1

All figures taken from Vector Mechanics for Engineers: Dynamics, Beer and Johnston, 2004

ENGR 214Chapter 16

Plane Motion of Rigid Bodies:Forces & Accelerations

2

Equations of Motion for a Rigid Body

GF ma

GG HM

Equations of motion:

remember:

GH

= angular momentum about G

G GH I

G G GH I I

GI = mass moment of inertia about G

G GM I

3

Plane Motion of a Rigid Body

x x y y G gF ma F ma M I

Motion of a rigid body in plane motion is completely defined by the resultant & moment resultant of the external forces about G.

Free-body diagram

4

Sample Problem 16.1

At a forward speed of 30 ft/s, the truck brakes were applied, causing the wheels to stop rotating. It was observed that the truck skidded to a stop in 20 ft.

Determine the magnitude of the normal reaction and the friction force at each wheel as the truck skidded to a stop.

5

Sample Problem 16.1

2 20 0

2

2

0 30 2 20

v v a x x

a

22.5 ft sa

x GxF maFree-body diagram:

22.5A BF F m

y GyF ma A BW mg N N

G GM I 7 4 4 5B B A AN F F N

But ,A A B BF N F N

22.5A BN N m

A BN N mg 0.699

0.35 , 0.65A BN mg N mg

, ,A BN N Unknowns:

6

Sample Problem 16.2

The thin plate of mass 8 kg is held in place as shown.

Neglecting the mass of the links, determine immediately after the wire has been cut (a) the acceleration of the plate, and (b) the force in each link.

7

Sample Problem 16.2

t tF ma

cos30

cos30t

t

mg ma

a g

28.50 m sta

n nF ma

sin 30 0

39.24AE DF

AE DF

F F mg

F F

47.9 NAEF

GM I

sin 30 250 cos30 100

sin 30 250 cos30 100 0

AE AE

DF DF

F F

F F

8.7 NDFF

Solving: tensile

compressive

8

Sample Problem 16.3 (SI units)

A pulley weighing 5.44 kg and having a radius of gyration of 0.203 m is connected to two blocks as shown.

Assuming no axle friction, determine the angular acceleration of the pulley and the acceleration of each block.

25.4 cm

15.24 cm

4.536 kg

2.268 kg

9

Sample Problem 16.3

20.36 /

A Aa r

m s

20.6 /

B Ba r

m s

2 20.2246I mk kgm note:

For the entire system:

GM I

25.4 cm

15.24 cm

4.536 kg2.268 kg

2 2

0.1524 0.254

0.2246 0.1524 0.254

B A

B A

m g m g

m m

4.536 kg2.268 kg

5.44 kg22.37 /rad s

10

Sample Problem 16.3Obtain the tension in the chord

11

Sample Problem 16.4

A cord is wrapped around a homogeneous disk of mass 15 kg. The cord is pulled upwards with a force T = 180 N.

Determine: (a) the acceleration of the center of the disk, (b) the angular acceleration of the disk, and (c) the acceleration of the cord.

12

Sample Problem 16.4

y yF ma

2

180 (15)(9.81) 15

2.19 /

y

y

a

a m s

GM I

2 212

2

1.875

180(0.5) 1.875

48 /

I mr kgm

I

rad s

x xF ma0 xma 0xa

+

13

Sample Problem 16.4

2

2.19 0.5 48

26.19 /

cord A Gt A Gt ta a a a

m s

14

Sample Problem 16.5

A uniform sphere of mass m and radius r is projected along a rough horizontal surface with a linear velocity v0. The coefficient of kinetic friction between the sphere and the surface is k.

Determine: (a) the time t1 at which the sphere will start rolling without sliding, and (b) the linear and angular velocities of the sphere at time t1.

SOLUTION:

• Draw the free-body-diagram equation expressing the equivalence of the external and effective forces on the sphere.

• Solve the three corresponding scalar equilibrium equations for the normal reaction from the surface and the linear and angular accelerations of the sphere.

• Apply the kinematic relations for uniformly accelerated motion to determine the time at which the tangential velocity of the sphere at the surface is zero, i.e., when the sphere stops sliding.

15

Sample Problem 16.5SOLUTION:

• Draw the free-body-diagram equation expressing the equivalence of external and effective forces on the sphere.

• Solve the three scalar equilibrium equations.

effyy FF

0 WN mgWN effxx FF

mg

amF

k ga k

2

32 mrrmg

IFr

k

r

gk2

5

effGG MM

NOTE: As long as the sphere both rotates and slides, its linear and angular motions are uniformly accelerated.

16

Sample Problem 16.5

ga k

r

gk2

5

• Apply the kinematic relations for uniformly accelerated motion to determine the time at which the tangential velocity of the sphere at the surface is zero, i.e., when the sphere stops sliding.

tgvtavv k 00

tr

gt k

2

500

110 2

5t

r

grgtv k

k

g

vt

k0

1 7

2

g

v

r

gt

r

g

k

kk

0

11 7

2

2

5

2

5

r

v01 7

5

r

vrrv 0

11 7

5 075

1 vv

At the instant t1 when the sphere stops sliding,

11 rv

17

Constrained Plane Motion

Motions with definite relations between acceleration components

18

Constrained Motion: Noncentroidal RotationRotation of a body about an axis that does not pass through its mass center

2,t na r a r

O OM I

19

G

G

P F ma

N W

Fr I

Frictional Rolling Problems

Wheels, cylinders or spheres rolling on rough surfaces

P

W

NF

Gr

xy

Unknowns: , , ,GF a N

20

Rolling Motion• For rolling without sliding:

x r a r

• For rolling with no sliding:NF s a r

For rolling with impending sliding:NF s a r

For rolling and sliding:NF k ra , are independent

• Geometric center of an unbalanced disk:

raO

Acceleration of the mass center:

