Linear momentum equation problems

7/26/2019 Linear momentum equation problems www.engineering.purdue.edu.pdf

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Practice Problems on the Linear Momentum Equations

C. Wassgren, Purdue University Page 1 of 49 Last Updated: 2010 Sep 16

COLM_01

A frequently used hydraulic brake consists of a movable ram that displaces water from a slightly larger cylinder, asshown in the figure. The cross-sectional area of the cylinder isAcand the cross-sectional area of the ram isAr. Theram velocity, V, remains constant. Assume that the gap between the cylinder and the ram is much smaller than thedisplacement of the ramx.

a. Determine the water velocity as it leaves the cylinder.

b. Determine the force Fon the ram in terms ofAr,Ac, and V. Assume that the cylinder is initially full of water,that gravitational effects are negligible, and that the water exits the cylinder to atmospheric pressure.

Answer(s):1

out 1cr

c r r

V AA

V A A A

2 3

2 2c r

r c

c r

A AF V A AA A

x

V

FAc Ar

water filledcylinder


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COLM_02

A sonar buoy is being tested in a wind tunnel with a circular test section as shown in the figure. The air in the tunnelhas a density of 1.2 kg/m3and the test section radius isR = 0.5 m. Measurements at location A indicate that the

incoming velocity is uniform with a velocity of U= 10 m/s while the downstream velocity, measured at location B,is zero at the centerline and increases linearly with radius, r, measured from the tunnels centerline. A U-tubemanometer filled with water (with a density of 1000 kg/m3) is used to measure the pressure difference betweensections A and B (the pressure is assumed uniform at each section). The elevation difference between the two legsof the manometer isH=0.03 m as shown in the figure.

a. Determine the maximum velocity, Umax, at section B.b. Determine the drag acting on the buoy.

Answer(s):

3max 2U U

2

2 2 2

drag H O air 8F gH R U R

Fdrag = 220 N

water

Rr

U

Section A Section B

H


3/49



COLM_03

An incompressible, viscous fluid with density, , flows past a solid flat plate which has a depth, b, into the page.The flow initially has a uniform velocity U, before contacting the plate. After contact with the plate at a distancexdownstream from the leading edge, the flow velocity profile is altered due to the no-slip condition. The velocity

profile at locationxis estimated to have a parabolic shape, u=U((2y/)-(y/)2), foryand u=Uforywhere

is termed the boundary layer thickness.

1. Determine the upstream height from the plate, h, of a streamline which has a height, , at the downstreamdistancex. Express your answer in terms of .

2. Determine the force the plate exerts on the fluid over the distancex. Express your answer in terms of , U, b,and . You may assume that the pressure everywhere isp. The force the drag exerts on the plate is called theskin friction drag.

Answer(s):23

h

2215

F U b

xu=U

u=U,y

u=U((2y/)-(

y/2),y

h

streamline

plate has depth, b,into the page

y


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COLM_04

Wake surveys are made in the two-dimensional wake behind a cylindrical body which is externally supported in auniform stream of incompressible fluid approaching the cylinder with velocity, U.

The surveys are made atxlocations sufficiently far downstream of the body so that the pressure across the wake is

the same as the ambient pressure in the fluid far from the body. The surveys indicate that, to a first approximation,the velocity in the wake varies with lateral position,y, according to:

1 cos

A xu y

U U b x

where

1 1

2 2

y

b x

The quantitiesA(x) and b(x) are the centerline velocity defect and wake width, respectively, both of which vary withposition,x. If the drag on the body per unit distance normal to the plane of the sketch is denoted byDand the

density of the fluid by , find the relation for b(x) in terms ofA(x), U, , andD.

Answer(s):

2

2

2

2

Db x

A xU A x

U U

x

y

b(x)

U

U

U

u= U-A(x) cos[y/b(x)]wake


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COLM_05A hydraulic jump is a sudden increase in the depth of a liquid stream (which in this case we assume is flowing overa horizontal stream bed with atmospheric pressure air everywhere above the liquid):

The depth increases suddenly from h1to h2downstream of the jump. The jump itself is often turbulent and involvesviscous losses so that the total pressure downstream is less than that of the upstream flow.

a. Find the ratio of the depths, h2/h1, in terms of the upstream velocity, U1, the depth, h1, and g, the accelerationdue to gravity. Assume the flows upstream and downstream have uniform velocity parallel to the stream bedand that the shear stress between the liquid and the stream bed is zero. The liquid is incompressible.

b. What inequality on the value of U12/(gh1) must hold for a hydraulic jump like this to occur?

