}$ is to be loaded into a barge. Determine the maximum force in the tie rope needed to hold the barge stationary. The waste can enter the barge at any point within the $12-\mathrm{m}$ region. Also, what is the speed of the waste exiting the pipe of the diameter $120 \mathrm{~mm}$ at $A$ when this occurs? The pipe has a diameter of $100 \mathrm{~mm}$.

James Kiss

Numerade Educator

Problem 18

There is a close duct of width $1.2 \mathrm{~m}$. Air flows through the duct with a uniform velocity of $0.4 \mathrm{~m} / \mathrm{s}$. Determine the horizontal force $\mathrm{F}$ required to hold the strap $\mathrm{C}$ it in place. Neglect any force at the slip joints $A$ and $B$. Take $\rho_{a}=1.22 \mathrm{~kg} / \mathrm{m}^{3}$

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 19

Crude oil flows through the horizontal tapered $45^{\circ}$ elbow at $0.02 \mathrm{~m}^{3} / \mathrm{s}$. If the pressure at $A$ is $300 \mathrm{kPa}$, determine the horizontal and vertical components of the resultant force the oil exerts on the elbow. Neglect the size of the elbow.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 20

The water is flowing with a velocity of $4 \mathrm{~m} / \mathrm{s}$ through a pipe of diameter $150 \mathrm{~mm}$ having an orifice plate of hole of diameter $50 \mathrm{~mm}$. If the pressure at $A$ is $250 \mathrm{kPa}$, and at $B$ it is $150 \mathrm{kPa}$, determine the force the water exerts on the plate.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 21

An oil is flowing through a pipe of length 25 -m-long and diameter of $250 \mathrm{~mm}$. The velocity of the oil is constant and equal to $3 \mathrm{~m} / \mathrm{s}$. Friction losses along the pipe cause the pressure at $B$ to be $10 \mathrm{kPa}$ less than the pressure at $A$. Determine the resultant friction force on this length of pipe. Take $\rho_{o}=880 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 22

Water flows into the tank at the rate of $0.05 \mathrm{~m}^{3} / \mathrm{s}$ from the 100 -mm-diameter pipe. If the tank is $500 \mathrm{~mm}$ on each side, determine the compression in each of the four springs that support its corners when the water reaches a depth of $h=1 \mathrm{~m}$. Each spring has a stiffness of $k=8 \mathrm{kN} / \mathrm{m}$. When empty, the tank compresses each spring $30 \mathrm{~mm}$.

Penny Riley

Numerade Educator

Problem 23

The toy sprinkler consists of a cap and a rigid tube having a diameter of $20 \mathrm{~mm}$. If water flows through the tube at $0.7\left(10^{-3}\right) \mathrm{m}^{3} / \mathrm{s}$, determine the vertical force the wall of the tube must support at $B$. Neglect the weight of the sprinkler head and the water within the curved segment of the tube. The weight of the tube and water within the vertical segment $A B$ is $4 \mathrm{~N}$.

James Kiss

Numerade Educator

Problem 24

Determine the flow through the tube of the toy sprinkler which consists of a cap and a rigid tube of diameter $25 \mathrm{~mm}$. The water creates a vertical force of $8 \mathrm{~N}$ in the tube at $B$. Neglect the weight of the sprinkler head and the water within the curved segment of the tube. The weight of the tube and water within the vertical segment $A B$ is $6 \mathrm{~N}$.

James Kiss

Numerade Educator

Problem 25

The disk valve is used to control the flow of $0.008 \mathrm{~m}^{3} / \mathrm{s}$ of water through the 40 -mm-diameter tube. Determine the force $\mathbf{F}$ required to hold the valve in place for any position $x$ of closure of the valve.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 26

Water flows through the pipe $C$ at $4 \mathrm{~m} / \mathrm{s}$. Determine the horizontal and vertical components of force exerted by elbow $D$ necessary to hold the pipe assembly in equilibrium. Neglect the size and weight of the pipe and the water within it. The pipe has a diameter of $60 \mathrm{~mm}$ at $C,$ and at $A$ and $B$ the diameters are $20 \mathrm{~mm}$.

