7. The pin P is forced to move in the fixed parabolic slot by the moving

vertical guide. The centerline of the slot is described by y = 0.015x2 [mm].

When x = 40 mm, the speed v of the guide is 100 mm/s and is not changing at

this instant. Determine the magnitudes N1 and N2 of the forces which the

guide and the vertical plate, respectively, exert on the 50 g pin.

mg+n

N1

N2q

qq

+t

slotparabolicbyPonactingForceN

guideverticalbyPonactingForceN

xdx

dy

o

x

:

:

19.50

2.103.0tan

2

1

40

q

q

Kinematics:

ja

smmxxxxy

xaxvjiv

smmxxyxy

P

xxP

dt

d

300

/300)040100(03.0)(03.0

0constant120100

/120)100)(40(03.003.0015.0

22

2

Kinetics:

1

60607850

300501950819050

0

12

22

2121

N

yyy

xx

N.NN.N

...cosN..macosNmgmaF

sinNNsinNNmaF

q

qq

mg+n

N1

N2q

qq

+t

P

8. A 0.8-kg slider is propelled upward at A along the fixed curved bar which

lies in a vertical plane. If the slider is observed to have a speed of 4 m/s as it

passes position B, determine, (a) the magnitude N of the force exerted by the

fixed rod on the slider and (b) the rate at which the speed of the slider is

increasing. Assume that friction is negligible.

Normal&Tangential Coordinates

+ t

+ n

+ t

+ n

mg

N

man

mat

FBD KD

2905430 s/m.amasinmgmaF tttt

N

at

N.N

..cos..amcosmgNmacosmgNmaF

v

nnnn

5314

60

48030819803030

2

2

nntt maFmaFamF ,

9. The small object is placed on the inner surface of the conical dish at the

radius shown. If the coefficient of static friction between the object and the

conical surface is 0.30, for what range of angular velocities w about the

vertical axis will the block remain on the dish without slipping? Assume that

speed changes are made slowly so that any angular acceleration may be

neglected.

Normal&Tangential Coordinatesy

+ n

30o

+ n FfmsN

mg

FfmsN N

30o

(for wmax)

(for wmin)

rmNNmaF

mgNNF

snn

sy

230cos30sin

030sin30cos0

wm

m

for wmax

s/rad.sincos

cossin

r

g

rm

mg

cosNsinN

sinNcosN

rmcosNsinN

mgsinNcosN

mgsinNcosN

s

smax

s

s

s

s

s

2173030

3030

3030

3030

3030

3030

03030

2

2

m

mw

wm

m

wm

m

m

1

2

1

2

for wmin

s/rad.sincos

cossin

r

g

rm

mg

cosNsinN

sinNcosN

rmcosNsinN

mgsinNcosNmgsinNcosN

s

smin

s

s

s

ss

4133030

3030

3030

3030

3030

303003030

2

2

m

mw

wm

m

wm

mm 3

4

3

4

ms=0.30

10. The 1 kg collar slides along the smooth parabolic rod in the vertical

plane toward point O. The spring whose stiffness is k=600 N/m has an

unstretched length of 1 m. If, at the position shown in the figure, velocity of

the collar is 3.5 m/s, determine the force acting on the collar by the

parabolic rod for this instant. Neglect the friction.

m=1 kg, k=600 N/m, lo= 1 m, v=3.5 m/s, the force on the collar by the parabolic rod. Normal&Tangential Coordinates

N.Nam..cos.cosmgN

s/m..

.vmacosFcosmgNmaF

nkg

nspnn

84196873644692254469

771324853

53

1

222

qq

mymx 5.0375.09

32375.0 2

Final length of the spring:

(Compressed)

NlkFsp 225

m.lll

m...l

o 3750

6250375050 22

o

.x

.xdx

dytan 4469

9

64

3750

qq

m

dx

yd

dx

dy

2485.3

1

2

2

2/32

oo 87.3613.5375.0

1tan

(Radius of curvature at x=0.375 m)

+ t

+ n

q

mgFsp

N

q

1 m

11. The slotted arm revolves in the horizontal plane about the fixed vertical

axis through point O. The 2 kg slider C is drawn toward O at the constant rate of

50 mm/s by pulling the cord S. At the instant for which r=225 mm, the arm has

a counterclockwise angular velocity w=6 rad/s and is slowing down at the rate

of 2 rad/s2. For this instant, determine the tension T in the cord and the

magnitude N of the force exerted on the slider by the sides of the smooth radial

slot. Indicate which side, A or B, of the slot contacts the slider.

