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