Ajay Singhal

Numerade Educator

Problem 27

A circular craft of the mass $350 \mathrm{~kg}$ is suspended 150 $\mathrm{mm}$ from the ground. For this to occur, air is to be drawn in at $20 \mathrm{~m} / \mathrm{s}$ through the 250 -mm-diameter intake and discharged to the ground as shown. Determine the pressure that the craft exerts on the ground. Take $\rho_{a}=1.22 \mathrm{~kg} / \mathrm{m}^{3}$.

Penny Riley

Numerade Educator

Problem 28

A dumper dumps is used to water on the ground such that it flows from the dumper through a $80-\mathrm{mm}$ -wide opening at an angle of $45^{\circ} .$ The length of the opening is $1.8 \mathrm{~m}$. Determine the friction force that all the wheels of the truck must exert on the ground to keep the dumper from moving at the instant the water depth in the truck is $2 \mathrm{~m}$.

Penny Riley

Numerade Educator

Problem 29

The oil is flowing through a vertical pipe assembly of $U$ - section as shown in the Fig. The diameter of the pipe is $60 \mathrm{~mm}$, and the pressure and velocity at $A$ is $200 \mathrm{kPa}$ and $2 \mathrm{~m} / \mathrm{s}$, respectively. Determine the horizontal and vertical components of force the pipe exerts on the U-section $A B$ of the assembly. The assembly and the oil within it have a combined weight of $40 \mathrm{~N}$. Take $\rho_{o}=900 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 30

The cylindrical needle valve is used to control the flow of $0.003 \mathrm{~m}^{3} / \mathrm{s}$ of water through the 20 -mm-diameter tube. Determine the force $\mathbf{F}$ required to hold it in place when $x=10 \mathrm{~mm}$.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 31

The cylindrical needle valve is used to control the flow of $0.003 \mathrm{~m}^{3} / \mathrm{s}$ of water through the 20 -mm-diameter tube. Determine the force $\mathbf{F}$ required to hold it in place for any position $x$ of closure of the valve.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 32

Air at a temperature of $30^{\circ} \mathrm{C}$ flows through the expansion fitting such that its velocity at $A$ is $15 \mathrm{~m} / \mathrm{s}$ and the absolute pressure is $250 \mathrm{kPa}$. If no heat or frictional loss occurs, determine the resultant force needed to hold the fitting in place.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 33

Air at a temperature of $30^{\circ} \mathrm{C}$ flows through the expansion fitting such that its velocity at $A$ is $15 \mathrm{~m} / \mathrm{s}$ and the absolute pressure is $250 \mathrm{kPa}$. If heat and frictional loss due to the expansion cause the temperature and absolute pressure of the air at $B$ to become $20^{\circ} \mathrm{C}$ and $7.50 \mathrm{kPa}$, determine the resultant force needed to hold the fitting in place.

James Kiss

Numerade Educator

Problem 34

Water is flowing through a nozzle. The velocity of water at the inlet of the nozzle is $3 \mathrm{~m} / \mathrm{s}$. Determine the force $\mathbf{F}$ needed to keep the circular plate moving to the right at $3 \mathrm{~m} / \mathrm{s}$.

James Kiss

Numerade Educator

Problem 35

Water is flowing through a nozzle. The velocity of water at the inlet of the nozzle is $5 \mathrm{~m} / \mathrm{s}$. Determine the force $\mathbf{F}$ needed to keep the circular plate moving to the left at $5 \mathrm{~m} / \mathrm{s}$

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 36

Flow from the water stream strikes the inclined surface of the cart. Determine the power produced by the stream if, due to rolling friction, the cart moves to the right with a constant velocity of $2 \mathrm{~m} / \mathrm{s}$. The discharge from the 50 -mm-diameter nozzle is $0.04 \mathrm{~m}^{3} / \mathrm{s}$. One-fourth of the discharge flows down the incline, and three-fourths flows up the incline.

James Kiss

Numerade Educator

Problem 37

A motor boat is powered by a fan, which develops a slipstream having a diameter of $1.5 \mathrm{~m}$. If the fan ejects air with an average velocity of $50 \mathrm{~m} / \mathrm{s}$ relative to the boat, and the boat is traveling with a constant velocity of $10 \mathrm{~m} / \mathrm{s}$, determine the force exerted by the fan on the boat. Assume that the air has a constant density of $\rho_{\alpha}=1.22 \mathrm{~kg} / \mathrm{m}^{3}$ and that the entering air at $A$ is essentially at rest relative to the ground.

