Problem 1 (25 points)

A diving board consists of a uniform

plank of length L = 1.5 m and a m

ass M = 3.0 kg. If

a man (m

ass m = 70 kg) stands on his toes at the end of the board, w

hat will be the force

exerted on the plank by a support at B and by the pivot at A?

A

B

L/4m

M

Problem 2 (25 points)

A uniform

plastic ball of mass m

= 0.50 kg and radius R = 0.020 m

is rolling down w

ithout slipping on an inclined plane. It has a linear velocity v = 0.8 m

/s when it is at a height h =

0.40 m.

a) C

alculate the velocity of the ball when it reaches the floor.

b) K

nowing that the plane is inclined by an angle ����= 30

0, find the linear acceleration of the ball on the inclined plane.

c) H

ow the answ

ers to a) and b) change if the radius R of the ball is two tim

es larger. d)

When the ball reaches the floor it m

akes a completely inelastic collision w

ith a block of m

ass M = 2.0 kg, w

hat is the velocity after the collision? (If you don’t know how to solve a), put a velocity by hand and you can work on the other questions.)

h

Problem 3 (25 points)

Problem 2 (25 points)

You design three pulley w

ith radius R = 0.160 m and m

ass M = 10.0 kg. The rope does not slip on

pulley rim. U

se the energy methods to calculate the speed of the 4.00-kg block just before it strikes

the floor if the pulley is (a) Solid disk, (b) Solid sphere, or (c) Thin-walled hollow

cylinder. (d) If you re-design the pulleys w

ith Rnew = 0.320 m

, but maintaining the sam

e mass, w

hich pulley design w

ould give the fastest speed for the 4.00-kg block?

m1 m

2

(a) Solid disk

(b) Solid sphere

(c) Thin-walled hollow

cylinder

Problem 2 (25 points) - C

hapter 9 Y

ou design three pulley with radius R = 0.160 m

and mass M

= 10.0 kg. The rope does not slip on pulley rim

. Use the energy m

ethods to calculate the speed of the 4.00-kg block just before it strikes the floor if the pulley is (a) Solid disk, (b) Solid sphere, or (c) Thin-w

alled hollow cylinder. (d) If

you re-design the pulleys with R

new = 0.320 m, but m

aintaining the same m

ass, which pulley design

would give the fastest speed for the 4.00-kg block?

m1 m

2

(a) Solid disk

(b) Solid sphere

(c) Thin-walled hollow

cylinder

ghm

U

vm

vm

IK

Kgh

mU

KU

KU

f f

ii

ff

ii

2

22

21

2

1

2 12 1

2 10

; where�

��

�

��

��

�

�

22

21

2

22

21

22

1

2 12 1

2 1J

98

2 12 1

2 1

J 98

vm

vm

R vI

vm

vm

Igh

mgh

m

KU

Uf

fi

��

�� ��

��

��

��

�

��

�

��

��

���

�

m/s

50

3)2

410

(2 198

(c)

m/s

43

4)2

4(4

2 198

5 2

(b)

m/s

22

4)2

45(

2 198

2 1

(a)

22

22

22

.v

vM

RI

.v

vM

RI

.v

vM

RI

�

��

�

�

�

��

�

�

�

��

�

�

I = 0.560 kg*m2

sphere

Solid.

of

independet is

answer

The

(d)new

�

R

From textbook

15 pts for correct setup

4 pts for correct concept &

answer

2pts 2pts 2pts

Problem 3 (25 points)

A large w

ooden turntable in the shape of a flat uniform disk has a radius of R

= 2.00 m and a total

mass of M

= 120.0 kg. The turntable is initially rotating at �� = 3.00 rad/s about a vertical axis though its center. Suddenly, a m

= 70.0-kg parachutist makes a soft landing on the turntable at a point near

the outer edge. (a) (15 pts) find the angular speed of the turntable after the parachutist lands. Assum

e that you can treat the parachutist as a particle. (b) (10 pts) com

pute the kinetic energy of the system

before and after the parachutist lands. Why are these kinetic energies not equal?

