Rectilinear Motion

1. Small objects are delivered to the 2 m inclined chute by a conveyor belt A

which moves at a speed v1=0.4 m/s. If the conveyor belt B has a speed

v2=0.9 m/s and the objects are delivered to this belt with no slipping,

calculate the coefficient of friction mk between the objects and the chute.

Rectilinear Motion

2. A force P is applied to the initially stationary cart. Determine the velocity

and displacement at time t=5 s for each of the force histories P1 and P2.

Neglect friction.

3. The sliders A and B are connected by a light rigid bar and move with

negligible friction in the slots, both of which lie in a horizontal plane. For the

position shown, the velocity of A is 0.4 m/s to the right. Determine the

acceleration of each slider and the force in the bar at this instant.

Rectilinear Motion

4. The collar slides down the rod. The

coefficient of friction is mk = 0.35. The

tension in the cable attached to the

collar is constant as 80 N and the angle

between the cable and the horizontal

remains the same as a throughout the

motion. The weight of the collar is 60

N. If the collar stops after 10 seconds it

starts sliding down the rod with an

initial velocity of 5 m/s, what should a

be?

Rectilinear Motion

Curvilinear Motion

5. 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 N and R of the forces which the guide

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

Curvilinear Motion

6. 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

2

2

905430

531460

480308198030

30

2

s/m.amasinmgmaF

N.N.

.cos..amcosmgN

macosmgNmaF

maF,maFamF

tttt

v

n

nnn

nntt

N

at

Curvilinear Motion

7. 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.

Curvilinear Motion

8. 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.

Curvilinear Motion

10. 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 FBD + q

N

maq

KD

N.N..NmaF

N.N.T

rrmTmaF

maF,maFamF

s/rad,s/rad,)constant(s/m.r,mmr

rr

rr

1260502222502

2166225002

26050225

2

2

2

qq

qq

q

qq

N

(Side B)

Curvilinear Motion

11. The 1.5 kg cylinder C travels along the path described by r =(0.6sin q )

[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

Curvilinear Motion

13. The slotted arm is used to move the

smooth 0.5 kg particle P around the

vertical path in the shape of a limaçon with

r = 0.3 (2 + cosq ) m. The spring with a

stiffness of k = 8 N/m in the slot ensures

that the particle P does not lose contact

with the path. The spring is unstretched

when q is equal to 90°. If q = (0.5t2) rad,

where t is in seconds, determine the forces

N and F which the rod and the slotted arm,

respectively, exerts on the particle at the

instant t = 1.5 s.

0.9 m

0.6 m

q

q

Curvilinear Motion

12. 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

+r

+q

q30o

b

+n

b

+t

2

0

22

222

0

2

4020122

041492028606434

643430201010

s/mrra

s/m...a

s/m.cos.cossin.r

rra

r

r

qq

qqqq

q

q

o..

arctan 997325

1

b

rv

qv

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

N.R

R.sin.cos.

maR.sinNcosmgmaF

74838

4019913242308191

9930

qqq

N.N

..cosNsin.

maF.cosNsinmgmaF rspringrr

13242

04149118699308191

9930

Curvilinear Motion

9. The horizontal platform is rotating

with a constant angular velocity of

w=5 rad/s. Two identical blocks, each

with a mass of 2 kg are held in the

smooth slot within the platform by a

pair of springs each with a stiffness

of k=250 N/m. If the unstretched

length of each spring is lo= 0.18 m,

what should angle q be for steady

state position? Springs are attached

to the platform at point A.

(Consider only one block for the

solution of the problem.)

w= 5 rad/s, m = 2 kg

2qqq rrmsinFcosN

maF

spring

rr

circular motion, steady state position

2sincos qqq mrxkN

xkFsp

q

q

qqq

qqqq

qq

sin

cosxkN

cosxkcosFsinN

rrmcosFsinN

maF

spring

spring

2

w

r q

0.3 m

x

y

N

Fspring

q x2

qsin2 rx

x y

N

Fspring

W

R

Putting into

qqqq

qqqq

qqqqq

q

sinsincos

sinsincos

sinsincossin

cos

2

1

22

222

2

mrxk

mrxkxk

mrxkxk

87.36

75.03.0

225.0

cos

sintan

225.0sin

45sin200,sin5045sin250

sin5218.0250sin250

sin18.0sin

18.0sin18.0

2

2

212

q

q

qq

q

qqq

qq

qqq

q

r

r

r

rrr

rr

mrrk

rxlengthinitiallengthfinalxxx

w

r q

0.3 m

x

y

N

Fspring

q x2

qsin2 rx

Curvilinear Motion

10. 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 N and R of the forces which the

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

mg +n

N1

N2 q

q

q

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

606.0785.0

)3.0(05.019.50cos)81.9)(05.0(cos

sin0sin

12

22

2121

N

yyy

xx

NNNN

NmaNmgmaF

NNNNmaF

q

qq