Answer(s):

212

1 1

21 1

2 4

Uhh gh

21

11

U

gh

U1

U2h1

h2

g

free surface


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COLM_06

A turbojet engine in a wind tunnel receives air at a velocity of Ui = 100 m/s and a density of i = 1 kg/m3. The

velocity is uniform and the cross-sectional area of the approaching stream which enters the engine isAi = 0.1 m2.

The velocity of the exhaust jet from the engine, however, is not uniform but has a velocity which varies over thecross-section according to:

uo= 2U(1-r2/R2)

where the constant U = 600 m/s andRis the radius of the jet cross-section. Radial position with the axi-symmetric

jet is denoted by r. The density of the exhaust jet is uniformly 0 = 0.5 kg/m3.

a. Determine the average velocity of the exhaust jet.b. Find the thrust of the turbojet engine.c. Find what the thrust would be if the exhaust jet had a uniform velocity, U.

Assume the pressures in both the inlet and exhaust jets are the same as the surrounding air and that mass isconserved in the flow through the engine (roughly true in practice).

Answer(s):

ou U

2 43

1i i ii

UT U AU

; T= 7000 N

2 1i i ii

UT U AU

; T= 5000 N

mounting stand

velocity, Ui,incoming area,Ai,

density, i

velocity, uo= 2U(1-r2/R2)

density, o

Rr


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COLM_07

A jet of water is deflected by a vane mounted on a cart. The water jet has an area,A, everywhere and is turned an

angle with respect to the horizontal. The pressure everywhere within the jet is atmospheric. The incoming jetvelocity with respect to the ground (axesXY) is Vjet. The cart has massM. Determine:a. the force components, Fxand Fy, required to hold the cart stationary,

b. the horizontal force component, Fx, if the cart moves to the right at the constant velocity, Vcart(Vcart


8/49



COLM_09

A fluid enters a horizontal, circular cross-sectioned, sudden contraction nozzle. At section 1, which has diameterD1, the velocity is uniformly distributed and equal to V1. The gage pressure at 1 isp1. The fluid exits into theatmosphere at section 2, with diameterD2. Determine the force in the bolts required to hold the contraction in place.

Neglect gravitational effects and assume that the fluid is inviscid.

Answer(s):22 2

2 1 1 1bolts 1 1,gage

2

14 4

D D DF V p

D

atmosphere

D1

V1

D2

p1

bolts


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COLM_10

Water is sprayed radially outward through 180as shown in the figure. The jet sheet is in the horizontal plane andhas thickness,H. If the jet volumetric flow rate is Q, determine the resultant horizontal anchoring force required tohold the nozzle stationary.

Answer(s):2

2xQ

F RHRH

H R

Q


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COLM_11

A variable mesh screen produces a linear and axi-symmetric velocity profile as shown in the figure. The staticpressure upstream and downstream of the screen arep1andp2respectively (and are uniformly distributed). If theflow upstream of the mesh is uniformly distributed and equal to V1, determine the force the mesh screen exerts onthe fluid. Assume that the pipe wall does not exert any force on the fluid.

Answer(s):

2 2 21 1 28F V R p p R

2R

Section 1 Section 2

V1

p1

p2

variable mesh

screen


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COLM_12

The tank shown rolls along a level track. Water received from a jet is retained in the tank. The tank is to acceleratefrom rest toward the right with constant acceleration, a. Neglect wind and rolling resistance. Find an algebraicexpression for the force (as a function of time) required to maintain the tank acceleration at constant a.

Answer(s):

2 210 2F V at A a M Vt at A

initial mass of cart and water,M0

U

A

V

F


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COLM_14

A cart with frictionless wheels holds a water tank, motor, pump, and nozzle. The cart is on horizontal ground andinitially still. At time zero the cart has a massM0and the pump is started to produce a jet of water with constant

areaAj, velocity Vjat an angle with respect to the horizontal. Find and solve the equations governing the mass andvelocity of the cart as a function of time.

Answer(s):

CV 0 j jM M V A t

0

cos ln 1 j j

j

V A tU V

M


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COLM_15

An incompressible, inviscid liquid flows steadily through a circular pipe of constant diameter,D. The pipe containsa section with porous wall of length,L. There, liquid is removed at constant radial velocity, v0, with no axialcomponent of velocity. The liquid speed and pressure at the entrance to the porous section are U0in the axialdirection andp0, respectively. Assume v0


14/49



COLM_17

A rocket cart that is initially at rest and has an initial mass ofM0is to be fired and is to have a constant acceleration

of a. To accomplish this, the exhaust gases will be deflected through an angle which varies as a function of time.The rocket exhausts gases with constant density through a constant area nozzle at a mass flow rate of m and aconstant speed, Ue, relative to the rocket.

a. Find an expression for cos() as a function of time t.b. Determine the time, tmin, at which cos() is a minimum.c. Find max.

