PROBLEMS

1. The 200-kg lunar lander is descending onto the moon’s surface with a

velocity of 6 m/s when its retro-engine is fired. If the engine produces a

thrust T for 4 s which varies with the time as shown and then cuts off,

calculate the velocity of the lander when t = 5 s, assuming that it has not yet

landed. Gravitational acceleration at the moon’s surface is 1.62 m/s2. (3/184)

SOLUTION

smv

v

v

vmg

mvmvFdt

/ 1.2

9.36

620016008001620

6200)800(221

)800(2)5(

2

2

2

2

12

mg

motion

T

m = 200-kg, v1 = 6 m/s, v2 = ? when t = 5 s, gmoon = 1.62 m/s2

PROBLEMS

2. The hydraulic braking system for the truck and trailer is set to produce

equal braking forces for the two units. If the brakes are applied uniformly for

5 seconds to bring the rig to a stop from a speed of 30 km/h down the 10

percent grade, determine the force P in the coupling between the trailer and

the truck. The mass of the truck is 10 Mg and that of the trailer is 7.5 Mg.

(3/199)

PROBLEMS The hydraulic braking system for the truck and trailer is set to produce equal braking forces for the two units, brakes are applied uniformly for 5 seconds to bring the rig to a stop from a speed of 30 km/h down the 10 percent grade, determine the force P in the coupling between the trailer and the truck. mtruck = 10 Mg, mtrailer = 7.5 Mg.

Wtruck

Fb

Wtrailer

Ntruck

Ntrailer Fb

71.5,1.0101

tan

NFF

F

vvmdtF

bb

b

x

23120,23120010

33.80105.710

5271.5sin)81.9(105.7103

3

12

smhkm /33.8/30 FBD of whole system

)(3.3

33000,165205

33.80105.7

52312071.5sin)81.9(105.73

3

12

tensionkNP

NPP

P

vvmdtFx

FBD of trailer

P

Wtrailer

Ntrailer Fb

PROBLEMS

3. The 140 gram projectile is fired with a velocity of 600 m/s and picks up

three washers, each with a mass of 100 gram. Find the common velocity v of

the projectile and washers. Determine also the loss |DE| of energy during the

interaction. (3/192)

PROBLEMS mp = 140 g, v1 = 600 m/s, mw = 100 g. Find the common velocity v of the projectile and washers. Determine the loss |DE| of energy during the interaction.

kJE

J

vmvmTTVVTE

smv

v

mvmvGGFdt

eg

t

18.17

74.17181)600(140.021

)91.190(440.021

21

21

/91.190

600140.0)100.0(3140.00

22

2

11

2

2212

00

2

2

1212

0

0

D

DDDD

PROBLEMS

4. A tennis player strikes the tennis ball with her racket while the ball is still

rising. The ball speed before impact with the racket is v1=15 m/s and after

impact its speed is v2=22 m/s, with directions as shown in the figure. If the

60-g ball is in contact with the racket for 0.05 s, determine the magnitude of

the average force R exerted by the racket on the ball. Find the angle b made

by R with the horizontal. Comment on the treatment of the ball weight during

impact. (3/211)

68.8tan02.43

49.6325.005.0

10sin1506.020sin2206.0)81.9(06.0

53.42127.205.0

10cos1506.020cos2206.0

05.0

0

05.0

0

10 2

05.0

0

10 2

bb

x

y

yy

y

yt

yy

xx

x

xt

xx

R

RNR

NRR

ttR

mvmvdtF

NRR

tR

mvmvdtF

SOLUTION

Rx

Ry R b

1v

2v

10°

20°

xv1

yv1

xv2

yv2

Rx

Ry R

W=0.59 N

in x direction

in y direction

v1 = 15 m/s, v2 = 22 m/s, m = 60-g, t = 0.05 s, determine R exerted by the racket on the ball and b made by R with the horizontal. Comment on the treatment of the ball weight during impact.

PROBLEMS

5. The 40-kg boy has taken a running jump from the upper surface and lands

on his 5-kg skateboard with a velocity of 5 m/s in the plane of the figure as

shown. If his impact with the skateboard has a time duration of 0.05 s,

determine the final speed v along the horizontal surface and the total normal

force N exerted by the surface on the skateboard wheels during the impact.

s/m.vvcos

vmmvmvm SBSxSBxB

853540030540

Linear momentum is conserved in x-direction

kNNorNN

N

dtgmmNvmvm SBSySByB

44.22440

005.081.94505.0030sin540

0

05.0

0

SOLUTION mB = 40-kg, mS = 5-kg, v1 = 5 m/s, t = 0.05 s, v2 = ?, determine the normal force N exerted by the surface on the skateboard wheels during the impact.