James Kiss

Numerade Educator

Problem 38

The car is used to scoop up water that is lying in a trough at the tracks. Determine the force needed to pull the car forward at constant velocity $\mathbf{v}$ for each of the three cases. The scoop has a cross-sectional area $A$ and the density of water is $\rho_{w}$

James Kiss

Numerade Educator

Problem 39

Water flows at $0.1 \mathrm{~m}^{3} / \mathrm{s}$ through the $100-\mathrm{mm}$ -diameter nozzle and strikes the vane on the $150-\mathrm{kg}$ cart, which is originally at rest. Determine the velocity of the cart 3 seconds after the jet strikes the vane.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 40

Water flows at $0.1 \mathrm{~m}^{3} / \mathrm{s}$ through the 100 -mmdiameter nozzle and strikes the vane on the $150-\mathrm{kg}$ cart, which is originally at rest. Determine the acceleration of the cart when it attains a velocity of $2 \mathrm{~m} / \mathrm{s}$.

James Kiss

Numerade Educator

Problem 41

A water jet of diameter $20 \mathrm{~mm}$ impacts on a blade with velocity $12 \mathrm{~m} / \mathrm{s}$ and is deflected $180^{\circ}$ as shown. If the blade is moving to the left at $3 \mathrm{~m} / \mathrm{s}$, determine the horizontal force $F$ of the blade on the water.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 42

Solve Prob. $6-41$ if the blade is moving to the right at $3 \mathrm{~m} / \mathrm{s}$. At what speed must the blade be moving to the right to reduce the force $F$ to zero?

James Kiss

Numerade Educator

Problem 43

Water flows through the Tee fitting at $0.02 \mathrm{~m}^{3} / \mathrm{s}$. If the water exits the fitting at $B$ to the atmosphere, determine the horizontal and vertical components of force, and the moment that must be exerted on the fixed support at $A$, in order to hold the fitting in equilibrium. Neglect the weight of the fitting and the water within it.

James Kiss

Numerade Educator

Problem 44

Water flows through the Tee fitting at $0.02 \mathrm{~m}^{3} / \mathrm{s}$. If the pipe at $B$ is extended and the pressure in the pipe at $B$ is $75 \mathrm{kPa},$ determine the horizontal and vertical components of force, and the moment that must be exerted on the fixed support at $A$, to hold the fitting in equilibrium. Neglect the weight of the fitting and the water within it.

James Kiss

Numerade Educator

Problem 45

Water flows through the curved pipe at a speed of $5 \mathrm{~m} / \mathrm{s}$. If the diameter of the pipe is $150 \mathrm{~mm}$, determine the horizontal and vertical components of the resultant force, and the moment acting on the coupling at $A$. The weight of the pipe and the water within it is $450 \mathrm{~N}$, having a center of gravity at $G$.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 46

The chute is used to divert the flow of water. If the flow is $0.4 \mathrm{~m}^{3} / \mathrm{s}$ and it has a cross-sectional area of $0.03 \mathrm{~m}^{2}$, determine the horizontal and vertical force components at the pin $A$, and the horizontal force at the roller $B$, necessary for equilibrium. Neglect the weight of the chute and the water on it.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 47

Air enters into the hollow propeller tube at $A$ with a mass flow of $3 \mathrm{~kg} / \mathrm{s}$ and exits at the ends $B$ and $C$ with a velocity of $400 \mathrm{~m} / \mathrm{s}$, measured relative to the tube. If the tube rotates at $1500 \mathrm{rev} / \mathrm{min}$, determine the frictional torque $\mathbf{M}$ on the tube.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 48

Water flows into the bend fitting with a velocity of $3 \mathrm{~m} / \mathrm{s}$. If the water exits at $B$ into the atmosphere, determine the horizontal and vertical components of force, and the moment at $C,$ needed to hold the fitting in place. Neglect the weight of the fitting and the water within it.

James Kiss

Numerade Educator

Problem 49

Water flows into the bend fitting with a velocity of $3 \mathrm{~m} / \mathrm{s}$. If the water at $B$ exits into a tank having a gage pressure of $10 \mathrm{kPa}$, determine the horizontal and vertical components of force, and the moment at $C,$ needed to hold the fitting in place. Neglect the weight of the fitting and the water within it.