Answer(s):

0cos

e

M mt a

mU

0min

Mt

m

max maxcos 02

Ue


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COLM_18

In an attempt to model the speed of a tsunami wave in the deep ocean, consider the propagation of a smallamplitude, solitary wave front moving with speed, c, from right to left as shown in the figure below. Neglect theeffects of surface tension. The liquid is initially at rest but after the wave passes by, the fluid behind the wave has asmall velocity, dV, in the same direction as the wave.

Derive an expression for the wave speed, c. You may neglect the shear forces the channel bed and the atmosphere

exert on the liquid. Hint: Consider choosing a steady frame of reference when analyzing the problem.

Answer(s):

c gh

c

g

dV

h


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COLM_19A flat plate of mass,M, is located between two equal and opposite jets of liquid as shown in the figure. At time t=0,the plate is set into motion. Its initial speed is U0to the right; subsequently its speed is a function of time, U(t). Themotion is without friction and parallel to the jet axes. The mass of liquid that adheres to the plate is negligiblecompared toM.

Obtain algebraic expressions (as functions of time for t>0) for:a. the velocity of the plate and

b. the acceleration of the plate.c. What is the maximum displacement of the plate from its original position?

Express all of your answers in terms of (a subset of) U0, V,A,,M, and t.

Answer(s):

0

4exp

U VAt

U M

04 4expU VAdU VAt

adt M M

0max

4

MUx VA

U(t)

AA

,V,V

plate with mass,M


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COLM_22

A spherical droplet of liquid, with density, d, moves horizontally (no gravity) through a stationary vapor cloud. Asit moves, additional liquid accumulates on the upstream side of the droplet at a mass rate, m , given by:

2dm R V

where is a dimensionless constant,Ris the droplet radius, and Vis the droplet velocity. In addition, there is a dragforce acting on the droplet given by:

2 212d vD c V R

where cd is the dimensionless drag coefficient (assumed to be a constant) and v is the surrounding vapor clouddensity (also assumed constant).

a. Determine the time rate of change of the droplet radius as a function of and V.b. Determine (but do not solve) the differential equation describing the time rate change of the drop radius in

terms of cd, , v, and d. Do not include Vin this differential equation.

Answer(s):

4

dRV

dt

22

22

13 1 0d v

d

cd R dR

R dtdt


18/49



COLM_23Two parallel plates of width, 2a, are separated by a gap of height, h. The upper plate approaches the lower plate at a

constant speed, V. The space between the plates is filled with a frictionless, incompressible gas of density,.Assume that the velocity is uniform across the gap width (ydirection) so that u=u(x, t).

Obtain algebraic expressions for:

a. the velocity distribution, u(x, t).b. the pressure distribution in the gap,p(x,t). The pressure outside of the gap is atmospheric pressure. Note:You do not need to use Bernoullis equation to solve this problem.

Answer(s):

xu V

h

2 2

atm

212

2p p a x

h hV

a

Vh

x

y

a

u(x, t)patm patm


19/49



COLM_24

A weir discharges into a channel of constant breadth as shown in the figure. It is observed that a region of still waterbacks up behind the jet to a height a. The velocity and height of the flow in the channel are given as Vand h,

respectively, and the density of the water is . You may assume that friction and the horizontal momentum of thefluid falling over the weir are negligible.

What is the height ain terms of the other parameters?

Answer(s):

21 2Fr a

h where Fr = V/(gh)1/2

ah V


20/49



COLM_25

A tank with a reentrant orifice called a Borda mouthpiece is shown in the figure. The reentrant orifice essentiallyeliminates flow along the tank walls, so the pressure there is nearly hydrostatic. Calculate the contractioncoefficient, Cc=Aj/A0. You may reasonably assume that the fluid is inviscid and incompressible.

