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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.
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
N
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.
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.
ax
ax
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.
+
na
ta
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.
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
ppqppqppq
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
+
na
ta
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. In the compressor shown, crank AB
revolves with a constant angular velocity
of w1 = 10 rad/s. The homogeneous piston
arm BC has a mass of 2 kg, while piston C
has a mass of 1 kg. If it is known that the
pressure acting on the piston is p = 10 kPa,
determine the forces acting on pins B and
C. The compressor operates on the
vertical plane. The diameter of the piston
is D = 15 cm.
PROBLEMS w1 = 10 rad/s (cst). mBC = 2 kg, mpiston = 1 kg, p = 10 kPa.
Determine the forces acting on pins B and C. D = 15 cm.
PROBLEMS 8. The 6 kg uniform rod BC connects a 10 kg uniform thin disk centered at A to a 5 kg uniform rod CD. The motion of the system is controlled by the couple M applied to disk A. Knowing that at the instant shown disk A has an angular velocity of 36 rad/s clockwise and an angular acceleration of 150 rad/s2 counterclockwise, determine (a) the couple M, (b) the components of the force exerted at C on rod BC. (Mechanism lies on the vertical plane)
PROBLEMS mBC = 6 kg, mdisk = 10 kg, mCD = 5 kg. Determine (a) couple M, (b) components of the force exerted at C on rod BC.
PROBLEMS
9. 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
10. The 45 kg uniform flywheel shown
is rotating with an angular velocity of
30 rad/s and an angular acceleration
of 72 rad/s2 both in counter clockwise
direction. Bar AB has a mass of 15 kg
and a radius of gyration with respect
to its mass center of 325 mm. The
coefficient of friction between the pin
at C and the slot in bar AB is 0.10.
When bar AB is in the position shown,
determine the force exerted on the
bar by the pin at support A and the
support reactions exerted on the
flywheel at point O.
PROBLEMS mfw = 45 kg, wfw = 30 rad/s, afw = 72 rad/s2 (ccw), mAB =15 kg,
kAB = 325 mm, m =0.10. Determine the force exerted on the bar
by the pin at support A and the support reactions exerted on the
flywheel at point O.