Download PHY61-Practice Questions-T1 (Part 2)-Teachers-AY2020-2021

Term 1 (Part 2)– Practice Questions
Grade 11 Advanced – Physics (PHY61)
Learning Outcome
Topic 2: Forces
Subtopic 2.2– Newton’s Second Law
Subtopic 2.3– Newton’s Third Law
(KPIs 2.2.1 – 2.3.4)
# of
KPI’s
Number
of
Periods
12
4
8
10
Chapter
Chapter 4
• Practice Questions
Multiple choice questions
1. A force of 100.0 N is applied to an object of mass
85.0 kg. What is the object's acceleration?
√ A.
1.18 𝑚/𝑠 !
B.
2.49 𝑚/𝑠 !
C.
4.91 𝑚/𝑠 !
D.
9.81 𝑚/𝑠 !
4. Which of the following observations about the
friction force is incorrect?
2. An object whose mass is 0.092 kg is initially at rest
and then attains a speed of 75.0 m/s in 0.028 s. What
average net force acted on the object during this time
interval?
1.2 × 10 𝑁
√ B.
2.5 × 10! 𝑁
C.
2.8 × 10! 𝑁
D.
4.9 × 10! 𝑁
√ B.
The magnitude of the static friction is
always proportional to the normal force
C.
The magnitude of the static friction is
always proportional to the external
applied force
D.
The direction of the kinetic friction force is
always opposite the direction of the
relative motion of the object with respect
to the surface the object moves on.
5. A car of mass 𝑀 travels in a straight line at
constant speed along a level road with a coefficient of
friction between the tires and the road of 𝜇 and a
drag force of 𝐷. The magnitude of the net force on the
car is __.
3. You push a large crate across the floor at constant
speed, exerting a horizontal force F on the crate.
There is friction between the floor and the crate. The
force of friction has a magnitude that is ____.
A.
less than F
B.
greater than F
D.
The magnitude of the kinetic friction is
always proportional to the normal force
!
A.
√ C.
A.
A.
𝜇𝑀𝑔
B.
𝜇𝑀𝑔 + 𝐷
C.
4(𝜇𝑀𝑔)! + (𝐷)!
√ D.
equal to F
zero
1
Zero
6. A box with a mass of 10 kg is sliding along the top
of a rough, horizontal table at a speed of 30 m/s. The
friction between the box and the table brings the box
to rest in a time of 6.0 s. What is the coefficient of
friction between the box and the table?
A.
0.20
B.
0.31
√ C.
0.51
D.
10. A normal force is a contact force that acts at the
surface between two objects. Which of the following
statements concerning the normal force is not
correct?
The normal force is always equal to the
√ A.
force of gravity
√ A.
0 𝑚/𝑠 !
B.
1 𝑚/𝑠 !
C.
(𝑚𝑔) 𝑚/𝑠 !
D.
(2𝑚𝑔) 𝑚/𝑠 !
1.4 𝑚/𝑠 !
C.
2.1 𝑚/𝑠 !
D.
6.4 𝑚/𝑠 !
9. The figure below shows the directions and relative
magnitudes of all three forces that are acting on an
object. Which of the following best shows the
direction of the object’s acceleration?
C.
D.
The normal force is not necessarily equal
to the force of gravity
D.
The normal force is perpendicular to the
plane of the contact surface between the
two objects
√ A.
𝑚𝑔
B.
𝑀𝑔
C.
𝑚! 𝑔/𝑀
D.
𝑀! 𝑔/𝑚
12. A force is applied to a 100 kg go-cart to accelerate
it from 10.0 m/s to 20.0 m/s over a 7.00 s time frame.
If the acceleration is assumed to be constant over the
time frame, what is the magnitude of the force
applied?
A. 50.0 N
0 𝑚/𝑠 !
B.
B.
C.
11. A person stands on the surface of the Earth. The
mass of the person is 𝑚, and the mass of the Earth is
𝑀. The person jumps upward, reaching a maximum
height ℎ above the Earth. When the person is at this
height ℎ, the magnitude of the force exerted on the
Earth by the person is ______.
8. A 100 kg object rests on a level surface with a
coefficient of friction (both static and kinetic) of 0.40.
If a 250 N force is applied, find the magnitude of the
acceleration of the object.
√ A.
The normal force is just large enough to
keep the two objects from penetrating
each other
1.0
7. A horizontal force equal to the object’s weight is
applied to an object resting on a table. What is the
acceleration of the moving object when the
coefficient of kinetic friction between the object
and floor is 1 (assuming the object is moving in the
direction of the applied force)?
√ A.
B.
B.
123 N
√ C.
143 N
D.
150 N
13. There are only two forces on an object that has a
mass of 10.0 kg. Both forces have a magnitude of 10.0
N. The angle between the forces is 60.0 degrees. What
is the magnitude of the object's acceleration?
↗
↙
↑
↓
2
A.
1.41 𝑚/𝑠 !
B.
1.50 𝑚/𝑠 !
√ C.
1.73 𝑚/𝑠 !
D.
2.00 𝑚/𝑠 !
14. What determines the magnitude of the force of
static friction on an object at rest at all times on a flat
level horizontal surface being pushed to the right by
another solid object?
A.
the magnitude of the normal force in the
vertical direction
√ B.
the magnitude of the normal force in the
horizontal direction
C.
the object's weight
D.
None of the above
18. An object of mass m is at rest on a horizontal
surface, and the coefficient of static friction is µ" and
kinetic friction µ# . If a force F is applied to the object
as shown and it remains at rest, then it must be true
that ___.
15. A package rests on the back seat of your car. The
coefficient of friction between the seat and the
package is 0.24. Assuming you drive on a level road,
what is the maximum acceleration your car can have
if the package is to remain in place relative to your
back seat?
A.
1.0 𝑚/𝑠 !
√ B.
2.4 𝑚/𝑠 !
C.
3.0 𝑚/𝑠 !
D.
3.5 𝑚/𝑠 !
A.
𝐹 = µ" 𝑚𝑔
√ B.
𝐹 ≤ µ" 𝑚𝑔
C.
𝐹 ≥ µ" 𝑚𝑔
D.
𝐹 = µ# 𝑚𝑔
19. A block with mass m is being pushed by a
constant force F that makes an angle of q with the
horizontal as shown below. The block is moving with
constant velocity on a level surface. The coefficient of
kinetic friction between the block and the surface is
µk. Which one of the following equations is correct
for the magnitude of F?
Questions 16 and 17
A 20 kg box moving at an initial speed of 10 m/s
slides 25 m to the right on a horizontal floor before it
comes to a complete stop.
A.
16. What is the coefficient of friction between the box
and the floor?
A. 0.10
√ B.
0.20
C.
0.40
D.
0.60
√ B.
C.
D.
17. Which of the following best describes the
frictional forces exerted on the box and on the floor
while the box is sliding?
Box
Floor
µ# 𝑚𝑔
cos 𝜃 + µ# sin 𝜃
µ# 𝑚𝑔
𝐹=
cos 𝜃 − µ# sin 𝜃
µ# 𝑔
𝐹=
cos 𝜃 + 2𝑚𝑔 sin 𝜃
𝐹=
𝐹=
µ# 𝑚𝑔
tan 𝜃
20. An 85.0 kg snowboarder slides down a
frictionless slope inclined at an angle of 30°. What is
her acceleration?
B.
To the right
To the right
A.
1.18 𝑚/𝑠 !
C.
To the right
To the left
B.
2.49 𝑚/𝑠 !
D.
To the left
None
√ C.
4.91 𝑚/𝑠 !
√ E.
To the left
To the right
D.
9.81 𝑚/𝑠 !
3
21. In the figure, a force F is being applied to the
block with a magnitude such that it moves up along
the frictionless inclined place with a constant velocity
of 5 m/s. The angle in the figure is 30°, and the mass
of the block is 50 kg. What is the normal force exerted
by the inclined plane on the block?
A.
50 N
B.
240 N
√ C.
425 N
D.
500 N
25. A block slides down a frictionless incline and then
flies into the air as shown
22. Three blocks with masses of 1 kg, 2 kg, and 3 kg
are sliding down a plane inclined at an angle q = 25°.
If the coefficient of kinetic friction is the same for
each of the blocks, which block has the greatest
acceleration along the plane?
A.
1 kg block
B.
2 kg block
C.
3 kg block
√ D.
They all have the same acceleration
The acceleration of the 4-kg mass is 2.5
times greater than the acceleration of the
10-kg mass
C.