Problem 3 (25 points) – C

hapter 10 A

large wooden turntable in the shape of a flat uniform

disk has a radius of R = 2.00 m

and a total m

ass of M = 120.0 kg. The turntable is initially rotating at �� = 3.00 rad/s about a vertical axis though

its center. Suddenly, a m = 70.0-kg parachutist m

akes a soft landing on the turntable at a point near the outer edge. (a) (15 pts) find the angular speed of the turntable after the parachutist lands. A

ssume

that you can treat the parachutist as a particle. (b) (10 pts) compute the kinetic energy of the system

before and after the parachutist lands. W

hy are these kinetic energies not equal?

15 pts

2pts off per major m

istake

10 pts

Problem 4 (25 points)

A loaded cement m

ixer truck drives onto an old drawbridge (length of L = 40.0 m

and mass of M

B = 18,000 kg), w

here it stalls with its center of gravity three-quarters of the w

ay across the span. The truck driver sets the handbrake. Then the bridge is raised by m

eans of a cable attached to the end opposite to the hinge. The truck w

ith the driver has a mass of M

T = 30,000 kg. When the bridge has

been raised to an angle of 30.0o above the horizontal, the cable m

akes an angle of 70.0o w

ith the surface of the bridge. (a) W

hat is the tension T in the cable when the bridge is held in the position.

(b) What are the horizontal and vertical com

ponents of the force the hinge exerts on the span?

Problem 4 (25 points) – C

hapter 11 A loaded cem

ent mixer truck drives onto an old draw

bridge (length of L = 40.0 m and m

ass of MB =

18,000 kg), where it stalls w

ith its center of gravity three-quarters of the way across the span. The

truck driver sets the handbrake. Then the bridge is raised by means of a cable attached to the end

opposite to the hinge. The truck with the driver has a m

ass of MT = 30,000 kg. W

hen the bridge has been raised to an angle of 30.0

o above the horizontal, the cable makes an angle of 70.0

o with the

surface of the bridge. (a) What is the tension T in the cable w

hen the bridge is held in the position. (b) W

hat are the horizontal and vertical components of the force the hinge exerts on the span?

15 pts

2pts off per major m

istake

10 pts

Pro

blem

1

A w

ire of length l0 and cross-sectional area A supports a hanging w

eight W. (a) Show

that if the wire obeys H

ooke’s law, behaving like

a spring of force constant AY/l0 , where Y is Y

oung’s modulus for the

material of w

hich the wire is m

ade. (b) What w

ould the force constant be for a 75.0-cm

length of 16-gauge (diameter = 1.291 m

m) copper

wire? See Table 11.1. (c) W

hat would W

have to be to stretch the wire

in part (b) by 1.25 mm

?

11

Pro

blem

1 S

olu

tion

A

wire of length l0 and cross-sectional area A supports a hanging

weight W

. (a) Show that if the w

ire obeys Eq. (11.7), it behaves like a spring of force constant AY/l0 , w

here Y is Young’s m

odulus for the m

aterial of which the w

ire is made. (b) W

hat would the force constant

be for a 75.0-cm length of 16-gauge (diam

eter = 1.291 mm

) copper w

ire? See Table 11.1. (c) What w

ould W have to be to stretch the w

ire in part (b) by 1.25 m

m?

22

Pro

blem

2

In lab tests on a 9.25-cm cube 0f a certain

material, a force of 1375 N

directed at 8.50o to

the cube causes the cube to deform through an

angle of 1.24o. W

hat is the shear modulus of

the material?

33

���

�tan

h x

��

S = (F|| /A

)/(x/h).

Pro

blem

2 S

olu

tion

In lab tests on a 9.25-cm

cube 0f a certain m

aterial, a force of 1375 N directed at 8.50

o to the cube causes the cube to deform

through an angle of 1.24

o. What is the shear m

odulus of the m

aterial?

44

[11.38]

���

�tan

h xS = (F

|| /A)/(x/h).

A. m

ore stress and more strain.

B. the sam

e stress and more strain.

C. the sam

e stress and less strain.

D. less stress and less strain.

E. the sam

e stress and the same

strain.