Answer(s):

0

1

2

j

c

AC

A

A0

Aj

free surface


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COLM_26

A wedge with a vertex angle of 2is inserted into a jet of water of width, b, density, , and velocity, U, as shown inthe sketch. The angle of attack, , of the wedge is also defined in the sketch. After impinging on the wedge, thesingle incident jet is divided into two jets, both of which leave the back edges of the wedge with velocity, U. The

widths of the two departing jets are band (1-)bas indicated in the figure. It is assumed that the flow is planar,gravity may be neglected, and the pressure in the surrounding air is everywhere atmospheric.

a. Find the lift and drag on the wedge per unit length normal to the sketch as functions of , U, b, , , and .Note that drag and lift are defined as the forces on the wedge that are, respectively, parallel and perpendicularto the direction of the incident jet.

b. While holding the other parameters fixed, find the angle of attack, at which the lift is zero.c. If the wedge is moved in a direction perpendicular to the incident jet while, U, b, , and remain fixed then

will vary. There is one such position at which the lift is zero; what is the value ofat this position in termsof and ? If the wedge were free to move in such a way, would this position represent a position of stableor unstable equilibrium?

Answer(s):

2 1 cos 1 cosD U b

2 1 sin sinL U b

1

tan 2 1 tan

1 tan

12 tan

is an unstable equilibrium point

, U

b

b

b

U

U


22/49



COLM_27

Gravel is dumped from a hopper, at a rate of 650 kg/s, onto a moving belt. The gravel then passes off the end of thebelt as shown in the figure. The drive wheels are 80 cm in diameter and rotate clockwise at 150 rpm. Neglectingsystem friction and air drag, estimate the power required to drive this belt.

Answer(s):

2

12

W FV m D


23/49



COLM_28

A block of mass,M=10 kg, with rectangular cross-section is arranged to slide with negligible friction along ahorizontal plane. As shown in the sketch, the block is fastened to a spring that has stiffness such that F=kxwherek=500 N/m. The block is initially stationary. At time, t=0, a liquid jet begins to impinge on the block (the jet

properties are also shown in the sketch). For t>0, the block moves laterally with speed, U(t).a. Obtain a differential equation valid for t>0 that could be solved for U(t) andX(t). Do not solve.

b. State appropriate boundary conditions for the differential equation of part (a).

c. Evaluate the final displacement of the block.

Answer(s):22

20

d X A dX k V X

M dt Mdt

2

f

AX V

k

=1000 kg/m3V=30 m/s

A=100 mm2M

k

X


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COLM_32

A cart hangs from a wire as shown in the figure below. Attached to the cart is a scoop of width W(into the page)which is submerged into the water a depth, h, from the free surface. The scoop is used to fill the cart tank with water

of density, .

a. Show that at any instant V=V0M0/MwhereMis the mass of the cart and the fluid within the cart.b. Determine the velocity, V, as a function of time.

Answer(s):

00

MV V

M

0 0

0

1

21

V

V hWV t

M

V

stagnant water

h

scoop has width, W, into the page

cart has initial mass,M0and initial velocity, V0


25/49



COLM_34

Two water jets strike each other and merge into a single jet as shown in the figure. Determine the speed, V, and

direction, , of the resulting combined jet. Gravity is negligible.

Answer(s):2 2

1 1

2 2

2 2tan

V d

V d

2 2

2 23 2 2

1 1 2 2cos

V dV

V d V d

V1

d1

d2V2

V

V1= 10 ft/sV2= 15 ft/sd1= 0.1 ft

d2= 0.05 ft

90


26/49



COLM_35

Police attempt to control demonstrators by using a water cannon mounted on a moving truck. The cannon shootsa steady stream of water, with a diameter of 25 mm and speed of 10 m/s (relative to the truck). The truck moves atspeed of 3 m/s toward the demonstrators. Calculate the maximum force that could be exerted on a demonstrator bythe water stream assuming that the water jet is not deflected back toward the water cannon and the jet hits thedemonstrator perpendicularly. Describe the manner in which the force would vary if the stream were to impinge atan angle other than perpendicular.

Answer(s):

2

2 jetjet truck

4

DF V V

2

jetjet truck jet truckcos cos

4

DF V V V V


27/49



COLM_36

Liquid falls steadily and vertically into a short horizontal rectangular open channel of width b. The total volumeflow rate, Q, is distributed uniformly over area bL. Neglect viscous effects.a. Obtain an expression for h1in terms of h2, Q, and b.

b. Plot the dimensionless surface profile, h/h1, as a function of the dimensionless position,x/L, for a variety ofparameters.