Both masses accelerate at the same rate
D.
The answer will depend on the length of
the time interval
1.60 N
√ C.
0.62 N
D.
1.24 N
B.
The acceleration changes sign when the
block leaves the incline
C.
The magnitude of the acceleration is a
maximum just after the block leaves the
incline and then increases
√ D.
The magnitude of the acceleration is a
maximum just after the block leaves the
incline and then stays constant
A.
12.3 N
B.
16.3 N
C.
19.4 N
√ D.
37.3 N
27. A wooden crate is at rest on a wooden ramp. (µs =
0.50; µk = 0.20; g = 9.8 m/s2). Find the angle q of the
ramp for which the crate will start sliding down the
ramp.
24. A 3 kg block slides at constant velocity down a
32° inclined plane. What is the coefficient of friction
between the block and the plane?
A. 0.31 N
B.
The magnitude of the acceleration is the
same during the entire motion including
while it is in the air
26. An object with mass 5.00 kg is being pushed up an
inclined plane with a constant force F. The plane
makes an angle of 30.0° with respect to the
horizontal. The object is moving with constant
velocity. The coefficient of kinetic friction between
the object and the surface is 0.300. What is the
magnitude of this force?
23. A 10 kg and a 4 kg mass are acted on by the same
size net force (which remains constant) for the same
period of time. Which one of the following statements
is true?
The acceleration of the10-kg mass is 2.5
A. times greater than the acceleration of the
4-kg mass
√ B.
A.
4
√ A.
26.6˚
B.
32.6˚
C.
57.4˚
D.
63.4˚
28. A 2.0 kg block is released from rest at the top of a
rough plane inclined at 37° to the horizontal, as
shown below. The block slides down the incline with
an acceleration of 4.0 𝑚/𝑠 ! . What is the approximate
magnitude of the force of friction on the block as it
slides down the incline?
√ A.
4.0 N
B.
6.0 N
C.
10 N
D.
12 N
Questions 32 and 33:
A 15 kg crate is put on a 25° inclined plane, as shown
in the figure below.
32. Find the minimum coefficient of static friction
between the crate and the incline in order to keep the
crate at rest.
A. 0.38
29. An object of mass 𝑚 moves with acceleration 𝑎
down a frictionless incline that makes an angle with
the horizontal, as shown below. If 𝑁 is the normal
force exerted by the plane on the block, which of the
following is correct?
√ B.
0.47
C.
0.55
D.
0.60
33. What force should be applied on the crate so that
it accelerates down the incline at 1.5 m/s2, knowing
that the coefficient of kinetic friction is 0.35?
√ A.
7.00 N
B.
9.23 N
A.
𝑁 = 𝑚𝑔
B.
𝑎 = 𝑚𝑔𝑠𝑖𝑛𝜃
C.
11.4 N
𝑎 = 𝑔𝑠𝑖𝑛𝜃
D.
13.1 N
√ C.
D.
𝑎 = 𝑚𝑔𝑐𝑜𝑠𝜃
34. Consider two blocks hung by a massless string
from a frictionless pulley as shown in the figure. The
mass of block 𝑚$ is 5 kg. The bottoms of the blocks
are 20 meters above the top of the table. Mass 𝑚!
accelerates downward at 3.5 m/s2. What is the mass
of 𝑚! ?
31. A block of mass4.00-kg rests on a 30.0° incline as
shown in the figure below. What is the minimum
horizontal force F on the block that will start moving
it up the incline if the coefficient of static friction
between the block and the incline is 0.700?
A.
√
5.0 kg
√ B.
10.5 kg
B.
51.1 N
C.
12.3 kg
C.
54.7 N
D.
15.0 kg
D.
76.4 N
E.
84.0 N
5
35. Two blocks, of masses m1 and m2 are connected
by a rope that passes over a massless, frictionless
pulley as shown in the figure below. Given m2 > m1.
Which of the following expressions gives the
acceleration of each block once the system is
released?
A.
√ B.
C.
D.
38. A block of mass 𝑀$ on a horizontal table is
connected to a hanging block of mass 𝑀! by a
string that passes over a pulley, as shown below.
The acceleration of the blocks is 0.6𝑔. Assume that
friction and the mass of the string are negligible. The
tension 𝑇 in the string is ___.
𝑚! + 𝑚$
O
P𝑔
𝑚! − 𝑚$
𝑚! − 𝑚$
O
P𝑔
𝑚! + 𝑚$
𝑚!
O
P𝑔
𝑚! + 𝑚$
𝑚$
O
P𝑔
𝑚! + 𝑚$
A.
2.14
√ B.
3.27
C.
4.98
D.
6.12
2.68 N
B.
5.91 N
C.
14.7 N
D.
39.2 N
0.4 𝑀! 𝑔
C.
0.6 𝑀! 𝑔
D.
1.0 𝑀! 𝑔
A.
1594 N
B.
2426 N
√ C.
2860 N
D.
3354 N
40. A person with a mass of 50 kg is standing on a
scale in an elevator that is accelerating upwards at a
rate of 3.2 m/s2. What is the reading on the scale (the
apparent weight)?
37. A mass 𝑚$ of 4.00 kg slides on a frictionless
surface. This mass is connected to another mass 𝑚!
of 1.50 kg by a massless string over a frictionless
pulley. The masses are held motionless and then
released. After the masses are released, what is the
tension in the string?
√ A.
√ B.
39. An elevator, with a total mass of 454 kg,
accelerates downward at 3.51 𝑚/𝑠 ! . During this time,
the tension in the elevator cable is _____.
36. A mass 𝑚$ of 3.00 kg slides on a frictionless
surface. This mass is connected to another mass 𝑚!
of 1.50 kg by a massless string over a frictionless
pulley. The masses are held motionless and then
released. What is the acceleration of 𝑚$ in m/s2?
A.
Zero
√
A.
50 N
B.
160 N
C.
490 N
D.
650 N
41. A student with a mass of 50 kg is standing on
a bathroom scale while riding in an elevator. If the
reading on the scale is 400 N, which of the following
is a correct description of the elevator’s motion?
A.
Moving upward with increasing speed
B.
Moving upward with constant speed
C.
Moving downward with constant speed
√ D.
6
Moving downward with increasing speed
42. A 75 kg man rides an elevator which is
accelerating upward with a uniform acceleration of
1.4 m/s2. What is the maximum force of static friction
if the coefficient of friction between him and the floor
is 0.42?
A.
260 N
B.
310 N
√ C.
350 N
D.
840 N
46. A 7 𝑘𝑔 block and a 2 𝑘𝑔 block are in contact with
each other on a horizontal frictionless surface. The
7 𝑘𝑔 block is pushed by a 36 𝑁 force. What is the
magnitude of the force exerted by the 2 𝑘𝑔 block on
the 7 𝑘𝑔 block?
√
43. You are riding an elevator that is moving upward
with 3 m/s and slowing down at 2 m/s2. Your real
weight is 490 N. Your apparent weight is
√ A.
390 N
B.
490 N
C.
680 N
D.
980 N
𝑇 = 𝑚𝑔
B.
𝑇 = 𝑚𝑔 − 𝑎
√ C.
𝑇 = 𝑚(𝑔 − 𝑎)
D.
𝑇 = 𝑚(𝑔 + 𝑎)
2N
B.
8N
C.
20 N
D.
28 N
47. Three blocks, 𝐴, 𝐵, and 𝐶, of masses 1, 2, and 3 kg,
respectively, are initially at rest on a frictionless
surface as indicated in the figure below. What force 𝐹
has to be applied on block 𝐶 to accelerate the three
blocks at 2.2 𝑚/𝑠 ! ?
44. An object with mass m is hanging from a wire
attached to the ceiling of an elevator. The elevator is
moving downward and its speed is increasing at rate
of a = 5 m/s2. Which one of the following statements
is true?
A.
A.
A.
1.5 N
B.
3.0 N
C.
6.0 N
√ D.
12 N
48. A force 𝐹 accelerates a system of two blocks, 𝑋
and 𝑌, on a horizontal frictionless surface, as shown
below. The acceleration is 4 𝑚/𝑠 ! .
45. Two masses, 𝑚$ = 3.0 𝑘𝑔 and 𝑚$ = 4.0 𝑘𝑔 rest
on a frictionless surface as shown. A force of 7.0 N is
applied to 𝑚$ . What is the acceleration of the two
masses
√ A.
1.0 𝑚/𝑠 !
B.
2.5 𝑚/𝑠 !
C.
3.0 𝑚/𝑠 !
