KINETICS OF RIGID BODIES

PROBLEMS

PROBLEMS

1. The 6 kg frame AC and the 4 kg uniform slender bar AB of length l

slide with negligible friction along the fixed horizontal bar under the

action of the 80 N force. Calculate the tension T in wire BC and the x and

y components of the force exerted on the bar by the pin at A. The x-y

plane is vertical. (6/13)

PROBLEMS mAC = 6 kg, mAB = 4 kg, calculate the tension T in wire BC and the x and y components of the force exerted on the bar by the pin at A.

rectilinear translation

0 GMamF

for the whole system mAC g = 6(9.81) = 67.97 N

mAB g = 4(9.81)

= 39.24 N

80 N

FBD

a

KD

Ξ

amAC

amAB

2/8,)46(80 smaa

amFx

bar AB

FBD KD

Ξ

mAB g = 39.24 N

T

60°

Ax

Ay

G a G amAB

NAA

amF

NTNAA

llA

ldamM

TA

TAF

xx

x

yy

yC

y

yy

34.18,)8(460cos33.27

33.27,57.15,86.1343.29

60sin2

)8(44

324.39

60sin24.39

060sin24.39,0

2/l 2/l

2/l

2/l

dl 60sin2/

+

G

PROBLEMS

2. The block A and attached rod

have a combined mass of 60 kg and

are confined to move along the 60°

guide under the action of the 800

N applied force. The uniform

horizontal rod has a mass of 20 kg

and is welded to the block at B.

Friction in the guide is negligible.

Compute the bending moment M

exerted by the weld on the rod at

B. (6/22)

SOLUTION

x

W=60(9.81) N

60o

N

FBD Kinetic Diagram

x

2/84.4

6060sin)81.9(60800

sma

amaF

x

xxx

FBD of rod

Bx

By

M

W1=20(9.81) N

KD of rod

mTax=60ax

m1ax=20ax 60°

mNM

MdmaM xB

196

)7.0)(60sin84.4)(20(7.0)81.9(20+

mtotal=60 kg, mrod= 20 kg, compute the bending moment M exerted by the weld on the rod at B.

PROBLEMS

3. The uniform 100 kg log is supported by the two cables and used as a

battering ram. If the log is released from rest in the position shown, calculate

the initial tension induced in each cable immediately after release and the

corresponding angular acceleration a of the cables.

2/45.22

905.4sradrat aa

SOLUTION

W=100(9.81) N

FBD KD

TA TB

nam

tam

+n

+t

+n

+t

When it starts to move, v=0, w = 0 but a ≠ 0 02 ran w

2/905.430sin

57.849030cos0

smaammgamF

TTmgTTF

tttt

BABAn

NTNT

TTTTM

BA

BABAG

17.63739.212

30)5.0(60sin)5.1(60sin0

The motion of the log is curvilinear translation.

*

*

1.5 m 0.5 m

G G

m=100 kg, log released from rest, calculate the initial tension in each cable and corresponding angular acceleration a of the cables.

+

PROBLEMS

4. The parallelogram linkage is used to transfer crates from platform A to

platform B and is hydraulically operated. The oil pressure in the cylinder is

programmed to provide a smooth transition of motion from q = 0 to q = q0 =

p/3 rad given by where t is in seconds. Determine the force at

2cos1

6

tppq

D on the pin when

t = 1 s. The crate

and platform have

a combined mass

of 200 kg with

mass center at G.

The mass of each

link is small and

may be neglected.

(6/29)

PROBLEMS q = 0 to q = q0 = p/3 rad, , m total=200 kg, determine the force at D for t = 1 s.

2cos1

6

tppq

)(2178

)30sin48.030cos6.0(6.162)30cos6.0()6.0(1962

/6.162)144/)(2.1(200,0

0,/12

,6

,1

2cos

24,

2sin

12,

2cos1

6

2.1

242

2

32

ncompressioNF

F

damM

smmrama

sradradst

ttt

mCDr

D

D

nF

nt

pq

qpqpq

ppq

ppq

ppq

m g = 200(9.81) = 1962 N

na

0ta

Fn

Ft

FD

FBD KD

namΞ

+

q =30°

q =30°

q =30°

PROBLEMS

5. The spring is uncompressed when the uniform slender bar is in the vertical

position shown. Determine the initial angular acceleration a of the bar when it is

released from rest in a position where the bar has been rotated 30° clockwise from

the position shown. Neglect any sag of the spring, whose mass is negligible.

PROBLEMS Spring uncompressed when uniform slender bar in vertical position, determine initial angular acceleration a of bar when released from rest in a position where the bar has been rotated 30° clockwise from the position shown.

