kinetics of rigid bodies problems - ki??isel...
TRANSCRIPT
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