D.
!
3.5 𝑚/𝑠
The force with which block 𝑌 pushes on block 𝑋 has
magnitude ___.
7
A.
6N
B.
8N
√ C.
16 N
D.
24 N
49. Blocks X and Y of masses 3.0 kg and 5.0 kg,
respectively, are connected by a light string and are
both on a level horizontal surface of negligible
friction. A force 𝐹 = 12 𝑁 is exerted on block Y, as
shown in the figure above.
Questions 52 and 53:
A 6.0kg box is prevented from sliding down a vertical
wall by applying a horizontal force of 75N as shown
in the figure below.
6.0 kg
75 N
What is the tension in the string connecting the two
blocks?
A. 4.0 N
√ B.
4.5 N
C.
7.5 N
D.
12 N
52. What are the directions of the friction and normal
force on the box?
Force of friction Normal force
Upward
A. Left
√
50. A horizontal force F pushes a block of mass m
against a vertical wall. The coefficient of friction
between the block and the wall is μ. What value of F
is necessary to keep the block from slipping down the
wall?
A. mg
B.
√ C.
mg /μ
mg(1 - μ)
E.
mg(1 + μ)
√
51. The Tornado is a carnival ride that consists of a
hollow vertical cylinder that rotates rapidly about its
vertical axis. As the Tornado rotates, the riders are
pressed against the inside wall of the cylinder
by the rotation, and the floor of the cylinder drops
away. The force that points upward, preventing the
riders from falling downward, is _____.
√ A.
Upward
Right
C.
Upward
Left
D.
Downward
Right
E.
Downward
Left
53. What is the minimum coefficient of static friction
required to prevent the box from slipping down the
wall?
A. 0.08
μmg
D.
B.
B.
0.50
C.
0.61
D.
0.78
E.
1.30
54. Four weights, of masses 𝑚$ = 3.00 𝑘𝑔, 𝑚! =
1.00 𝑘𝑔, 𝑚% = 4.00 𝑘𝑔, 𝑚& = 2.00 𝑘𝑔, are hanging
from the ceiling as shown. They are connected to
each other with ropes. The tension in the rope
connecting the masses 𝑚$ and 𝑚! is ______.
friction force
B.
a normal force
A.
19.6 N
C.
gravity
B.
58.9 N
D.
a tension force
√ C.
68.7 N
D.
98.1 N
8
55. A sky diver has jumped from a plane and achieved
a nearly constant velocity as she falls. Which
statement below correctly explains how this
happened.
She was high enough from the Earth that g
A.
was negligibly small
B.
She had reached equilibrium; the frictional
force exceeded her weight
C.
Terminal velocity occurred when the air
was so thin that friction became negligible.
√ D.
59. A 2.0 kg object, initially at rest at the origin of x-y
coordinate system, is subjected to two forces, Fx in
the positive x-direction, and Fy in the positive
y-direction, whose time-varying magnitudes are
shown in graphs below. Calculate the x and y
components of the object's velocity at t = 2 s.
She reached equilibrium; the frictional
force equaled her weight
56. An object is released from rest from a great height
and reaches its terminal velocity. Which of the
following statements is true of the object while it is
falling with terminal velocity?
A.
B.
C.
√ D.
There is no longer a gravitational force on it
There is no longer a drag (air resistance)
force on it
Its acceleration is upward
The magnitudes of the gravitational and drag
forces on it are equal
57. Find the acceleration of the skydiver when his
speed reaches 12.0 m/s.
A. 7.25 m/s2
7.74 m/s2
√ C.
8.32 m/s2
D.
9.80 m/s2
E.
11.3 m/s2
53.6 m/s
C.
60.3 m/s
D.
68.1 m/s
√ E.
79.6 m/s
√ B.
𝑣' = 2.5 𝑚/𝑠,
𝑣( = 8.0 𝑚/𝑠
C.
𝑣' = 8.0 𝑚/𝑠,
𝑣( = 2.5 𝑚/𝑠
D.
𝑣' = 8.0 𝑚/𝑠,
𝑣( = 8.0 𝑚/𝑠
A.
the work done by gravity is zero
B.
the work done by air resistance is zero
√ C.
D.
58. Find the terminal velocity of the skydiver.
A. 47.6 m/s
B.
𝑣' = 2.5 𝑚/𝑠,
𝑣( = 2.5 𝑚/𝑠
60. A skydiver is subject to two forces: gravity and air
resistance. Falling vertically, she reaches a constant
terminal speed at some time after jumping from a
plane. Since she is moving at a constant velocity from
that time until her chute opens, we conclude from the
work–kinetic energy theorem that, over that time
interval,
Question 57 and 58:
A 65.0 kg skydiver, falling through the air with a
speed of 15.0 m/s, opens his parachute. As a result,
he experiences a drag force of magnitude
𝐹 = 8𝑣, where 𝑣 is the speed of the skydiver.
B.
A.
9
the work done by gravity equals the negative
of the work done by air resistance
the work done by gravity equals the work
done by air resistance
Answer the following questions:
1. Multiselect: (indicate all possibilities)
Only two forces, 𝐹$ and 𝐹$ , are acting on a block. Which of the following can be the magnitude of the net force, 𝐹,
acting on the block?
A.
𝐹 > 𝐹$ + 𝐹!
√ B.
𝐹 = 𝐹$ + 𝐹!
√ C.
𝐹 < 𝐹$ + 𝐹!
2. Multiselect: (indicate all possibilities)
An SUV of mass 3250 kg has a head-on collision with a 1250 kg subcompact. Identify all the statements that are
incorrect.
√ A.
The SUV exerts a larger force on the subcompact than the subcompact exerts on the SUV
√ B.
The subcompact exerts a larger force on the SUV than the SUV exerts on the subcompact
C.
The subcompact experiences a larger acceleration than the SUV
√ D.
The SUV experiences a larger acceleration that the subcompact
3. True or False?
a. To move a book along a table, you need to apply a force to overcome its weight. (False)
b. A massless string passes over a massless pulley. A 50 N weight is attached to one end of the string and 100
N to the other end. Tension on one side of the string is twice that on the other side. (False)
Learning Outcome
# of
KPI’s
Number
of Periods
4
2
Topic 3– Work, Energy and Power
Subtopic 3.1: Work
(KPIs 3.1.1 – 3.1.4)
Chapter
Chapter 5
• Practice Questions
Multiple choice question
1. If negative work is being done by an object, which
one of the following statements is true?
B.
An object is moving in the negative xdirection
An object has negative kinetic energy
C.
Energy is being transferred from an object
A.
√ D.
2. Which of the following is not a unit of energy?
A.
Newton-meter
B.
Joule
C.
Kg m2/s2
√ D.
Energy is being transferred to an object
10
All the above are units of energy
3. Kathleen climbs a flight of stairs. What can we say
about the work done by gravity on her?
√ A.
Gravity does negative work on her
B.
Gravity does positive work on her
C.
Gravity does no work on her
We can’t tell what work gravity does on
her
D.
6. A box of mass m is pulled by a force F, which makes
an angle 𝜙 with the horizontal. A frictional force f is
applied on the box as it slides on a horizontal surface,
as shown in the figure below. Which of the following
represents the net work on the box when it covers a
distance d?
3. A student pushes an object along a horizontal
surface a distance d with a force F at an angle 𝜃 . If the
velocity is constant at a value v, then ____.
√ A.
A.
𝐹𝑑
B.
𝐹 cos 𝜙 𝑑
√ C.
(𝐹 cos 𝜙 − 𝑓)𝑑
D.
(𝐹 sin 𝜙 − 𝑓)𝑑
7. You pull sled with a rope which makes an angle of
30° to the horizontal and the tension in the rope is
40.0 N. Calculate the work you do on the sled if it
moves 100 m.
there is no net work done
B.
the net work done is 𝐹𝑑 cos 𝜃
A.
1.22 kJ
C.
the net work done is 𝐹𝑑 sin 𝜃
B.
2.00 kJ
D.
the net work done is 𝐹𝑣
√ C.
3.46 kJ
D.
5.00 kJ
4. Jack is holding a box that has a mass of 𝑚 𝑘𝑔. He
walks a distance of 𝑑 𝑚 at a constant speed of 𝑣 𝑚/𝑠.
How much work, in joules, has Jack done on the box?
A.
−𝑚𝑔𝑑
B.
𝑚𝑔𝑑
1
𝑚𝑣 !
2
Zero
C.
√ D.