G O G O

O G O G Ot n

a a a

a a a

Rolling Friction

Rolling Friction

• Rolling an object on a soft surface (grass or sand) requires more effort than rolling it on a hard surface (concrete)

• No sliding in either case

• No object is truly rigid

• Rolling causes continuous deformation of both surfaces giving rise to internal friction

Pr Wa

r

aP W W

r

Coefficient of rolling resistance

Steel on steel: 10-3 - 10-6

Tire on concrete: 0.01 – 0.015

23

Sample Problem 16.6

The portion AOB of the mechanism is actuated by gear D and at the instant shown has a clockwise angular velocity of 8 rad/s and a counterclockwise angular acceleration of 40 rad/s2.

Determine: a) tangential force exerted by gear D, and b) components of the reaction at shaft O.

kg 3

mm 85

kg 4

OB

E

E

m

k

m

24

Sample Problem 16.6

rad/s 8

2srad40

kg 3

mm 85

kg 4

OB

E

E

m

k

m

x E Ex OB OBx

y E Ey OB OBy

G E OB

F m a m a

F m a m a

M I I

2

2 2 2112

3 40 0.2 24

(3 9.81) (4 9.81) 3 (8) 0.2

107.07

0.12 4 (0.085) 40 ( 3 0.4 3 0.2 ) 40

62.97

170.04

x

y

y

y

R N

R F

R F

F

F N

R N

25

Sample Problem 16.7

If pin B is suddenly removed, find the angular acceleration of the plate and the pin reactions. Very similar to sample problem 16.2, please read on your own.

26

Sample Problem 16.8

A sphere of weight W is released with no initial velocity and rolls without slipping on the incline.

Determine: a) the minimum value of the coefficient of friction, b) the velocity of G after the sphere has rolled 10 ft and c) the velocity of G if the sphere were to move 10 ft down a frictionless incline.

27

Sample Problem 16.8

a r

2 22sin

5

7sin

5

C CM I

mg r mr mr

g r

For rolling:

sin ( )x xF ma

mg F m r

But cosF N mg

sin cos ( )

5sin cos sin

75

cos sin 17

mg mg m r

2tan 0.165

7

2

2 2 2

2

52

5

G

C G

I mr

I I mr mr mr

For a sphere:

28

Sample Problem 16.8

2 20 02

0 2 11.5 10 230

v v a x x

15.17 ft sv

25sin 11.5 /

7a r g ft s (uniform)

For no friction, no rolling (pure sliding)sin

sin 16.1 /

mg ma

a g ft s

2 20 02

0 2 16.1 10 322

v v a x x

17.94ft sv

29

Sample Problem 16.9

A cord is wrapped around the inner hub of a wheel and pulled horizontally with a force of 200 N. The wheel has a mass of 50 kg and a radius of gyration of 70 mm. Knowing s = 0.20 and k = 0.15, determine the acceleration of G and the angular acceleration of the wheel.

30

Sample Problem 16.9

2

22

mkg245.0

m70.0kg50

kmI

2

2

2

200 0.04 0.245 50 0.1

10.74 rad s

10.74 0.1 1.074 m sa r

C CM I

y yF ma0

50 1.074 490.5 N

N W

N mg

x xF ma

If we assume pure rolling:Do we have pure rolling or rolling + sliding?

200 50 1.074

146.3

F

F N

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Sample Problem 16.9

N3.146F N5.490NWithout slipping,

• Compare the required tangential reaction to the maximum possible friction force.

N1.98N5.49020.0max NF s

F > Fmax , rolling without slipping is impossible.

• Calculate the friction force with slipping and solve the equations for linear and angular accelerations.

N6.73N5.49015.0 NFF kk

G GM I

2

73.6 0.100 200 0.0.06

0.245

18.94 rad s

x xF ma 200 73.6 50 a 22.53m sa

32

Sample Problem 16.10

The extremities of a 4-ft rod weighing 50 lb can move freely and with no friction along two straight tracks. The rod is released with no velocity from the position shown.

Determine: a) the angular acceleration of the rod, and b) the reactions at A and B.

SOLUTION:

• Based on the kinematics of the constrained motion, express the accelerations of A, B, and G in terms of the angular acceleration.

• Draw the free-body-equation for the rod, expressing the equivalence of the external and effective forces.

• Solve the three corresponding scalar equations for the angular acceleration and the reactions at A and B.

33

Sample Problem 16.10SOLUTION:

• Based on the kinematics of the constrained motion, express the accelerations of A, B, and G in terms of the angular acceleration.

Express the acceleration of B as

ABAB aaa

With the corresponding vector triangle and the law of signs yields

,4ABa

90.446.5 BA aa

The acceleration of G is now obtained from

AGAG aaaa

2 where AGa

Resolving into x and y components,

732.160sin2

46.460cos246.5

y

x

a

a

34

Sample Problem 16.10• Draw the free-body-equation for the rod, expressing

the equivalence of the external and effective forces.

69.2732.12.32

50

93.646.42.32

50

07.2

sftlb07.2

ft4sft32.2

lb50

12

1

2

22

2121

y

x

am

am

I

mlI

• Solve the three corresponding scalar equations for the angular acceleration and the reactions at A and B.

2srad30.2

07.2732.169.246.493.6732.150

effEE MM

2srad30.2

effxx FF

lb5.22

30.293.645sin

B

B

R

R

lb5.22BR

45o

effyy FF

30.269.25045cos5.22 AR

lb9.27AR