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1. The sphere P travels in a straight line with a constant
speed of v=100 m/s. For the instant shown, determine
the corresponding values of as measured
relative to the fixed Oxy coordinate system.
,,,,, rrr
+ r+ Position
Velocity
o
m.r
45
137113280
v
rv
v
s/rad..
.
r
v
s/m.sinvrv
s/m.cosvrv
evevv
r
rr
2290137113
88225
8822515
5939615
The sphere P travels in a straight line with a constant speed of v=100 m/s.
+ r+
Acceleration
v
rvv
2
222
22
39102
02
9250
00
s/rad.r
rrra
s/m.rrrra
aaaaaacstv
r
rr
2. As the hydraulic cylinder rotates around O, the exposed length l of the piston rod P is
controlled by the action of oil pressure in the cylinder. If the cylinder rotates at the constant
rate =60 deg/s and l is decreasing at the constant rate of 150 mm/s, calculate the magnitudes
of velocity and acceleration of end B when l =125 mm.
r = 375 + l when l =125 mm
r = 500 mm
0
31806060
0
150
)cst(s/radsdeg/
r
)cst(s/mmlr
Velocity
s/mm..vve.ev
s/mm.rvs/mmrvevevv
r
rrr
665445952315059523150
595233
500150
22
Acceleration
222
222
2
856313143154831431548
3143
150202315483
5000
s/mm..aaee.a
s/mmrras/mm.rra
r
r
eaeaa rr
3. At the bottom of a loop in the vertical (r-) plane at an altitude of 400 m,
the airplane P has a horizontal velocity of 600 km/h and no horizontal
acceleration. The radius of curvature of the loop is 1200 m. For the radar
tracking at O, determine the recorded values of and for this instant. r
+ r+
o.tana
m.r
8211000
400
0310774001000 22
Position
Velocity
s/rad.r
vrv
s/m..sin.sinvv
s/m..cos.cosvrv
s/m..
v
r
05750
8966182167166
7515482167166
6716663
600
v
v
rv
+ r+
222
15231200
67166s/m.
.va
o.
m.r
821
031077
s/rad.
s/m.r
05750
75154
v
a
Acceleration (no horizontal acceleration)
a
1200 m (radius of curvature – in normal & tangential coordinates)r= 1077.03 m (radial distance measured from a fixed point (pole) to particle – in polar coordinates)
22
2222
2
/036.02
2/49.218.21cos15.23cos
/158.120575.003.1077597.8
/597.88.21sin15.23sin
sradr
rarrasmaa
smrarrra
smaa
rr
r
4. The hydraulic cylinder gives pin A a constant velocity v=2 m/s along its axis for
an interval of motion and, in turn,causes the slotted arm to rotate about O.
Determine the values of and for the instant when =30o . ,, rr
v = 2 m/s (cst), determine when = 30°. ,, rr +r
r
+
=30°
b
vvr
v
b
Geometry:
B
s/rad..r
vs/msinsinvrv
s/m.coscosvrvr
33330
11302
7321302
b
b
Acceleration:
2
2222
453830
333732122202
323333300
00
s/rad..
..
r
rrrrra
s/m.r..rrrra
aaa
r
r
Velocity:
Pin A: (Piston: rectilinear motionAO: in polar coordinates)
3012030180 b
r
300 mm
30°
r = 300 mm
30°O B
A
isosceles triangle
5. At time t=0, the ball is thrown with an initial speed of 30 m/s at an angle of
30o to the horizontal. Determine the quantities and , all
relative to the x-y coordinate system shown, at time t=0.5 s.
,,,, rrr
Determine the quantities at time t=0.5 s. ,,,r,r,r
in cartesian coordinates
x y
m.x
.costvxx xoo
9912
503030
m....siny
gttvyy yoo
278508192
15030302
2
1
2
2
s/m.vs/m.v
..singtvvcosvv
yx
yoyxox
095109825
5081930303030
y=8.27 m
x=12.99 m
=32.48o
o..
.tana
m...r
48329912
278
4152789912 22
+r+
vx
vya
v
v
a
o
x
y
yx
.v
vtana
s/m.vvv
2321
872722
a
//x
Determine the quantities at time t=0.5 s. ,,,r,r,r
in polar coordinates
=32.48o
+r+
v
a21.23o
//x
vr
v
s/rad..