James Kiss

Numerade Educator

Problem 50

The 5-mm-diameter arms of a rotating lawn sprinkler have the dimensions shown. Water flows out relative to the arms at $6 \mathrm{~m} / \mathrm{s}$, while the arms are rotating at $10 \mathrm{rad} / \mathrm{s}$. Determine the frictional torsional resistance at the bearing $A$, and the speed of the water as it emerges from the nozzles, as measured by a fixed observer.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 51

When operating, the air-jet fan discharges air with a speed of $V=18 \mathrm{~m} / \mathrm{s}$ into a slipstream having a diameter of $0.5 \mathrm{~m}$. If the air has a density of $1.22 \mathrm{~kg} / \mathrm{m}^{3}$, determine the horizontal and vertical components of reaction at $C,$ and the vertical reaction at each of the two wheels, $D$. The fan and motor have a mass of $25 \mathrm{~kg}$ and a center of mass at $G$. Neglect the weight of the frame. Due to symmetry, both of the wheels support an equal load. Assume the air entering the fan at $A$ is essentially at rest.

James Kiss

Numerade Educator

Problem 52

If the air has a density of $1.22 \mathrm{~kg} / \mathrm{m}^{3},$ determine the maximum speed $V$ at which the air-jet fan can discharge air into the slipstream having a diameter of $0.5 \mathrm{~m}$ at $B$, so that the fan does not topple over. The fan and motor have a mass of $25 \mathrm{~kg}$ and a center of mass at $G .$ Neglect the weight of the frame. Due to symmetry, both of the wheels support an equal load. Assume the air entering the fan at $A$ is essentially at rest.

James Kiss

Numerade Educator

Problem 53

The waterwheel consists of a series of flat plates that have a width $b$ and are subjected to the impact of water to a depth $h$, from a stream that has an average velocity of $V$. If the wheel is turning at $\omega,$ determine the power supplied to the wheel by the water.

James Kiss

Numerade Educator

Problem 54

A boat has a 250-mm-diameter propeller that discharges $0.6 \mathrm{~m}^{3} / \mathrm{s}$ of water as the boat travels at $35 \mathrm{~km} / \mathrm{h}$ in still water. Determine the thrust developed by the propeller on the boat.

Narayan Hari

Numerade Educator

Problem 55

A ship has a $2.5-\mathrm{m}$ -diameter propeller with an ideal efficiency of $40 \%$. If the thrust developed by the propeller is $1.5 \mathrm{MN},$ determine the constant speed of the ship in still water and the power that must be supplied to the propeller to operate it.

Penny Riley

Numerade Educator

Problem 56

A fan of mass $10 \mathrm{~kg}$ and blade diameter $0.6 \mathrm{~m}$ is used to develop a breeze at $12 \mathrm{~m} / \mathrm{s}$. Determine the smallest dimension $d$ for the support so that the fan does not tip over. Take $\rho_{a}=1.20 \mathrm{~kg} / \mathrm{m}^{3}$.

Penny Riley

Numerade Educator

Problem 57

The wind turbine has a rotor diameter of $40 \mathrm{~m}$ and an ideal efficiency of $50 \%$ in a $12 \mathrm{~m} / \mathrm{s}$ wind. If the density of the air is $\rho_{a}=1.22 \mathrm{~kg} / \mathrm{m}^{3},$ determine the thrust on the blade shaft, and the power withdrawn by the blades.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 58

The wind turbine has a rotor diameter of $40 \mathrm{~m}$ and an efficiency of $50 \%$ in a $12 \mathrm{~m} / \mathrm{s}$ wind. If the density of the air is $\rho_{a}=1.22 \mathrm{~kg} / \mathrm{m}^{3},$ determine the difference between the pressure just in front of and just behind the blades. Also find the mean velocity of the air passing through the blades.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 59

An airplane having a $3 \mathrm{~m}$ diameter propeller is flying with a constant speed of $300 \mathrm{~km} / \mathrm{h}$ in still air. Determine the force acting on the plane if the speed of the air behind the propeller relative to the plane, is $800 \mathrm{~km} / \mathrm{h}$. Also, what is the ideal efficiency of the propeller, and the power produced by the propeller? Take $\rho_{a}=0.910 \mathrm{~kg} / \mathrm{m}^{3}$.