Answer(s):

22

1 2 2

2

2Qh h

gh b

32

2

2

1 Fr

dh

dx x h

h x

wherex=x/L, h= h/h1, and Fr2= gb2h1

3/Q2(this is a Froude number).

h1 h2

L

Qy

x

Q


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COLM_39

A stream of incompressible liquid moving at low speed leaves a nozzle pointed directly upward. Assume the speedat any cross section is uniform and neglect viscous effects. The speed and area of the jet at the nozzle exit are V0andA0, respectively. Apply conservation of mass and the momentum equation to a differential control volume oflength dzin the flow direction. Derive expressions for the variations of jet speed and area as functions ofz.Evaluate the vertical distance required to reduce the jet speed to zero. (Take the origin of coordinates at the nozzleexit.)

Answer(s):

2

0 2V V gz ; zV=0= V02/(2g)

0 0

2

02

V AA

V gz


29/49



COLM_40

Incompressible fluid of negligible viscosity is pumped, at total volume flow rate Q, through a porous surface into thesmall gap between closely spaced parallel plates as shown. The fluid has only horizontal motion in the gap.Assume uniform flow across any vertical section. Obtain an expression for the pressure variation as a function ofx.

Answer(s):

2

atm

2

1

4

p p x

LQ hw

Q

x V(x)

L

Assume a depth winto the page.h


30/49



COLM_42

Flying Elvi, 20 of them in all, jump out of an airplane at a rate of one Elvis, weighing 255 lbm, every 5 seconds. If

the airplane is flying horizontally at a velocity of 120 mph and tries to accelerate at a rate of 1 ft/s2, determine the

change in the thrust that must be supplied by the airplane propellers as a function of time until all of the Elvi haveleft the building plane.

Assume that there is a drag force acting on the plane that can be modeled by FD= -kV2where k= 0.2 lbf/(ft

2/s2) and

Vis the velocity of the plane relative to the air. The air density at an altitude of 2.5 miles is approximately 0.61times the air density at sea level. You may assume that the mass rate at which fuel is burned is very small incomparison to the mass rate at which Elvi leave the plane. The plane weight at altitude not counting Elvi is 30000

lbm.

Answer(s):

0dV

T M mt dt

T = 1090 lbf (1.6 lbf/s) t (0 t100 s since there are 20 Elvi)

120 mph

altitude = 2.5 miles

Elvis weight = 255 lbm


31/49



COLM_43

A model solid propellant rocket has a mass of 69.6 gm, of which 12.5 gm is fuel. The rocket produces 1.3 lbfofthrust for a duration of 1.7 sec. For these conditions, calculate the maximum speed and height attainable in theabsence of air resistance. Plot the rocket speed and the distance traveled as functions of time.

Answer(s):

Umax= U(t = t= 1.7 sec) = 139.2 m/s (h(t= t) = 114 m)hmax= h(t= tm= 15.9 sec) = 1100 m


32/49



COLM_46

A jet of water sprays into a container as shown in the figure. The water jet is deflected as it enters the container sothat all of the water that enters the container remains in the container. The initial mass of the container and the waterwithin it isM0. The velocity of the container is given by U.

a. Determine the velocity, V2, of the fluid just as it enters the container in terms of V1, g, and h.

For the remaining questions, express your answers in terms of V1, V2, U,A1, , g, and h.

b. Determine the mass flow rate of the water entering the container.c. Determine the equation for the mass,M, of the container plus the water within the container at time t. You do

not need to solve any integrals you might have.d. Write out (but do not solve) the differential equation describing the container acceleration, dU/dt. You do not

need to substitute your result from part (c) into this differential equation, i.e.leave terms involving mass interms ofM.

e. Solve (numerically) for the container height, h, as a function of time, t, for the following conditions:M0= 1 kgV1= 10 m/sA1= 4.0*10

-4m2

= 1000 kg/m3

g= 9.81 m/s2h0= 1 mU0= 1 m/s

Answer(s):

2

2 1 2V V gh

2 1in 1 12

1

22

Vm V gh U A

V gh

; 2 1CV 0 1 12

0 1

22

t t

t

VM M V gh U A dt

V gh

2

2 2

CV

V U AdUg

dt M

g

V1A1

h

V2

U


33/49



COLM_48

a. Construct from first principles an equation for conservation of mass governing the planar flow (in thexyplane)of an incompressible liquid lying on a flat horizontal plane. The depth, h(x,t), is a function of position,x, andtime, t. Assume that the velocity of the fluid in the positivex-direction, u(x,t), is independent ofy.

b. Now apply conservation of linear momentum in thex-direction using the same control volume. Assume that theground does not exert a shear stress on the fluid.

The resulting partial differential equations are part of what is known as Shallow Water Wave Theory.