Polar Coordinates(Horizontal plane)+ r

T

mar

+ q

N

maq

NNNmaF

NTT

rrmTmaF

maFmaFamF

sradsradsmrmmr

rr

rr

1.2605.022225.02

2.166225.002

,

/2,/6,)constant(/05.0,225

2

2

2

qq

qq

q

qq

N

(Side B)

12. The 1.2 N particle B is forced to move in the circular slot DE by the slotted arm OC.

Determine the forces P and Q exerted by the arm OC and circular slot DE on the particle B,

respectively at the instant when q =20°. Motion occurs in the horizontal plane. (b=50 cm).

At this instant, .225015 s/rad,s/rad qq

b

b

q

C

O

BD

E

225015 s/rad,s/rad qq

b

bq

O

B

Polar Coordinates(Horizontal plane)

q

r

r

Q

Rq

q

q

N.Q

...

.cosQ

maF rr

0166

1594094296819

2120 2

N.R

...

.sinQR

maF

4732

151352250940819

2120

qq

22 942962

1352

9402

s/m.rcossinbr

s/m.rsinbr

m.rcosbr

qqqq

qq

q

for b=50 cm=0.5 m and q=20o

qq

q

q

rra

rrar

2

2

13. The 1.5 kg cylinder C travels along the path described by r=(0.6sinq)

[meter]. If arm OA rotates counter clockwise with a constant angular velocity of

=3 rad/s, determine the force exerted by the smooth slot in arm OA on the

cylinder at the instant q = 60°. The spring has a stiffness of 100 N/m and is

unstretched when q =30°. Motion occurs in the vertical plane.

q

m=1.5 kg r =(0.6sin q ). =3 rad/s (cst), the force exerted by the slot on the cylinder at

q = 60°. K=100 N/m , spring is unstretched when q =30°. Motion in the vertical plane.

q

03 qq )cst(s/rad

Polar Coordinates(Vertical plane)

m.sin.rsin.r 5190606060 q

22

3

6764609060 s/m.sincos.rs/m.cos.r qqqqqq

222 4523479 s/m.rras/m.rrar qqq q

Spring is unstretched when q =30°.

m.rwhen

m.rwhen

o

o

519060

3030

q

q

N..rkF

m...r

spring 9212190100

2190305190

(Spring is compressed)

222 4523479 s/m.rras/m.rrar qqq q

mg

Fspring

N

+r

+q

R

q60o30o

30o

N.N

..cosNcos...

macosNcosmgF

maF

rspring

rr

81323

347951303081951921

3030

N.R

..Rsin.sin..

maRsinNsinmg

maF

3627

1455130813233081951

3030

q

qq

14. The 1 kg ball A, which slides in a

slot in the rotating arm OB, is kept in

contact with the stationary cam C by a

spring. The spring has a stiffness of

1000 N/m and is unstretched when

the arm OB is in position OP. If the

arm rotates with the constant angular

velocity =20 rad/s (ccw), calculate

the normal force N which the cam

exerts on A and also the force R

exerted on A by the sides of the slot

when q=30o. All surfaces are smooth.

The problem occurs in vertical plane.

r=0.1(2+cos q) m

0.2 m

0.1 m 0.3 m

q

P

C

r

y

x

B

A

Q

q

r=0.1(2+cos q) m

0.2 m

0.1 m 0.3 m

q

P

C

r

y

x

B

A

Q

+r

+q

q30o

b

b?+n

30o

q

q

q

qq

q

e.ev

s/m..r

s/msin.r

m.cos.r

ererv

r

r

73251

7325202860

110

286030210

mg

R

N

Fspring

Kinematics

o..

arctan 997325

1

b

+r

+q

q30o

b

+n

b

+t

qv

rv

v

mg

R

N

Fspring

The spring is unstretched when the arm OB is in position OP.

r=0.1 m when the spring is unstretched.r=0.286 m when q=30o.

N..rkFspring 1861028601000

30o

30o

+r

+q

b

+n

NRR

maRNmgmaF

748.384019.9sin132.4230cos81.91

9.9sin30cos

qqq

NNN

maFNmgmaF rspringrr

132.4204.14911869.9cos30sin81.91

9.9cos30sin

2

0

22

222

0

/4020122/04.14920286.064.34

/64.3430cos201.0cossin1.0

smrrasma

smr

r

qq

qqqq

q

Kinetics