8. How much work do movers do (horizontally) in
pushing a 150 kg crate 12.3 m across a floor at
constant speed if the coefficient of friction is 0.70?
5. How much work is done when a 75.0 kg person
climbs a flight of stairs 10.0 m high at constant
speed?
A.
75 J
B.
750 J
√ C.
D.
A.
1.3 × 10% 𝐽
B.
1.8 × 10% 𝐽
√ C.
1.3 × 10& 𝐽
D.
1.8 × 10& 𝐽
9. A boy pushes a 35 kg crate across a horizontal floor
with a force of 250 N. The coefficient of kinetic
friction between the crate and the floor is 0.30. How
far does the crate move if the net work on it is
2400 J?
7360 J
A.
8.5 m
736000 J
B.
12 m
C.
16 m
D.
21 m
√
11
10. A 30.0 kg crate is pushed across a rough floor by a
horizontal force of 100 N. The coefficient of kinetic
friction between the crate and the floor is 0.100. If a
total of 80.5 J of work is done on the crate, how far
was it moved along the floor?
A.
14. In the figure, a force F is being applied to the
block with a magnitude such that it moves up along
the frictionless inclined place with a constant velocity
of 5 m/s. The angle in the figure is 30°, and the mass
of the block is 50 kg. How much work (in Joules) is
done by the force F in moving the block 10 meters
along the surface of the inclined plane?
0.840 m
√ B.
1.14 m
C.
1.64 m
D.
2.04 m
A.
11. A person pushes a box of mass m a distance d
across a floor. The coefficient of kinetic friction
between the box and the floor is μk. The person then
picks up the box, raises it to a height h, carries it back
to the starting point, and puts it back down on the
floor. How much work has the person done on the
box?
A. zero
√ B.
µ# 𝑚𝑔𝑑 + 2𝑚𝑔ℎ
D.
µ# 𝑚𝑔𝑑 − 2𝑚𝑔ℎ
√
2500 J
C.
4200 J
D.
5000 J
Questions 15 to 17
A 10 𝑘𝑔 block is released from rest at the top of an
inclined plane of height h = 2m, as shown in the
figure below. The coefficient of kinetic friction
between the block and the surface of the incline is
0.2.
12. A crane lifts a crate of mass 425 kg vertically
upward by a distance of 117 m. How much work does
the crane do on the crate to accelerate it upward at
1.8 m/s2? Neglect frictional forces.
B.
√ B.
µ# 𝑚𝑔𝑑
C.
A.
500 J
15. The work done by gravity is ______________.
)
4.0 × 10 𝐽
)
4.9 × 10 𝐽
√
)
C.
5.8 × 10 𝐽
D.
7.2 × 10) 𝐽
B.
−200 𝐽
C.
0𝐽
D.
200 𝐽
E.
350 𝐽
16. The work done by friction is ________________.
13. A 25.0 kg box is pulled by a 125 N force, which
makes an angle 20.0° with the horizontal. A frictional
force of 55.0 N is applied on the box as it slides on a
horizontal surface. What is the work done on the box
when it slides 30.0 m?
√
A.
−85 𝐽
B.
−68 𝐽
C.
0𝐽
D.
68 𝐽
√ A.
1870 J
B.
2140 J
C.
3220 J
B.
69 J
D.
4560 J
C.
85 J
D.
200 J
17. The work done by the normal force is _____.
√ A. 0 J
12
21. The following graph shows the force 𝐹⃑ exerted on
a 2 kg object as a function of the distance d that the
object travels. The object is at rest at 𝑑 = 0 and
travels on a horizontal, frictionless surface along the
line of action of the force. The work done on the
object by the force 𝐹⃑ during the first 10 m of travel is
most nearly ____.
18. An 800-N box is pushed up an inclined plane that
is 4.0 m long. It requires 3200 J of work to get the box
to the top of the plane, which is 2.0 m above the base.
What is the magnitude of the average friction force
on the box? (Assume the box starts at rest and ends
at rest.)?
A.
0N
√ B.
400 N
A.
10 J
C.
800 N
B.
50 J
D.
1600 N
√ C.
75 J
D.
19. 541 J of work is required to slide a 20-kg ice block
up a frictionless slope. The incline of the slope with
respect to horizontal is 9.71°. What is the length of
the slope?
A.
12 m
√ B.
16 m
C.
10 m
D.
23 m
100 J
3. The net force on an object is represented in the
force-position graph shown below. Find the work
done on the object as it moves from x = –40 m to x =
40 m.
20. A particle moves parallel to the x-axis. The net
force on the particle increases with x according to the
formula 𝐹𝑥 = (120 𝑁/𝑚)𝑥, where the force
is in newtons when x is in meters. How much work
does this force do on the particle as it moves from
x = 0 to x = 0.50 m?
A.
7.5 J
√ B.
15 J
C.
30 J
D.
60 J
√
Answer the following questions.
1. Is each of the following statements true or false?
a. Work cannot be done in the absence of motion. (True)
b. A force is required to do work. (True)
13
A.
90 J
B.
60 J
C.
40 J
D.
30 J
2. A 95.0 kg refrigerator rests on the floor. How much work is required to move it at constant speed for 4.00 m
along the floor against a friction force of 180. N?
𝑊 = 𝐹* 𝑑 cos 𝜃
𝑊 = (180 𝑁)(4.0 𝑚) cos 0 = 720 𝐽
3. Suppose you pull a sled with a rope that makes an angle of 30.0˚ to the horizontal. How much work do you do if
you pull with 25.0 N of force and the sled moves 25.0 m?
Only the component of the force parallel to
the displacement does work.
𝑊 = 𝐹𝑑 cos 𝜃
𝑊 = (25.0 𝑁)(25.0 𝑚) cos 30˚
𝑊 = 5.41 × 10! 𝐽
4. A particle of mass m is subjected to a force acting in the x-direction, 𝐹' = (3.00 + 0.500𝑥) 𝑁. Find the work done
by the force as the particle moves from x = 0.00 to x = 4.00 m.
𝒙𝒇
&
𝑊 = d 𝐹(𝑥)𝑑𝑥 = d (3.00 + 0.500𝑥)𝑑𝑥
𝒙𝒊
'-&
+
1
𝑊 = e3𝑥 + 𝑥 ! f
4
'-+
1
𝑊 = 3(4) + (4)! − 0 = 16 𝐽
4
5. The graph shows the component (𝐹 cos 𝜃) of the net force that acts on a 2.00 kg block as it moves along a flat
horizontal surface.
a. Find the net work done on the block.
𝑊./0 = ∑𝑊 = 𝐹$ 𝑑$ + 𝐹! 𝑑! + 𝐹% 𝑑%
+ 𝐹& 𝑑&
𝑊./0 = (0.0 𝑁)(1.0 𝑚)
+ (2.0 𝑁)(4.0 𝑚)
+ (−1.0 𝑁)(2.0 𝑚)
+ (0.0 𝑁)(1.0 𝑚)
𝑊./0 = 6.0 𝑁𝑚
b. Find the final speed of the block if it
starts from rest at s = 0
1
1
𝑊./0 = 𝑚𝑣* ! − 𝑚𝑣1 !
2
2
𝑣* = h
2
1
O𝑊./0 + 𝑚𝑣1 ! P
𝑚
2
𝑣* = h
2
(6.0 𝑁𝑚 + 0) = 2.4 𝑚/𝑠
2.0 𝑘𝑔
14
Learning Outcome
# of
KPI’s
Number
of Periods
3
2
Topic 3– Work, Energy and Power
Subtopic 3.2: Work-Energy Theorem
(KPIs 3.2.1 – 3.2.3)
Chapter
Chapter 5
• Practice Questions
Multiple choice question
5. A 3.0 kg crate rests at the bottom of a plane
inclined at an angle of 15.0° above the horizontal. The
crate is given a push up the plane and when it has
travelled a distance of 2.0 m up the plane its speed is
3.0 m/s. How much work is done on the crate by
gravity?
1. The work–kinetic energy theorem is equivalent to
A. Newton’s first law
√ B.
Newton’s second law
C.
Newton’s third law
D.
None of Newton’s laws
2. Initially an object of mass 1.00 kg is moving to the
left at 10.0 m/s. If 150 J of work is done on the object,
then how fast will it be moving?
A. 17.3 m/s
√ B.
20.0 m/s
C.
25.0 m/s
D.
27.3 m/s
√ A.
–15 J
B.
–30 J
C.
–45 J
D.
