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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. Pin A moves in a circle of 90 mm radius as crank AC revolves at the constant rate
.The slotted link rotates about point O as the rod attached to A moves in and
out of the slot. For the position b=30o, determine and . ,, rr
srad /6.0b
Crank AC (Pin A: Circular Motion)
In Normal and Tangential Coordinates:
Velocity :
s/mm.ACvA 549060 b
Acceleration:
222 4329060
00
s/mm..ACa
cst:ACa
aaa
nA
tA
nAtAA
b
bbb
O C
A
+t
+n
vA
aA=(aA)r
o.sin
.
sin
mm.rcosrfor
461130
5722690
57226309030029030030 222
b
30bGeometry
for b=30o, determine and . ,, rr
AO (Pin A: In Polar Coordinates)
s/mmvA 54
Acceleration:
2432 s/mm.aA
Velocity :
O C
A
+
+r
vA
aAr
11.46o 30b
11.46o
30o
s/rad..
.
r
vrvs/mm..sinvv
s/mm..cosvrv
A
Ar
1786057226
474047405448
75355448
2
2
2222
2
151057226
1786075352452122
45214641
5073117860572262824
28244641
s/rad..
...
r
rarra
s/mm..sinaa
s/mm.r...rarrra
s/mm..cosaa
A
rr
Ar
6. 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
7. 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)8. 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
9. 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
10. Particle P moves along a curvilinear path as shown in the figure. At the instant when r = 2 m,
= 30°, the magnitude of its velocity is 3.2 m/s and the velocity vector makes an angle of b = 60°
with the horizontal. The y-component of the acceleration of particle P is ay = 5 m/s2; its r -
component is ar = 1.83 m/s2. Determine components of the velocity and acceleration in
a) Cartesian Coordinates,
b) Normal and Tangential Coordinates, also radius of curvature of the path for the same instant,
c) Polar Coordinates.
r = 2 m, = 30°, v=3.2 m/s , b = 60° ay = 5 m/s2, ar = 1.83 m/s2
Velocity
+r
+
30o
e.e.v
esin.ecos.v
j.i.v
jsin.icos.v
e.v
s/m.v
r
r
t
61772
30233023
77261
60236023
23
23
in normal and tangential coordinates
in Cartesian coordinates
in polar coordinates
+t
+n
r = 2 m, = 30°, v=3.2 m/s , b = 60° ay = 5 m/s2, ar = 1.83 m/s2
Acceleration
+r
+
30o
nt
nt
rr
e.e.a
esin.ecos.a
e.e.ecos.e.a
jijisin.a
82968291
6007760077
8296831077831
55530077
aa
a
a
in normal and tangential coordinates
in Cartesian coordinates
in polar coordinates
+t
+n
ay
ar
//x
// a
30o
a
2077
152680
732508660
90907323030
9073230
s/m.a
.cos
sin
sin.sin.cos.
sinsincoscos.sinsincoscos
cos.cos
o
aa
a
aaa
aaaa
aa
83190
8316030
530
.cosa
.cosaa
cosaa
r
y
a
a
a
73290
30
831
5
90
30.
cos
cos
.cosa
cosa
a
a
a
a
m..
.
a
v
n
518296
23 22
Radius of curvature
11. The slotted arm AB rotates about the support A. For the instant when /6 radians, the
angular velocity of the arm is rad/s and its angular acceleration is rad/s2. As arm
AB forces pin C to move while staying inside the channel, the spring ensures that the pin does not
loose contact with the channel surface. For the position indicated in the figure determine the
velocity and acceleration of pin C in Cartesian Coordinates. Also determine the values of and .
30.
r r
78224204221
1750350350
1750
2 ..x.dx
dyx.y
m.sin.xm.l
.x
b
+r
//x
30o
b22.78o
30o
+t
//y
v
Velocity:
In Cartesian Coordinates:
s/m..cosvrv
s/m..sin
vv
s/m..rv
m.rs/rad
.sinvv.cosvv
r
r
44812237
81812237
0991350
350
22372237
j.i.v
.sinvv.cosvv yx
703906761
78227822
Acceleration:
222
232
2
2
232
2
2
21765320
8181
532042
42011
42894481230350
230
s/m..
.va
m..
.
dx
yd
dx
dy
s/m....a
rrarra.
n
/
/
r
*
*
+r
//x
30o
b22.78o
30o
//y
+t+n
an
//t
//r
b22.78o
a
a
222222
2
2
2
138263506842268422
93147822
504197822
565246614312260480
s/m...rrras/m.aaa
s/m.).sin(aa
s/m.).cos(aa
s/m.a.sin.cos.
rr
y
x
o
a
a
aaa
5161
2176
428978527852
2
1
22176
1428978527852
42897852
7852
..
.
sina
.sinsin.coscosa
.sinaasina
..sinsin.coscosa
..cosa
a.cosa
n
a
aa
aa
aa
a
a