Answer(s):

0uhh

t x

2u huh hgh

t x x

or

u u hu g

t x x

xy

h(x,t)

u(x,t)

free surface

liquid

g


34/49



COLM_51

A cart travels at velocity, U, toward a liquid jet that has a velocity, V, relative to the ground, a density, , and aconstant area,A. The mass of the cart and its contents at time t= 0 isM0and the carts initial velocity is U0towardthe jet. The resistance between the carts wheels and the surface is negligible.

a. Determine the mass flow rate into the cart in terms of (a subset of),A, V, U, g, and .

b. Determine the acceleration of the cart, dU/dt, in terms of,A, V, U, g, , andM(t) whereM(t) is the mass ofthe cart and water at time t. You neednt solve any integrals or differential equations that appear in youranswer.

Answer(s):

into relcart

CS

m d U V A u A

2

U V AdU

dt M

V,A,

U

g


35/49



COLM_52

A rocket powered sled travels with velocity, U(with respect to the ground), up an incline that is at an angle, withrespect to the horizontal. The rocket exhaust, directed in the horizontal direction, has a constant velocity, V(withrespect to the rocket), and a mass flow rate, m , that varies with time, t, according to:

0 1

0

tm t T

m T

t T

where0m is a constant and Tis the time at which all of the fuel has been expended.

Assuming that the initial mass of the rocket sled and fuel isM0and the initial rocket velocity up the slope is zero,determine:

a. the mass of the rocket sled as a function of time,b. the cart acceleration in theXdirection (shown in the figure) for all times, andc. the cart velocity in theXdirection for all times (you neednt solve any integrals that occur),

d. Determine the time (tT) at which the cart velocity will be zero.

You may neglect aerodynamic drag and rolling resistance.

Answer(s):

2

0 0

10 02

1

2

tM m t t T

M t T

M m T t T

02

0 0

1 cossin

1

2

sin

tm VT

g t TdU t

M m tdt T

g t T

0

0

1sin cos ln 1 1

2

sin

m t tg t V t T

M TU t

U T g t T t T

0

sinU

U Tt T

g

,V mg

U

X


36/49



COLM_54

a. Using an integral approach, write the differential equation governing the motion of an inviscid,

incompressible fluid (with density ) oscillating within the U-tube manometer shown. The manometer cross-sectional area isA.

b. What is the natural frequency of the fluid motion?c. What are the implications of this result for making time-varying pressure measurements using a manometer?

Answer(s):2

2

20

d z gz

Ldt

0

2 2sin cos

2

L g gz V t z t

g L L

;

2g

L

Assume this distance is negligible

compared to h1+h2.

h2

h1

g

incompressible, inviscid fluid

with density

tube ends are open to theatmosphere


37/49



COLM_55

The toy boat (shown below) travels at a constant speed, Vboat. The boat is propelled using a compressed air tank that

issues air at a pressure of 1 atm, temperature of 15 C, velocity of 340 m/s, and diameter 5 mm. Although air drag isnegligible, the hull drag due to the water is significant and varies as kVboat

2where k15 Ns2/m2. Determine thespeed of the boat.

Answer(s):

2jet jet2

jetjet

boat

4

p DV

RTV

k

Vboat

compressed air jet


38/49



COLM_56

Air leaves a nozzle with a diameter of 15 mm and strikes the center of a vertically oriented circular plate with adiameter of 50 mm. The force required to move the plate toward the jet at a constant speed of 0.5 m/s is 10 N.Determine the pressure in the air supply pipe, which has a diameter of 20 mm.

Answer(s):

ppipe = 120 kPa (abs) = 19.3 kPa (gage)

pressure = ?

pipe diameter = 20 mm

exit diameter = 15 mm

plate diameter = 50 mm

force = 10 N

plate velocity = 0.5 m/s


39/49



COLM_57

Consider the four carts shown below. Each of the carts rests on a frictionless surface, is initially stationary, and isrestricted to move only in the horizontal direction. The pressure surrounding the cart is atmospheric and the flow issteady and incompressible. In what direction will each device move when it is released? You must provide supportfor your answers.