–60 J
6. An object slides without friction down an incline
and loses height h = 131 m. The incline makes an
angle of 35.0° with respect to the horizontal. If the
object started with a speed of 45.0 m/s, what will be
its final speed in m/s?
3. A 20.0 kg object starts from rest and slides down
an inclined plane. The change in its elevation is 2.5 m
and its final speed is 5.00 m/s. How much energy did
the object lose due to friction?
√ A. 241 J
A.
21.1
B.
57.2
B.
250 J
C.
45.0
C.
491 J
√ D.
67.8
D.
741 J
7. How does the work required to accelerate a
particle from 10𝑚/𝑠 to 20𝑚/𝑠 compare to that
required to accelerate it from 20𝑚/𝑠 to 30𝑚/𝑠?
4. A curling stone of mass m is given an initial velocity
v on ice, where the coefficient of kinetic friction is μk.
The stone travels a distance d. If the initial velocity is
doubled, how far will the stone slide?
A. 𝑑/2
√ A.
It is less
B.
It is the same
B.
𝑑
C.
It is greater
C.
2𝑑
D.
It cannot be determined without knowing the
magnitude of the force exerted on the particle
√ D.
4𝑑
15
8. The following graph shows the force 𝐹⃑ exerted on a
2 kg object as a function of the distance d that the
object travels. The object is at rest at 𝑑 = 0 and
travels on a horizontal, frictionless surface along the
line of action of the force.
Questions 11 & 12
A 2 kg block is initially at rest on a horizontal
frictionless table. A force of 15 N is then exerted on
the block at an angle of 37° to the horizontal, as
shown below.
11. The change in the kinetic energy of the block after
moving a distance of 3 m is most nearly_____.
The kinetic energy of the 2 kg object when 𝑑 equals
20 m is the same as when 𝑑 is most nearly ____.
A.
60 J
A.
0m
B.
45 J
√ B.
5m
√ C.
36 J
D.
27 J
C.
10 m
D.
12.5 m
12. The magnitude of the force exerted on the block
by the table is most nearly ___.
Questions 9 & 10.
A 20 kg box moving at an initial speed of 10 𝑚/𝑠
slides 25 m to the right on a horizontal floor before it
comes to a complete stop.
9. What is the coefficient of friction between the box
and the floor?
A.
0.10
√ B.
0.20
C.
0.40
D.
0.60
35 N
B.
32 N
√ C.
29 N
D.
20 N
13. A student pushes a box across a rough horizontal
floor. If the amount of work done by the student on
the box is 100 J and the amount of energy dissipated
by friction is 40 J, what is the change in kinetic energy
of the box?
10. Which of the following best describes the
frictional forces exerted on the box and on the floor
while the box is sliding?
Box
Floor
A.
None
None
B.
To the right
To the right
C.
To the right
To the left
To the left
To the right
√ D.
A.
16
A.
0J
B.
40 J
√ C.
60 J
D.
100 J
E.
140 J
Answer the following questions
1. A force given by 𝐹(𝑥) = 5𝑥 % (𝑖𝑛 𝑁/𝑚% ) acts on a 1.00 kg mass moving on a frictionless surface. The mass moves
from x = 2.00 m to x = 6.00 m.
a. How much work is done by the force?
𝒙𝒇
&
𝑊 = d 𝐹(𝑥)𝑑𝑥 = d (5𝑥 % )𝑑𝑥
𝒙𝒊
+
𝒙𝒇
5
5
𝑊 = e 𝑥 & f = i𝑥* & − 𝑥1 & j
4
4
𝒙𝒊
5
𝑊 = 𝑁/𝑚% [(6 𝑚)& − (2 𝑚)& ] = 1600 𝐽
4
b. If the mass has a speed of 2.00 m/s at x = 2.00 m, what is its speed at x = 6.00 m?
1
1
1
𝑊 = ∆𝐾 = 𝐾* −𝐾1 = 𝑚𝑣* ! − 𝑚𝑣1 ! = 𝑚i𝑣* ! − 𝑣1 ! j
2
2
2
2𝑊
2(1600 𝐽)
𝑣1 = h
+ 𝑣1 ! = h
+ (2.00 𝑚/𝑠)! = 56.6 𝑚/𝑠
𝑚
1.00 𝑘𝑔
2. The 125 kg cart in the figure starts from rest and rolls with negligible friction. It is pulled by three ropes as
shown. It moves 100. m horizontally. Find the final velocity of the cart.
𝐹$' = 𝐹$ cos 𝜃$ = (300) cos 0˚ = 300 𝑁
𝐹!' = 𝐹! cos 𝜃! = (300) cos 40˚ = 193 𝑁
𝐹%' = 𝐹% cos 𝜃% = (200) cos 150˚ = −173 𝑁
𝐹' = 𝐹$' + 𝐹!' + 𝐹%' = 320𝑁
1
1
𝑊 = 𝐹' ∙ ∆𝑥 = 𝑚𝑣* ! − 𝑚𝑣1 !
2
2
𝑣* = h
2𝐹' ∙ ∆𝑥
2(320 𝑁)(100 𝑚)
=h
= 23 𝑚/𝑠
𝑚
125 𝑘𝑔
17
Learning Outcome
# of
KPI’s
Number
of Periods
3
1
Topic 3– Work, Energy and Power
Subtopic 3.3: Power
(KPIs 3.3.1 – 3.3.3)
Chapter
Chapter 5
• Practice Questions
Multiple choice question
1. Which of the following is the correct unit for
power?
√
A.
Kg m/s2
B.
J/s
C.
N
D.
m/s2
3. An electrical motor provides 0.50 W of mechanical
power. How much time will it take the motor to lift a
0.1 kg mass at constant speed from the floor to a shelf
2.0 m above the floor?
1. A 1000 W electric motor lifts a 100 kg safe at
constant velocity. The vertical distance through
which the motor can raise the safe in 10 s is most
nearly_____.
√
A.
1m
B.
3m
C.
10 m
D.
32 m
A.
0.40 s
B.
1.0 s
C.
2.0 s
√ D.
4.0 s
4. An engine pumps water continuously through a
hose. If the speed with which the water passes
through the hose nozzle is v and if k is the mass per
unit length of the water jet as it leaves the nozzle,
what is the power being imparted to the water?
A.
B.
2. A man of mass 60 kg runs up a flight of 60 steps in
40 seconds. If each step is 20 cm high, his power is __.
√
A.
18 W
B.
134 W
C.
177 W
D.
240 W
C.
√
D.
1
𝑘𝑣
2
1 !
𝑘𝑣
2
1 !
𝑣 /𝑘
2
1 %
𝑘𝑣
2
5. A 1500 kg car accelerates from 0 to 25 m/s in 7.0 s.
What is the average power delivered by the engine
(1 hp = 746 W)?
√
18
A.
60 hp
B.
70 hp
C.
90 hp
D.
180 hp
6. A 1000 kg car starts from rest and accelerates to
27 m/s in 6.0 s. What is the power required for this to
occur?
√
A.
4.5 × 10% 𝑊
B.
3.0 × 10& 𝑊
C.
6.1 × 10& 𝑊
D.
1.2 × 10) 𝑊
8. A lift is used to raise objects to the back of a
moving truck. If the maximum power the lift is
capable of delivering is 98 W, what is the maximum
constant speed with which this lift can raise a 5.0 kg
crate straight up from the ground to the back of the
moving truck? The back of the moving truck is 5.0 m
above the ground.
√
7. A 5.0 kg crate is lifted straight up from the ground
at a constant speed of 1.5 m/s to the back of a moving
truck which is at a height of 5.0 m above the ground.
What power is needed to accomplish this task?
√
A.
25 W
B.
53 W
C.
74 W
D.
87 W
A.
2.0 m/s
B.
3.2 m/s
C.
7.0 m/s
D.
19 m/s
Answer the following questions
1. True or false?
More power is required to lift a box slowly than to lift a box quickly. (False)
2. A horse draws a sled horizontally on snow at constant speed. The horse can produce a power of 791 W. The
coefficient of friction between the sled and the snow is 0.115, and the mass of the sled, including the load, is 204.7
kg. What is the speed with which the sled moves across the snow?