Penny Riley

Numerade Educator

Problem 60

The rocket has an initial total mass $m_{0}$, including the fuel. When it is fired, it ejects a mass flow of $\dot{m}_{e}$ with a velocity of $v_{e}$ measured relative to the rocket. As this occurs, the pressure at the nozzle, which has a crosssectional area $A_{c}$ is $p_{e}$. If the drag force on the rocket is $F_{D}=c t,$ where $t$ is the time and $c$ is a constant, determine the velocity of the rocket if the acceleration due to gravity is assumed to be constant.

James Kiss

Numerade Educator

Problem 61

A jet engine is to be tested with the braking deflector in place. It is mounted on the stand. If the exhaust has a velocity of $900 \mathrm{~m} / \mathrm{s}$ and the pressure just outside the nozzle is assumed to be atmospheric, determine the horizontal force that the supports exert on the engine. The fuel-air mixture has a flow of $12 \mathrm{~kg} / \mathrm{s}$.

Penny Riley

Numerade Educator

Problem 62

If a jet engine as discussed in Prob. $6-61$ is attached to a jet plane, and it operates the braking deflector with the conditions stated in that problem, determine the speed of the plane in 8 seconds after it lands with a touchdown velocity of $40 \mathrm{~m} / \mathrm{s}$. The plane has a mass of $6 \mathrm{Mg}$. Neglect rolling friction from the landing gear.

James Kiss

Numerade Educator

Problem 63

A helicopter is used to extinguish the fire. Its mass is $10 \mathrm{Mg}$ and is hovering over a lake as the suspended bucket collects $4 \mathrm{~m}^{3}$ of water used to extinguish a fire. Determine the power required by the engine to hold the filled water bucket over the lake. The horizontal blade has a diameter of $12 \mathrm{~m}$. Take $\rho_{\alpha}=1.23 \mathrm{~kg} / \mathrm{m}^{3}$.

Penny Riley

Numerade Educator

Problem 64

Plot Eq. $6-15$ and show that the maximum efficiency of a wind turbine is $59.3 \%$ as stated by Betz's law.

James Kiss

Numerade Educator

Problem 65

A jet engine is flying with velocity $150 \mathrm{~m} / \mathrm{s}$ in still air and draws in air at standard atmospheric temperature and pressure through a 0.4 -m-diameter inlet. If $2.5 \mathrm{~kg} / \mathrm{s}$ of fuel is added and the mixture leaves the $0.2-\mathrm{m}$ -diameter nozzle at $500 \mathrm{~m} / \mathrm{s}$, measured relative to the engine, determine the thrust provided by the turbojet.

Penny Riley

Numerade Educator

Problem 66

A jet engine is to be tested. It is mounted on the stand. If the exhaust has a velocity of $1800 \mathrm{~m} / \mathrm{s}$ and the pressure just outside the nozzle is assumed to be atmospheric, determine the horizontal force that the supports exert on the engine. The fuel-air mixture has a flow of $12 \mathrm{~kg} / \mathrm{s}$.

Penny Riley

Numerade Educator

Problem 67

The jet plane has a constant velocity of $750 \mathrm{~km} / \mathrm{h}$. Air enters its engine nacelle at $A$ having a cross-sectional area of $0.8 \mathrm{~m}^{2}$. Fuel is mixed with the air at $\dot{m}_{e}=2.5 \mathrm{~kg} / \mathrm{s}$ and is exhausted into the ambient air with a velocity of $900 \mathrm{~m} / \mathrm{s},$ measured relative to the plane. Determine the force the engine exerts on the wing of the plane. Take $\rho_{a}=0.850 \mathrm{~kg} / \mathrm{m}^{3}$

Penny Riley

Numerade Educator

Problem 68

The boat of mass of $180 \mathrm{~kg}$ is swimming in a river with a constant velocity of $60 \mathrm{~km} / \mathrm{h}$, relative to the river. The river is flowing in opposite to the river at $4 \mathrm{~km} / \mathrm{h}$. If a tube is placed in the water, as shown, and it collects $35 \mathrm{~kg}$ of water in the boat in $100 \mathrm{~s}$, determine the horizontal thrust $T$ on the tube that is required to overcome the resistance due to the water collection.