Answer(s):---

12

43


40/49



COLM_59

A bucket of water (with radius,R, and initial mass,M0) is filled by the falling water jet shown in the figure. Thewater jet falls under the action of gravity from a large tank of depth, s, and exit area,AE. The bucket can slide

vertically within the tube on a thin lubricating layer of fluid with thickness, t, and dynamic viscosity, . Thepressure everywhere is atmospheric.

a. Determine the velocity, Vj, of the water jet (relative to the ground) just before it enters the bucket. Assumethat the bucket at this instant in time is located a distance hbelow the free surface of the large tank. Expressyour answer in terms of the elevation, h, and other pertinent parameters.

b. Determine the mass flow rate of water entering the bucket, into bucketm , if the bucket speed is Urelative to the

ground. Express your answer in terms of the bucket velocity, U, the tank exit area,AE, the elevations sand h,and other pertinent parameters.

c. Determine the mass of the bucket with water,M, at time tassuming the initial bucket/water mass isM0. You

need not solve any integrals that appear in your derivation. You may leave your answer in terms of into bucketm

and other pertinent parameters.d. Determine the total force (body and surface) acting on the bucket with water if the bucket with water has

velocity Uand massM.e. Derive an expression for the bucket/water acceleration in terms of the bucket velocity, U, bucket mass,M, jet

velocity, Vj, mass flow rate, into bucketm , and other pertinent parameters.

Answer(s):

2jV gh

intobucket

2 Es

m gh U Ah

0 intobucket0

t t

t

M M m dt

total,2

z

UF Mg RL

t

intobucket

2j

UMg RL m V U

dU t

dt M

hg

U

bucket with mass,M, radius,R, and length,L

large tank with exit area,AEs

Vj

viscous fluid with dynamic

viscosity, , and thickness, t

L

Z


41/49



COLM_60

A railroad freight car, with an empty mass ofM0and a cross-sectional opening area ofA, is accidentally released

down an incline of angle with respect to the horizontal. As luck would have it, a steady rain is falling at a terminal

velocity V(with respect to the ground) and density. The rain enters the car as it rolls downhill. You may neglectdrag forces and friction acting on the car.

a. Determine, in vector form, the velocity of the rain relative to the car using a clearly defined frame of

reference.b. Determine the opening area vector using the same frame of reference used in part (a).c. Determine the mass flow rate of water enteringthe car.d. Determine the mass of the cart and water,M, as a function of time.

e. Determine the acceleration of the car down the incline in terms ofM, g, A, V,, and U.

Answer(s):

rel, sin cosxyz V U V u i j (using a coordinate system fixed to the cart and oriented so thatxpoints

downhill)

xyz AA j

rel

CScosd AV u A

0 cosM t M VAt

sin cossin

V U VAdUg

dt M

g

U

A

V,


42/49



COLM_61

Consider a viscous, incompressible, Newtonian, laminar liquid film of depth h, density, , and constant dynamicviscosity, ,flowing steadily under the influence of gravity down an inclined surface as shown in the figure below.Assume that the liquids velocity profile, u(y), is fully developed meaning that it is not a function of distance downthe incline (thexdirection).

a. Write a differential equation for the fluids velocity gradient, du/dy, using an analysis based on a differentialcontrol volume of thickness, dy, length,L, and depth into the page of W, as indicated in the figure. (Do notuse the Navier-Stokes equations for this problem.)

b. Write the appropriate boundary conditions for solving this differential equation.

c. Solve for the velocity profile, u(y).

Answer(s):2

2sin

d ug

dy

0 0u y

0du

y hdy

Re

Fr

1sin 2

2

h ghu y y

h hgh

y

x

u(y)

g

hdy

L Assume that theflow has a depth, W,into the page.


43/49



COLM_64

A cart of empty mass,M0, planform (i.e.cross-sectional) area,AC, and horizontal speed, U, moves on a thin viscous

layer of liquid with dynamic viscosity, , density,, and layer thickness, h. A horizontal liquid jet with velocity V1

(with respect to the ground), density,1, and areaA1, is directed into the cart. In addition, the cart moves through a

continuous liquid spray, with density,2, that has a downward vertical speed, V2(with respect to the ground).

a. Determine the rate at which the mass inside the cart is changing with time in terms ofM0,1, V1,A1,2, V2,

AC, U, h, ,, g, or a subset of these variables. You need not evaluate any integrals.

b. Determine the acceleration of the cart in terms ofM0,1, V1,A1,2, V2,AC, U, h, ,, g, or a subset of thesevariables. You need not evaluate any integrals.

Answer(s):

CV 1 1 1 2 2 CdM

V U A V Adt

2

1 1 1 2 2

CV

C C

UV U A V UA A

dU h

dt M

U

h viscous Newtonian fluid with

constant dynamic viscosity, anddensity,

1,V1A1

2,V2

AC

g


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COLM_65

Two long trains carrying coal are traveling in the same direction side by side on separate tracks. One train ismoving at 40 ft/s (train 1) and the other at 50 ft/s (train 2). In each coal car a man is shoveling coal and pitching itacross to the neighboring train. The rate of coal transfer from train 1 to train 2 is 8000 lbm/min for each 100 ft oftrain length, and from train 2 to train 1 is 6000 lbm/min for each 100 ft of train length. Find the extra force that must

be exerted on each train (per unit length) in order to maintain constant train speeds.