𝑃 = 𝐹* 𝑣
𝑣=
𝑃
𝑃
=
𝐹* µ𝑚𝑔
𝑣=
791 𝑊
= 3.43 𝑚/𝑠
(0.115)(204.7 𝑘𝑔)(9.81 𝑚/𝑠 ! )
19
Learning Outcome
# of
KPI’s
Number
of Periods
5
7
Topic 4– Potential Energy and Energy Conservation
Subtopic 4.1: Force and Potential Energy
(KPIs 4.1.1 – 4.1.5)
Chapter
Chapter 6
• Practice Questions
Multiple choice question
4. Suppose that the potential energy of a particle
constrained to move along the 𝑥 -axis can be
$
described by the function 𝑈(𝑥) = ! 𝑘𝑥 ! − 𝛼𝑥, where
1. Which of the four drawings represents a stable
equilibrium point for the ball on its supporting
surface?
both 𝑘 and 𝛼 are positive constants. Stable
equilibrium points, about which the particle
oscillates, are located at _____.
A.
√
A.
𝑥 = 0 only
√ B.
𝑥 = only
B.
𝑥=
D.
𝑥 = 0 and #
#
only
2
Questions 5 and 6
A particle moving on the x-axis has a potential energy
given by the equation below.
𝑈 = 8𝑥 ! − 4𝑥 + 400.
D.
2. A 1210 kg car travels 1.20 km up an incline at
constant velocity. The incline is 15° measured with
respect to the horizontal. The change in the car's
potential energy is _____.
5. The force on the object at x = 1.00 m is _______.
A.
0.00 N
B.
12..0 N in the (+) x-axis
A.
4.37 J
B.
1.92 kJ
√ C.
12.0 N in the (–) x-axis
C.
1.92 MJ
D.
20.0 N in the (+) x-axis
D.
3.68 MJ
6. Its state of equilibrium will be at ____.
3. The potential energy of an object as a function of
position is 𝑈(𝑥) = 𝑥 ! − 𝑥 − 6, where 𝑈 is measured
in joules and x is measured in meters. Where is the
object be located if it is in a stable equilibrium?
√
#
!2
C.
C.
√
2
A.
𝑥 = −2 𝑚
B.
𝑥 =0𝑚
C.
𝑥 = 0.5 𝑚
D.
𝑥 =3𝑚
√
20
A.
𝑥 = 0.025 𝑚
B.
𝑥 = 0.25 𝑚
C.
𝑥 = 2.5 𝑚
D.
𝑥 = 25 𝑚
7. A particle of mass m moving in the x-y plane is
confined by a two-dimensional potential. The net
force on the mass is 𝐹(𝑥, 𝑦) = −𝑘(2𝑥 % + 4𝑦 % ). What
is the potential energy?
A. 𝑈(𝑥, 𝑦) = 𝑘(𝑥 & + 2𝑦 & )
√
B.
C.
D.
Questions 11 and 12
The variable x represents the position of particle A in
a two-particle system. Particle B remains at rest. The
graph below shows potential energy U of the system
as a function of x .
1
𝑈(𝑥, 𝑦) = 𝑘(𝑥 & + 2𝑦 & )
2
1
𝑈(𝑥, 𝑦) = 𝑘(3𝑥 & + 2𝑦 & )
2
1
𝑈(𝑥, 𝑦) = 𝑘(6𝑥 & + 12𝑦 & )
2
8. A certain one-dimensional conservative force is
given as a function of 𝑥 by the expression 𝐹 = −𝑘𝑥 % ,
where 𝐹 is in newtons and 𝑥 is in meters. A possible
potential energy function 𝑈 for this force is ____.
1
A. 𝑈 = − 𝑘𝑥 !
2
1 !
B. 𝑈 = 𝑘𝑥
2
1
C. 𝑈 = − 𝑘𝑥 &
4
1 &
√ D. 𝑈 = 𝑘𝑥
4
E. 𝑈 = −3𝑘𝑥 !
11. If the total energy of the system is –2 0 J, which
of the following statements is true?
A.
The system has zero potential energy
B.
Particle A has 2.0 J of kinetic energy
C.
The system has 2.0 J of total mechanical
energy
D.
Particle A is always at x = 0.40 m
√ E.
9. The net force acting on a particle moving on the xaxis is given by:
𝐹 = −9𝑥 ! + 4𝑥 + 2
Find the change in the potential energy of the particle
as it moves from x = 0.00 m to x = 2.00 m.
A. -36.00 J
Particle A oscillates between x = 0.20 m
and 0.65 m
12. The x-component of the force on particle A when
it is at x = 0.15 m is most nearly ___.
A. −20 𝑁
B.
−2.0 𝑁
-12.00 J
C.
−1.0 𝑁
C.
0.00 J
D.
2.0 𝑁
D.
12.00 J
√ E.
20 𝑁
E.
36.00 J
√ B.
13. Which of the following is not a valid potential
energy function for the spring force 𝐹 = – 𝑘𝑥?
1 !
A.
𝑘𝑥
2
1 !
B.
𝑘𝑥 + 10 𝐽
2
1 !
C.
𝑘𝑥 − 10 𝐽
2
1
√ D. − 𝑘𝑥 !
2
10. A spring has a spring constant of 80 N/m. How
much potential energy does it store when stretched
by 1.0 cm?
√ A. 0.004 J
B.
0.40 J
C.
0.80 J
D.
80 J
21
14. As a particle moves along the 𝑥 −axis, it is acted
upon by a conservative force. The potential energy is
shown as a fuction of the coordinate 𝑥 of the particle.
Rank the labeled regions according to the magnitude
of the force, least to greatest.
17. The graph below shows a conservative force 𝐹' as
a function of position 𝑥 acting on an object in a closed
system. If this is the only force acting on the object,
what happens to the potential energy of the system
as the object moves from 0 m to 0.10 m?
A.
AB, BC, CD
√ A.
It increases only
B.
AB, CD, BC
It decreases only
C.
BC, CD, AB
B.
√ D.
BC, AB, CD
C.
It increases and then decreases
D.
It decreases and then increases
15. A particle is released from rest at the point
𝑥 = 𝑎 and moves along the 𝑥 axis subject to the
potential energy function 𝑈(𝑥) shown.
18. The force exerted by a spring on a block attached
to it is 𝐹(𝑥) = −𝑘𝑥, where k is expressed in N/m and
x is expressed in meters. Find the work done by the
spring when it stretches from x = 0 to x = d.
√
The particle moves to_____.
a point to the left of x = e, stops, and
A.
remains at rest
√ B. a point to x = e, then moves to the left
C.
infinity at varying speed
D.
x = b, where it remains at rest
0.5𝑊3
B.
𝑊3
C.
2𝑊3
√ D.
4𝑊3
−𝑘𝑑!
B.
−𝑘𝑑! /2
C.
𝑘𝑑/2
D.
𝑘𝑑! /2
19. A spring with spring constant k is suspended
from the ceiling. When a mass of M kg is attached to
the spring, it stretches 𝑑$ 𝑚. The mass is then pulled
down an additional 𝑑! 𝑚 and let go. Which one of the
following statements about the resulting motion is
true?
16. If you compress a spring a distance ℎ from its
equilibrium position and do work 𝑊3 in the process,
how much work will be required to compress the
same spring a distance 2ℎ?
A.
A.
A.
B.
C.
√
22
D.
The maximum kinetic energy is
(1/2)𝑘𝑑$
The maximum kinetic energy is
(1/2)𝑘𝑑$ !
The maximum kinetic energy is
(1/2)𝑘𝑑!
The maximum kinetic energy is
(1/2)𝑘𝑑! !
20. The relationship between the magnitude of the
restoring force 𝐹 and the resultant displacement 𝑥
from equilibrium for a nonlinear spring is given by
the equation 𝐹 = 𝑘𝑥 ! . What is the potential energy of
the spring when it has been compressed a distance
𝑥+ ?
A.
B.
C.
√ D.
21. The system represented below consists of two
objects of unequal masses, 𝑀$ and 𝑀! , with 𝑀$ > 𝑀! .
The objects hang from the ends of a cord of negligible
mass that passes over a pulley with negligible mass
and friction. Which of the following is true about the
changes in the gravitational potential energy, ∆𝑈, and
kinetic energy, ∆𝐾, of the system soon after the
objects are released from rest?
1 !
𝑘𝑥
2 +
1 %
𝑘𝑥
2 +
1 !
𝑘𝑥
3 +
1 %
𝑘𝑥
3 +
√ A.
∆𝑈 < 0 and ∆𝐾 > 0
B.
∆𝑈 = 0 and ∆𝐾 > 0
C.
∆𝑈 < 0 and ∆𝐾 = 0
D.