Q11.5 Tw

o rods are made of the

same kind of steel and have

the same diam

eter. F

F

length 2L

F

F

length L

A force of magnitude F is applied to the end of each rod.

Com

pared to the rod of length L, the rod of length 2L has

5

Y = (F� /A

)(l0 /�l)

A. m

ore stress and more strain.

B. the sam

e stress and more strain.

C. the sam

e stress and less strain.

D. less stress and less strain.

E. the sam

e stress and the same

strain.

A11.5 Tw

o rods are made of the

same kind of steel and have

the same diam

eter. F

F

length 2L

F

F

length L

A force of magnitude F is applied to the end of each rod.

Com

pared to the rod of length L, the rod of length 2L has

6

Y = (F� /A

)(l0 /�l)

A. m

ore stress and more strain.

B. the sam

e stress and more strain.

C. the sam

e stress and less strain.

D. less stress and less strain.

E. the sam

e stress and the same

strain.

Q11.6 Tw

o rods are made of the

same kind of steel. T

he longer rod has a greater diam

eter. F

F

length 2L

F

F

length L

A force of magnitude F is applied to the end of each rod.

Com

pared to the rod of length L, the rod of length 2L has

7

Y = (F� /A

)(l0 /�l)

A. m

ore stress and more strain.

B. the sam

e stress and more strain.

C. the sam

e stress and less strain.

D. less stress and less strain.

E. the sam

e stress and the same

strain.

A11.6 Tw

o rods are made of the

same kind of steel. T

he longer rod has a greater diam

eter. F

F

length 2L

F

F

length L

A force of magnitude F is applied to the end of each rod.

Com

pared to the rod of length L, the rod of length 2L has

8

Y = (F� /A

)(l0 /�l)

Oscillations

11

Oscillations

xm k

ax

�

�x

kF

x

���

An

alyzin

g Sprin

g+B

lock

System

vmax , a

x =0

Mechanical Energy Conservation

2

Oscillations

Con

cepts o

f S.H

.M..

Dynam

ics & Kinematics

Spring+block �

F = m

a & (x, v, a)

Pendulum

� a part of circular m

otion���� =

I ���& (�, �, �)

Force: C

onservative force R

estoring force

Conservation:�

K + U = constant

+ S.H.M

. (��as angular frequency)

3

Oscillations

Example 2(A

)

+

+

+

+

+ 0

T/4

T/2

3T/4

T

Equilibrium

Positions

A

4

Oscillations

Example 2(B

)

Ea = E

b = Ec = E

d

5

Oscillations

Example 5

You holdthe block

at x = A (= 0.030 m

) by applying 6.0 N

. T

hen, the block was

released.

The m

otion of the block undergoes SHM

. C

an you show that a = –(k/m

) x ? A

lso find: (a) k

(b)��

(c) T

(d) vm

ax (where?, w

hen?)

(e) x, v and a at t = 2 sec

k 0.50 kg

6

An object on the end of a spring is oscillating in simple

harmonic m

otion. If the amplitude of oscillation is doubled,

how does this affect the oscillation period T and the object’s

maxim

um speed v

max ?

A. T and vm

ax both double.

B. T remains the sam

e and vm

ax doubles.

C. T and vm

ax both remain the sam

e.

D. T doubles and v

max rem

ains the same.

E. T remains the sam

e and vm

ax increases by a factor of

Q14.1

2.

An object on the end of a spring is oscillating in simple

harmonic m

otion. If the amplitude of oscillation is doubled,

how does this affect the oscillation period T and the object’s

maxim

um speed v

max ?

A. T and vm

ax both double.

B. T remains the sam

e and vm

ax doubles.

C. T and vm

ax both remain the sam

e.

D. T doubles and v

max rem

ains the same.

E. T remains the sam

e and vm

ax increases by a factor of .

A14.1

2.

This is an x-t graph for an object in sim

ple harmonic

motion.

A. t = T/4

B. t = T/2

C. t = 3T/4

D. t =

T

Q14.2

At which of the follow

ing times does the object have the

most negative velocity v

x ?

This is an x-t graph for an object in sim

ple harmonic

motion.