(mB+mS)g

N

y

x

PROBLEMS

6. The simple pendulum A of mass mA and length l is suspended from

the trolley B of mass mB. If the system is released from rest at = 0,

determine the velocity vB of the trolley when = 90°. Friction is

negligible. (3/213)

PROBLEMS pendulum of mass mA , length l suspended from trolley B of mass mB. System released from rest at = 0, determine velocity vB of trolley when = 90°.

x

y

x

lBv

Bv

= 0

= 90°

W=mg

T

A

BABBBBA

x

m

mmvlvmvlm

vvmdtF

,0

012

B

AB

AB

BAB

AB

ABB

BA

BAB

A

BABA

ABBBA

g

m

m

gl

m

mv

mmm

glmv

glmvm

vm

mmv

m

mmvm

glmvmvlm

VTU

DD

1

2

2

021

221

021

21

0

22

2

22

2

2

22

21

PROBLEM

7. A small 110 gram particle is

projected with a horizontal

velocity of 2 m/s into the top A of

the smooth circular guide fixed in

the vertical plane. Calculate the

time rate of change of angular

momentum about point B when the

partcle passes the bottom of the

guide at C. (3/225)

BH

PROBLEM m = 110 g, vA = 2 m/s, Calculate the time rate of change of angular momentum about point B when the partcle passes the bottom of the guide at C.

BH

= vA

vC

W=0.110(9.81)

t

n

N mNkjiMH

NN

N

smv

aa

maF

BB

W

Cny

yy

519.1079.116.725.0

16.7

)24.55(110.0)81.9(110.0

/24.55250.081.13

079.1

2

2

22

222

22

)/(81.13

)5.0()81.9(222

21

21

smv

ghvv

vmghmvm

TUT

C

CAAC

CCAA

CCAA

PROBLEM

8. A small 0.1 kg particle is given a speed of 2 m/s at point A on the

horizontal x-y plane and is guided by the fixed curved rail. Friction is

negligible. As the particle crosses the y-axis at A, its velocity is in the x-

direction and as it crosses the x-axis at point B, its velocity makes a 60°

angle with the x-axis. The radius of curvature of the path at B is 500 mm.

Determine the time rate of change of the angular momentum HO of the

particle about the z-axis through O at both A and B. (3/230)

PROBLEM m = 0.1 kg, vA = 2 m/s (//x), B = 500 mm. Determine the time rate of change of the angular momentum HO of the particle about the z-axis through O at both A and B.

A

B

x

y

ANsmv /2

mm200

mm300

smv /2

BN

60

C

mmb 150

O 30

mm500

mNbNHso

Nv

mNmaFfrom

directionzclockwiseinbNMBAt

HsoMAAt

HM

BOz

Bnn

BOz

OzOz

OzOz

12.0)150.0(8.0

8.05.0

21.0

),(

00

22

PROBLEM

9. A pendulum consists of two 3.2 kg

concentrated masses positioned as

shown on a light but rigid bar. The

pendulum is swinging through the

vertical position with a clockwise

angular velocity w = 6 rad/s when a 50-g

bullet traveling with velocity v=300 m/s

in the direction shown strikes the lower

mass and becomes embedded in it.

Calculate the angular velocity w which

the pendulum has immediately after

impact and find the maximum deflection

of the pendulum. (3/235)

Angular momentum is conserved during impact;

)( / 77.2

64.0cossin512.0cossin128.0798.1

formscalar in or

sin512.0cos512.0

sin128.0cos128.0638.5072.3768.0

)sin4.0cos4.0(25.3sin4.0cos4.0

)sin2.0cos2.0(2.3sin2.0cos2.0

)20sin30020cos300(05.04.0

)6)(4.0(2.34.0)6)(2.0(2.32.0

0

, 0

1

22

1

22

22

22

798.1

21

0 2112

ccwsrad

kk

kkkkk

jiij

jiij

jij

ijij

rvvmrvmr

M

HHHHdtM

k

O

OOOO

t

O

1 2

2

1

w

w´

v1

v1´

v2´

v2

O

SOLUTION mmass = 3.2 kg, w = 6 rad/s (cw), mbullet = 50 g, v=300 m/s. Calculate angular velocity w which the pendulum has immediately after impact and find the maximum deflection of the pendulum.

Energy considerations after impact;

0

21 eg VVTU DDD

(Datum at O)

o

gΔV

JT

T

TTT

1.52

605.0cos,0)cos1(47.6558.2

)cos1(47.6

)4.0cos4.0)(81.9(25.3)2.0cos2.0)(81.9(2.3

558.277.22.02.321

77.24.02.305.021

)deflection maximum( 0

22

1

2

12

D

1 2

2

1

w

w´

v1

v1´

v2´

v2

O

SOLUTION