Penny Riley

Numerade Educator

Problem 69

The jet boat takes in water through its bow at $0.03 \mathrm{~m}^{3} / \mathrm{s}$, while traveling in still water with a constant velocity of $10 \mathrm{~m} / \mathrm{s}$. If the water is ejected from a pump through the stern at $30 \mathrm{~m} / \mathrm{s}$, measured relative to the boat, determine the thrust developed by the engine. What would be the thrust if the $0.03 \mathrm{~m}^{3} / \mathrm{s}$ of water were taken in along the sides of the boat, perpendicular to the direction of motion? If the efficiency is defined as the work done per unit time divided by the energy supplied per unit time, then determine the efficiency for each case.

James Kiss

Numerade Educator

Problem 70

A commercial jet aircraft has a mass of $150 \mathrm{Mg}$ and is cruising at a constant speed of $850 \mathrm{~km} / \mathrm{h}$ in level flight $\left(\theta=0^{\circ}\right)$. If each of the two engines draws in air at a rate of $1000 \mathrm{~kg} / \mathrm{s}$ and ejects it with a velocity of $900 \mathrm{~m} / \mathrm{s}$ relative to the aircraft, determine the maximum angle $\theta$ at which the aircraft can fly with a constant speed of $750 \mathrm{~km} / \mathrm{h}$. Assume that air resistance (drag) is proportional to the square of the speed, that is, $F_{D}=c V^{2},$ where $c$ is a constant to be determined. The engines are operating with the same power in both cases. Neglect the amount of fuel consumed.

James Kiss

Numerade Educator

Problem 71

The balloon has a mass of $20 \mathrm{~g}$ (empty) and it is filled with air having a temperature of $20^{\circ} \mathrm{C}$. If it is released, it begins to accelerate upward at $8 \mathrm{~m} / \mathrm{s}^{2}$. Determine the initial mass flow of air from the stem. Assume the balloon is a sphere having a radius of $300 \mathrm{~mm}$.

Penny Riley

Numerade Educator

Problem 72

The $10-\mathrm{Mg}$ jet plane has a constant speed of $860 \mathrm{~km} / \mathrm{h}$ when it is flying horizontally. Air enters the intake $I$ at the rate of $40 \mathrm{~m}^{3} / \mathrm{s}$. If the engine burns fuel at the rate of $2.2 \mathrm{~kg} / \mathrm{s}$, and the gas (air and fuel) is exhausted relative to the plane with a speed of $600 \mathrm{~m} / \mathrm{s}$, determine the resultant drag force exerted on the plane by air resistance. Assume that the air has a constant density of $\rho_{a}=1.22 \mathrm{~kg} / \mathrm{m}^{3}$.

Penny Riley

Numerade Educator

Problem 73

The $12-\mathrm{Mg}$ jet airplane has a constant speed of $950 \mathrm{~km} / \mathrm{h}$ when it is flying along a horizontal straight line. Air enters the intake scoops $S$ at the rate of $50 \mathrm{~m}^{3} / \mathrm{s}$. If the engine burns fuel at the rate of $0.4 \mathrm{~kg} / \mathrm{s}$, and the gas (air and fuel) is exhausted relative to the plane with a speed of $450 \mathrm{~m} / \mathrm{s},$ determine the resultant drag force exerted on the plane by air resistance. Assume that air has a constant density of $1.22 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 74

The cart has a mass $M$ and is filled with water that has an initial mass $m_{0}$. If a pump ejects the water through a nozzle having a cross-sectional area $A$, at a constant rate of $v_{0}$ relative to the cart, determine the velocity of the cart as a function of time. What is the maximum speed of the cart, assuming all the water can be pumped out? The frictional resistance to forward motion is $F$. The density of the water is $\rho$.

James Kiss

Numerade Educator

Problem 75

The jet is traveling at a constant velocity of $400 \mathrm{~m} / \mathrm{s}$ in still air, while consuming fuel at the rate of $1.8 \mathrm{~kg} / \mathrm{s}$ and ejecting it at $1200 \mathrm{~m} / \mathrm{s}$ relative to the plane. If the engine consumes $1 \mathrm{~kg}$ of fuel for every $50 \mathrm{~kg}$ of air that passes through the engine, determine the thrust produced by the engine and the efficiency of the engine.

Penny Riley

Numerade Educator