.rvs/m...sin.sinvv
s/m.r..cos.cosvrvr
3530415
43754375232148328727
3327232148328727
a
a
Velocity
Acceleration (a=9.81 m/s2)
a
ar
a
22 27582685 s/m.cosaas/m.sinaar
2
222
7150415
353033272275822
349335304152685
s/rad..
...
r
rarra
s/m....rarrra rr
6. When the yoke A is at the position d = 0.27 m, it has a velocity of v = 2 m/s towards
right which is increasing at a rate of 0.6 m/s each second. Pin P is forced to move in the
vertical slot of the yoke and the parabolic surface. For the instant depicted, determine
the velocity and acceleration of pin P in
a) Cartesian Coordinates,
b) Normal and Tangential
Coordinates,
c) Polar Coordinates.
A
x = 2 m x (m)
y (m)7. Particle A is moving along a parabolicpath. At the instant when the abscissa of itsposition is x = 2 m, its velocity is 6.45 m/sand it decreases at a rate of 15 m/s persecond. Determine the velocity andacceleration of the particle for this instant in
a) Cartesian coordinates,
b) Normal and tangential coordinates,
c) Polar coordinates.
2
16
3xy
87.364
3
16
6tan
2
bb xdx
dy
x
ttt eaev
1545.6
Solution
(Given)
A
x (m)
y (m)
+n+t
t
tev
45.6
tt ea
15
na
b
b8
3
16
6
2
2
2
xdx
yd
m
dx
yd
dx
dy
2083.5
8
3
4
311
2/32
2
2
2/32
222
/98.72083.5
45.6sm
van
in normal and tangential coordinates
ntt e.eae.v
98715456
A
x (m)
y (m)
+n+t
t
tev
45.6
tt ea
15
na
b
b
2
16
3xy
222 /99.1698.715/45.6 smasmv
in Cartesian coordinates
jijiv
87.316.5sin45.6cos45.6 bb
2/78.16
87.36cos1587.36sin98.7cossin
sma
aaa
x
tnx
bb
2/616.2sincos smaaa tny bb
jia
616.278.16
in polar coordinates
A
x (m)
y (m)
+r
v
ta
na
b
b
my 75.0216
3 2
smvvr /19.6cos b
x = 2 m
y = 0.75 m
oa 55.202
75.0tan
smvv /812.1sin b
eev r
812.119.6
2/638.16cossin smaaa tnr bb
2/443.3sincos smaaa tn bb
eea r
443.3638.16
Magnitudes of velocity and acceleration of particle A
8. The peg moves in the curved slot defined by the equation r2 = 4sin(2) [m2], and
through the slot in the arm. At = 30°, the angular velocity and angular acceleration of
the arm are = 2 rad/s and = 1.5 rad/s2, respectively. Determine the magnitudes of the
velocity and acceleration of the peg at this instant,
a) in polar coordinates,
b) in Cartesian coordinates,
c) in normal and tangential
coordinates. Also determine
the radius of curvature
for this instant.
,
at = 30° = 2 rad/s , = 1.5 rad/s2
,
mrr 86.1302sin42
Solution
2cos42cos242 rrrrdt
d
smrsrad o /15.230,/2
smrvsmrv r /15.2,/72.3
smveev r /297.472.315.2
in polar coordinates
2sin22cos4 22
2
2
rrrdt
d
*
**
22 /77.15/5.1,/15.2,/2,86.1,30 smrsradsmrsradmro
2
22
/39.112
/11.23
smrra
smrrar
2/85.2539.1111.23 smaeea r
smveev r /297.472.315.2
in Cartesian coordinates
2/85.2539.1111.23 smaeea r
A
+r
v
b
30o
vr
v v
30o
oa 97.5715.2
72.3tan
b
b
jiv
jiv
294.4152.0
30sin297.430cos297.4
bb
jia
jia
695.179.25
30sin85.2530cos85.25
aa
a
30o
ar
aa
aa
oa 24.2611.23
39.11tan
a
in normal and tangential coordinates
+t
b+n
tev
297.4
nt
nt
eea
eea
718.25608.2
76.303.2cos85.2576.303.2sin85.25
o76.3
o03.2
ma
v
n
718.0718.25
297.4 22
o03.2
o76.3