∆𝑈 = 0 and ∆𝐾 = 0
Answer the following questions
1. True or False?
a. The kinetic energy of an object can be negative. (False)
b. The potential energy of an object can be negative. (True)
c. A force can be defined as a conservative force if the work done on an object by the force depends only on
the initial and final position of the object. (True)
d. The work done by a conservative force will be zero if the object undergoes a displacement that completes a
complete closed path.(True)
e. Friction is an example of conservative force (False)
f. The work done by a non-conservative force does not depend on the path taken. (False)
g. The work done by a non-conservative force appear in the system as internal energy rather than kinetic or
potential energy (True)
2. A spring with a spring constant of 238.5 N/m is compressed by 0.231 m. Then a steel ball bearing of mass 0.0413
kg is put against the end of the spring, and the spring is released. What is the speed of the ball bearing right
after it loses contact with the spring? (The ball bearing will come off the spring exactly as the spring returns to its
equilibrium position. Assume that the mass of the spring can be neglected.)
The ball starts at rest with 𝐾1 = 0. It returns to the equilibrium position 𝑥1 = 0, at
$
which time 𝐾* = ! 𝑚𝑣* ! . Using work-energy theorem.
1
1
1
1
𝑘𝑥1 ! − 𝑘𝑥* ! = 𝑚𝑣* ! − 𝑚𝑣* !
2
2
2
2
1
1
!
!
𝑘𝑥 − 0 = 𝑚𝑣* − 0
2 1
2
𝑘
238.5 𝑁/𝑚
𝑣* = 𝑥1 h = 0.231 𝑚h
= 17.6 𝑚/𝑠
𝑚
0.0413 𝑘𝑔
23
3. A spring with spring constant k is initially compressed a distance 𝑥+ from its equilibrium length. After returning
to its equilibrium position, the spring is then stretched a distance 𝑥+ from that position. What is the ratio of the
work that needs to be done on the spring in the stretching to the work done in the compressing?
1
!
𝑊" 2 𝑘𝑥+
=
=1
𝑊4 1 𝑘𝑥 !
+
2
$
4. A particle is moving along the x-axis subject to the potential energy function 𝑈(𝑥) = 𝑎 u' v + 𝑏𝑥 ! + 𝑐𝑥 – 𝑑,
where 𝑎 = 7.00 𝐽𝑚, 𝑏 = 10.0 𝐽/𝑚! , 𝑐 = 6.00 𝐽/𝑚, and 𝑑 = 28.0 𝐽.
a. Express the force felt by the particle as a function of 𝑥.
𝑑
1
𝐹(𝑥) = − e𝑎 O P + 𝑏𝑥 ! + 𝑐𝑥 – 𝑑f
𝑑𝑥
𝑥
1
𝑎
𝐹(𝑥) = − e−𝑎 O ! P + 2 𝑏𝑥 + 𝑐f = ! − 2𝑏𝑥 − 𝑐
𝑥
𝑥
b. Determine the net force on the particle at the coordinate 𝑥 = 2.00 𝑚.
𝑎
𝐹(2.00) = ! − 2𝑏𝑥 − 𝑐
𝑥
7.00 𝐽𝑚
𝐹(2.00) =
− 2(10.0 𝐽/𝑚! )(2.00 𝑚) − 6.00 𝐽/𝑚 = −44.3 𝐽/𝑚
(2.00 𝑚)!
Learning Outcome
Topic 4– Potential Energy and Energy Conservation
Section 4.2: Conservation of Energy
(KPIs 4.2.1 – 4.2.7)
# of
KPI’s
Number
of Periods
7
8
Chapter
• Practice Questions
Multiple choice question
2. A block of mass 5.0 kg slides without friction at a
speed of 8.0 m/s on a horizontal table surface until it
strikes and sticks to a horizontal spring (with
spring constant of k = 2000 N/m and very small
mass), which in turn is attached to a wall. How far is
the spring compressed before the mass comes to
rest?
A. 0.020 m
1. A ball of mass m is thrown vertically into the air
with an initial speed v. Which of the following
equations correctly describes the maximum height, h,
of the ball?
√
A.
𝑣
ℎ=h
2𝑔
B.
ℎ=
C.
D.
2𝑔
𝑣!
𝑣!
ℎ=
2𝑔
𝑚𝑣 !
ℎ=
𝑔
√
24
B.
0.30 m
C.
0.40 m
D.
0.67 m
3. A pendulum swings in a vertical plane. At the
bottom of the swing, the kinetic energy is 8 J and the
gravitational potential energy is 4 J. At the highest
position of the swing, the kinetic and gravitational
potential energies are ____.
√
A.
kinetic energy = 0 J
gravitational potential energy = 4 J
B.
kinetic energy = 12 J
gravitational potential energy = 0 J
C.
kinetic energy = 0 J
gravitational potential energy = 12 J
D.
kinetic energy = 4 J
gravitational potential energy = 8 J
6. A baseball is dropped from the top of a building.
Air resistance acts on the baseball as it drops. Which
of the following statements is true?
√
4. A ball of mass 0.50 kg is released from rest at point
A, which is 5.0 m above the bottom of a tank of oil, as
shown in the figure. At point B, which is 2.0 m above
the bottom of the tank, the ball has a speed of 6.0
m/s. The work done on the ball by the force of fluid
friction is _____.
√
A.
The change in potential energy of the
baseball as it falls is equal to the
kinetic energy of the baseball just before it
strikes the ground
B.
The change in potential energy of the
baseball as it falls is greater than
the kinetic energy of the baseball just
before it strikes the ground
C.
The change in potential energy of the
baseball as it falls is less than the
kinetic energy of the baseball just before
it strikes the ground
D.
The change in potential energy of the
baseball is equal to the energy lost
due to the friction from the air resistance
while the ball is falling
A.
+15 J
Questions 7 and 8
You use your hand to stretch a spring to a
displacement x from its equilibrium position and
then slowly bring it back to that position.
B.
+9 J
7. Which of the statements below is true?
C.
–9 J
D.
–5.7 J
5. A 2 kg object is released from rest from a height of
3 m above Earth’s surface. How much kinetic energy
does the object have when it reaches a height of 1 m?
A.
2.5 J
B.
10 J
C.
20 J
√ D.
40 J
√
A.
The spring’s ∆U is positive
B.
The spring’s ∆U is negative
C.
The hand’s ∆U is positive
D.
The hand’s ∆U is positive
E.
None of the above statements are true
8. What is the work done by the hand?
A.
B.
C.
√
25
D.
1
− 𝑘𝑥 !
2
1
+ 𝑘𝑥 !
2
1
𝑚𝑣 !
2
Zero
Answer the following questions
1. Two identical billiard balls start at the same height and the same time and roll along different tracks, as shown in
the figure.
a. Which ball has the highest speed at the end?
Explain.
The initial energies are the same for both
the balls as they have the same mass and
same initial heights. The final energy is due
to their kinetic energy, so by conservation
of energy, their kinetic energies are also the
same. So the billiard balls have the same
speed at the end.
b. Which one will get to the end first?
Ball B undergoes an acceleration of a and a deceleration of –a due to the dip in the track. The effects of the
acceleration and deceleration ultimately cancel. However, the ball rolling on track B will have a greater
speed over the dip. Therefore, ball B will reach the end first.
2. A roller coaster is moving at 2.00 m/s at the top of the first hill (ℎ+ = 40.0 𝑚). Ignoring friction and air
resistance, how fast will the roller coaster be moving at the top of a subsequent hill, which is 15.0 m high?
𝐾1 + 𝑈1 = 𝐾* + 𝑈*
1
1
𝑚𝑣1 ! + 𝑚𝑔ℎ+ = 𝑚𝑣* ! + 𝑚𝑔ℎ
2
2
𝑣* = 4𝑣1 ! + 2𝑔(ℎ+ − ℎ)
𝑣* = 4(2.00 𝑚/𝑠)! + 2(9.8 𝑚/𝑠 ! )(40.0 𝑚 − 15.0 𝑚) = 22.2 𝑚/𝑠
3. You are on a swing with a chain 4.00 m long. If your maximum displacement from the vertical is 35.0˚, how fast
will you be moving at the bottom of the arc?
From the diagram:
ℎ+ = −𝑙 cos 𝜃 and ℎ = −𝑙
𝐾1 + 𝑈1 = 𝐾* + 𝑈*
1
0 + 𝑚𝑔ℎ+ = 𝑚𝑣* ! + 𝑚𝑔𝑙
2
1
𝑔(−𝑙 cos 𝜃) = 𝑣* ! − 𝑔𝑙
2
𝑣* = 42𝑔𝑙(1 − cos 𝜃) = 42(9.81)(4.00)(1 − cos 35˚) = 3.77 𝑚/𝑠
26
4. Two masses are connected by a light string that goes over a light, frictionless pulley, as shown in the
figure. The 10.0 kg mass is released and falls through a vertical distance of 1.00 m before hitting the ground.