Unstrecthed length of the spring: llllo2

5)4/2( 22

When q=30o , length of the spring: llspring2

3

When q=30o , spring force:

2

3

2

5

2

3

2

5klllkFspring

(in compression)

l

g

mk

lamml

lF

lmg

damIM

l

tspring

tO

857.0864.0

4121

2460cos

4

2

a

a

a

aW

O

G +n

+t

On

Ot

30o

30o

l

Fspring

60o

.

lspring

G +n

+t

04

22 l

mmram n ww

tam

aI 60o

FBD KD

+

6. The 65 kg thin rod is held by cables AB and AC. If cable AC

suddenly breaks loose determine the initial angular acceleration of

the rod and the tension in cable AB.

PROBLEMS

C

40 cm

A

B 40 cm

30 cm

x

y

m = 65 kg, cable AC suddenly breaks loose, determine the initial angular acceleration of the rod and the tension in cable AB. PROBLEMS

jjijaia

jika

rraaaa

jiijka

rraaa

CBABABGyGx

CBCBBG

BGCB

restfromstarts

BGCBCBBGBGBG

ABABABB

ABAB

restfromstarts

ABABABABAB

aaa

aa

aww

aaa

aww

4.04.03.0

4.04.0

,

4.03.04.03.0

/

/

)(0

///

/

)(0

//

C

A

B

G

tBGa /

CBa

ABa

0/ nBGa

ABABB rat

a/

0/ ABABB ran

w

t

t

n

n

222 47.3)8.0(6512

1

12

1

4.04.03.0

kgmmlI

ajai CBABGyABGx

aaa

C 40 cm

A

B 40 cm

30 cm

x

y

m = 65 kg, cable AC suddenly breaks loose, determine the initial angular acceleration of the rod and the tension in cable AB. PROBLEMS

C 40 cm

A

B 40 cm

30 cm

x

y

G

T

mg ya

xaΞ

C 40 cm

A

B 40 cm

30 cm

x

y

G

yam

xamCBIaCBa

FBD KD

22 /71.12,/53.7,6.183

65.637473.3,867.1066.165.6366.0

262665.6366.0

4.04.065)81.9(655

3

041.0,5.198.0,3.0655

4

0692.0,47.3)4.0(5

3

sradsradNT

TTTT

T

TamF

TTTamF

TTIM

CBAB

CBAB

CBAByy

ABABABxx

CBCBCBG

aa

aa

aa

aaa

aaa+

PROBLEMS

7. Crank AB rotates with an angular velocity of wAB = 6 rad/s and angular acceleration

of aAB=2 rad/s2. Roller C can slide along the circular slot within the fixed plate. For

the position shown, angular velocity and angular acceleration of rod BC are wBC =2.9

rad/s (counterclockwise) and aBC=37.47 rad/s2 (counterclockwise) The masses of

uniform bars AB and BC are mAB=2 kg and mBC=5 kg. Mass of the roller C and friction

can be neglected. Determine the reactions by the pins B and C.

wAB = 6 rad/s aAB=2 rad/s2

mAB=2 kg

wBC =2.9 rad/s (ccw) , aBC=37.47 rad/s2 (ccw) mBC=5 kg

A

B

C

37o 45o

LBC=500 mm LAB=300 mm

r =150 mm

PROBLEMS

wAB = 6 rad/s aAB=2 rad/s2

mAB=2 kg

wBC =2.9 rad/s (ccw) , aBC=37.47 rad/s2 (ccw) mBC=5 kg

A

B

C

37o 45o

LBC=500 mm LAB=300 mm

r =150 mm

Mass of the roller C and friction can be neglected. Determine the

reactions by the pins B and C.

PROBLEMS

8. In the mechanism shown, member AB is being rotated with a constant

angular velocity of wAB = 10 rad/s by a torque (not shown in the figure).

Member AB sets member BC in motion (mass of member BC is 6 kg), which then

causes gear D with a mass of 3 kg to move. The radius of gyration of the gear

with respect to center C is 200 mm. The radius of the gear is given as r = 250

mm. For the instant shown, determine the forces acting on pins C and B.

wAB

PROBLEMS wAB = 10 rad/s (cst), mBC = 6 kg, mD = 3 kg, kgear = 200 mm, r = 250 mm,

Determine forces acting on pins C and B.

wAB

PROBLEMS

9. The nonhomogeneous 20 kg wheel with the mass center at G has a radius of

gyration about G of 202 mm. The wheel rolls down the 20o rough incline

without slipping. In the position shown, the wheel has an angular velocity of 3

rad/s. Calculate the normal force and the friction force acting on the wheel

from the surface at this position.

250 mm

75 mm

3

SOLUTION “General Motion”

FBD

N

mg

Ff

KD

=

am

aI222

816.0)202.0(20 kgmmkI

aa 25.0 rao

x x

y

jia

ikkikiaaa

yx

aa

G

OGOG

aa

aa

075.0675.025.0

075.033075.025.0/

604.805

675.025.02020sin

a

a

f

fxx

F

FmgamF

a

a

5.1367.184

075.02020cos

N

mgNamF yy

aa 816.0)25.0()075.0( fG FNIM

NN

NF

srad

971.160

617.2

/597.15 2

a

a

aO

20°

250

mm

75

mm

3