A. t = T/4

B. t = T/2

C. t = 3T/4

D. t =

T

A14.2

At which of the follow

ing times does the object have the

most negative velocity v

x ?

This is an x-t graph for an object in sim

ple harmonic

motion.

A. t = T/4

B. t = T/2

C. t = 3T/4

D. t =

T

Q14.3

At which of the follow

ing times does the object have the

most negative acceleration a

x ?

This is an x-t graph for an object in sim

ple harmonic

motion.

A. t = T/4

B. t = T/2

C. t = 3T/4

D. t =

T

A14.3

At which of the follow

ing times does the object have the

most negative acceleration a

x ?

A.t = T/8

B.t = T/4

C. t = 3T/8

D. t = T/2

E. more than one of the above

This is an x-t graph for an object connected to a spring and m

oving in sim

ple harmonic

motion.

Q14.6

At which of the follow

ing times is the potential

energy of the spring the greatest?

This is an x-t graph for an object connected to a spring and m

oving in sim

ple harmonic

motion.

A14.6

At which of the follow

ing times is the potential

energy of the spring the greatest?

A.t = T/8

B.t = T/4

C. t = 3T/8

D. t = T/2

E. more than one of the above

A.t = T/8

B.t = T/4

C. t = 3T/8

D. t = T/2

E. more than one of the above

This is an x-t graph for an object connected to a spring and m

oving in sim

ple harmonic

motion.

Q14.7

At which of the follow

ing times is the kinetic

energy of the object the greatest?

A.t = T/8

B.t = T/4

C. t = 3T/8

D. t = T/2

E. more than one of the above

This is an x-t graph for an object connected to a spring and m

oving in sim

ple harmonic

motion.

A14.7

At which of the follow

ing times is the kinetic

energy of the object the greatest?

To double the total energy of a mass-spring

system oscillating in sim

ple harmonic m

otion, the am

plitude must increase by a factor of

A. 4.

B.

C. 2.

D.

E.

Q14.8

�2

1.414.

�42

1.189.

�2

22.828.

A. 4.

B.

C. 2.

D.

E.

To double the total energy of a mass-spring

system oscillating in sim

ple harmonic m

otion, the am

plitude must increase by a factor of

A14.8

�2

1.414.

�42

1.189.

�2

22.828.

A simple pendulum

consists of a point mass

suspended by a massless, unstretchable string.

If the mass is doubled w

hile the length of the string rem

ains the same, the period of the

pendulum

A. becomes 4 tim

es greater.

B. becomes tw

ice as great.

C. becomes greater by a factor of .

D. rem

ains unchanged.

E. decreases.

Q14.9

2

A simple pendulum

consists of a point mass

suspended by a massless, unstretchable string.

If the mass is doubled w

hile the length of the string rem

ains the same, the period of the

pendulum

A. becomes 4 tim

es greater.

B. becomes tw

ice as great.

C. becomes greater by a factor of .

D. rem

ains unchanged.

E. decreases.

A14.9

2

This is an ax -t graph

for an object in sim

ple harmonic

motion.

A. t = 0.10 s

B. t = 0.15 s

C. t = 0.20 s

D. t =

0.25 s

Q14.4

At which of the follow

ing times does the object have the

most negative displacem

ent x?

This is an ax -t graph

for an object in sim

ple harmonic

motion.

A. t = 0.10 s

B. t = 0.15 s

C. t = 0.20 s

D. t =

0.25 s

A14.4

At which of the follow

ing times does the object have the

most negative displacem

ent x?

This is an ax -t graph

for an object in sim

ple harmonic

motion.

A. t = 0.10 s

B. t = 0.15 s

C. t = 0.20 s

D. t =

0.25 s

Q14.5

At which of the follow

ing times does the object have the

most negative velocity v

x ?

This is an ax -t graph

for an object in sim

ple harmonic

motion.

A. t = 0.10 s

B. t = 0.15 s

C. t = 0.20 s

D. t =

0.25 s

A14.5

At which of the follow

ing times does the object have the

most negative velocity v

x ?

Oscillations

2 25

Oscillations

2 26