Answer(s):F1= -0.311 lbf/ft and F2= 0.414 lbf/ft


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COLM_68

Consider a cart, with mass,M0, and an open top of area,A, rolling along a flat surface as shown in the figure below.

Rain, with a density,R, and falling vertically with speed, VR, with respect to the ground, is collected within the cart

through the carts open top. The water within the cart, with density,, discharges at an angle, , with respect to thehorizontal through a hole of area,Aout, located near the bottom of the cart. Note that the mass of water within thecart does not remain constant.

Neglecting drag forces and rolling friction, determine:a. the rateat which water mass accumulates within the cart, i.e.dMH2O/dt, as a function of the current mass of

water within the cart,MH2O, and a subset of the following parameters: R, VR, U,A,,Aout, ,M0, and g. You

do not need to solve any differential equations you derive. Hint: You may need to use Bernoullis equation inthis derivation.

b. the acceleration of the cart, dU/dt, in terms of a subset of the following parameters: R, VR, U,A,,Aout, ,M0,MH2O, and g. You do not need to solve any differential equations you derive.

Answer(s):

2

2out

2H OH O R R

dM gA M V A

dt A

out2 cosCV R R CV AdU

M UV A gMdt A

rain with density,R, and verticalspeedVR, with respect to the ground

cross-section area,A

cart velocity, U, with

respect to the ground

discharge hole with

area,Aout

g


46/49



COLM_71

Water flows steadily through a reducing pipe bend as shown in the figure. The inlet and outlet conditions areindicated on the diagram. Neglecting the weight of the bend and water, estimate the total force exerted by the flange

bolts.

Answer(s):2 22 2

2 1 1 2 1bolts 1 1,gage 2,gage

2 1

1

4 4

D D D DF V p p

D D

D1= 25 cmp1= 350 kPa (abs)

V1= 2.2 m/s

1

D2= 8 cmp1= 120 kPa (abs)

2

patm= 100 kPa (abs)


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COLM_72

A cart, with an empty massM0and depth binto the page, travels up an incline (angle with respect to the

horizontal). As the cart moves up the incline, it scoops a downward flow of water, with density, into the cartinterior as shown in the figure. The downward flow of water has a velocity profile approximated by:

Yu U

h

where Uis the velocity at the free surface (its constant) and his the depth of the water layer (also constant).

a. Determine the mass flux out of the cart in terms of (a subset of), Vcart, U, h, b, and g.b. Determine thex-momentum flux out of the control volume using thex-yframe of reference attached to the

control volume in terms of (a subset of), Vcart, U, h, b, , and g.

c. Determine the net force acting in thex-direction on the control volume in terms of (a subset of), Vcart,MCV, U,

h, b, , and g, whereMCVis the instantaneous mass in the control volume.

d. Determine the acceleration of the cart in terms of (a subset of), Vcart,MCV, U, h, b, , and g.

Answer(s):

1out cart2m U V hb

2 21rel cart cart3CS

xu d bh U V U V u A 21

net, 2sin cosx CVF M g g h b

2 2 21 1 cart cart2 3cart sin sinCVCV

M g g h b bh U V U VdV

dt M

YX

g

Vcart (rel. to ground)

Yu U

h

h

scoop has depth binto the page

x

y


48/49



COLM_73

A cart is propelled by water jet as shown in the figure below. The cart is filled with water to a height, h, has aninitial water mass ofM0,a cart mass,Mcart, a cross-sectional area,A, and a (variable) velocity, V, with respect to theground. The water within the cart leaks through a small hole of area,Aexit, located in the bottom of the cart. Thehole area is much smaller than the cross-sectional area of the cart, i.e.Aexit


49/49


COLM_74

A pump in a tank of water (with a density of 1000 kg/m3and dynamic viscosity 1.0*10-3kg/(ms)) directs a jet at 10m/s (relative to the cart) and with a volumetric flow rate of 1 m3/s against a vane as shown in the figure. At a giveninstant in time, the cart speed is U= 10 m/s and the mass of the cart and the water contained within it is 200 kg.Determine the acceleration of the cart at this instant in time if the jet follows:a. path A, or

b. path B.

Answer(s):

0dU

dt

cos

CV

dU QV

dt M

P

path A

path B

7060g

U

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Linear momentum equation problems