Determine how fast the 5.00 kg mass is moving just before the 10.0 kg mass hits the ground.
𝐾1 + 𝑈1 = 𝐾* + 𝑈*
(𝐾$1 + 𝐾!1 ) + (𝑈$1 + 𝑈$1 ) = i𝐾$* + 𝐾!* j + i𝑈$* + 𝑈!* j
1
1
(0 + 0) + (𝑚$ 𝑔ℎ + 0) = 𝑚$ 𝑣* ! + 𝑚! 𝑣* ! + (0 + 𝑚! 𝑔ℎ)
2
2
1 !
𝑣 (𝑚$ +𝑚! ) = 𝑔ℎ(𝑚$ − 𝑚! )
2 *
𝑣* = h
2𝑔ℎ(𝑚$ − 𝑚! )
2(9.81)(1.00)(10.0 − 5.00)
=h
= 2.56 𝑚/𝑠
(𝑚$ + 𝑚! )
(10.0 + 5.00)
5. A mass of 1.00 kg attached to a spring with a spring constant of 100 N/m oscillates horizontally on a smooth
frictionless table with an amplitude of 0.500 m.
When the mass is 0.250 m away from equilibrium, determine:
a. Determine its total mechanical energy
1
𝑈56' = 𝑘𝐴!
2
1
𝑈56' = (100 𝑁/𝑚)(0.500 𝑚)! = 12.5 𝐽
2
b. Find the system’s potential energy and the mass’s
kinetic energy
1
1
𝑈' = 𝑘𝑥 ! = (100 𝑁/𝑚)(0.250 𝑚)! = 3.13 𝐽
2
2
𝐾' = 𝑈56' − 𝑈' = 12.5 𝐽 − 3.13 𝐽 = 9.37 𝐽
c. Calculate the mass’s kinetic energy when it is at the
equilibrium point.
At the equilibrium 𝑥 = 0, all the energy is in the form
of kinetic energy. Therefore:
𝐾'-+ = 𝐾56' = 𝑈56' = 12.5 𝐽
d. Suppose there was friction between the mass and the
table so that the amplitude was cut in half after some
time.
i.
By what factor has the mass’s maximum kinetic energy changed?
At the moment when the amplitude is cur in half, the maximum kinetic energy is obtained by the
maximum potential energy:
$
8 !
$
𝐾56',* = ! 𝑘 u ! v = & 𝑘𝐴!
1
𝐾56',* = 𝐾56'
4
Therefore the kinetic energy reduces by a factor of ¼
ii.
By what factor has the maximum potential energy changed?
As described in part (i), the maximum potential energy decreases by a factor of ¼
27
6. A 1.00 kg block is resting against a light, compressed spring at the bottom of a rough plane inclined at an angle of
30.0˚; the coefficient of kinetic friction between block and plane is µk = 0.100. Suppose the spring is compressed
10.0 cm from its equilibrium length. The spring is then released, and the block separates from the spring and slides
up the incline a distance of only 2.00 cm beyond the spring’s normal length before stopping.
a. Determine the change in total mechanical energy of the system.
Since this is not a conservative system, the change in the total mechanical energy can be related to the
energy lost due to friction. This energy can be determined by calculating the work done by the force of
friction.
𝑊*91:01;. = 𝐹*91:01;. 𝑑
∆𝐸0;0 = −𝑊*91:01;. = −µ# 𝑚𝑔(cos 𝜃)𝑑
∆𝐸0;0 = −(0.100)(1.00 𝑘𝑔)(9.81 𝑚/𝑠 ! )(cos 30˚)(120 × 10<! 𝑚) = −0.102 𝐽
b. Determine the spring constant k.
From conservation of energy, the change in total energy, ∆𝐸0;0 determined in (a), is equal to ∆𝐾 + ∆𝑈. Since
𝐾 = 0 at both the initial and final points it follows that
1
∆𝐸0;0 = 𝑈*1.6= − 𝑈1.1016= = 𝑚𝑔𝑑 sin 𝜃 − 𝑘∆𝐿!
2
𝑚𝑔𝑑 sin 𝜃 − ∆𝐸0;0
𝑘=2
∆𝐿!
(1.00 𝑘𝑔)(9.81 𝑚/𝑠 ! )(0.120 𝑚) sin 30˚ − (−0.102 𝐽)
𝑘=2
= 138 𝑁/𝑚
(0.100 𝑚)!
28
Learning Outcome
# of
KPI’s
Number
of Periods
3
5
Topic 5– Momentum and Collisions
Section 5.1: System of Particles and Extended Objects
(KPIs 5.1.1 – 5.1.3)
Chapter
Chapter 8
• Practice Questions
Multiple choice questions
4. The magnitude of the position vector of the center
of mass of the system is ____.
1. The center of mass of an irregular rigid object is
always located ____.
√
at the geometrical center of the object
B.
1.2 m
B.
somewhere within the object
C.
2.9 m
C.
both of the above
D.
3.6 m
D.
none of the above
√
5. The direction of the position vector of the center of
mass of the system measured from the positive xaxis is _____.
A system consists of two particles: particle 1, having
the mass 𝑚$ = 4.0 𝑘𝑔 is located at (–4.0 m, 1.0 m)
and particle 2, having the mass 𝑚! = 5.0 𝑘𝑔 is
located at (2.0 m, 1.0 m).
2. The x-coordinate of the center of mass of the
system is ____.
A.
–2.0 m
B.
–1.3 m
C.
–0.67 m
D.
1.0 m
√
A.
–79.2˚
B.
101˚
C.
112˚
D.
123˚
6. A system is made of three identical particles, A, B
and C, distributed on a (x,y) coordinate system as
follows:
𝐴 (0.0 𝑚; −5.0 𝑚)
𝐵 (2.0 𝑚; −1.0 𝑚)
𝐶 (4.0 𝑚; 8.0 𝑚)
Find the coordinates of the center of mass of the
system.
3. The y-coordinate of the center of mass of the
system is ____.
√
0.92 m
A.
Question 2 to 5
√
A.
A.
–2.0 m
A.
(0.67 m; 2.0 m)
B.
–1.3 m
B.
(1.5 m; 1.8 m)
C.
–0.67 m
C.
(1.7 m; 2.4 m)
D.
1.0 m
D.
(2.0 m; 0.67 m)
√
29
7. Three objects are located along the 𝑥-axis as shown
below.
10. A 4.0 m rod of negligible mass connects two small
spheres at its ends. The mass of one sphere is 3.0 kg
and the mass of the other is unknown. What is the
unknown mass if the center of mass of this system is
1.4 m to the right of the 3.0 kg sphere as shown in the
figure below?
The center of mass of the objects is at 𝑥 =?
A.
1.0 m
B.
1.5 m
C.
2.0 m
√ D.
2.5 m
8. A system consists of 2 kg objects located at
coordinates (0,2), (0,0), (2, 0), as shown in the figure
below. What are coordinates above (𝑥, 𝑦) of the
center of the mass of the system?
√
$
$
!
!
!
!
%
%
%
%
A.
( 𝑚,
√ B.
( 𝑚,
A.
4.2 kg
B.
3.4 kg
C.
2.7 kg
D.
1.6 kg
11. Find the coordinates of the center of mass of a
uniform metallic sheet if we remove a (2cm×2cm)
square from the middle of its lower half as shown in
the figure below. Note that the mass of every segment
of the sheet is proportional to the area of that
segment.
𝑚)
𝑚)
C.
(& 𝑚, & 𝑚)
D.
(1 𝑚, 1 𝑚)
9. A thin straight wire of uniform density is oriented
in the xy-plane. The ends of the wire are located at
(𝑥, 𝑦) = (−2.00 𝑚, −1.00 𝑚) and
(𝑥, 𝑦) = (2.00 𝑚, 3.00 𝑚)
√
Where is its center of mass located?
√
A.
(𝑥, 𝑦) = (0 𝑚, 0 𝑚)
B.
(𝑥, 𝑦) = (0 𝑚, 1.00 𝑚)
C.
(𝑥, 𝑦) = (1.00 𝑚, 0 𝑚)
D.
(𝑥, 𝑦) = (1.00 𝑚, 1.00 𝑚)
30
A.
(4.00 cm; 3.00 cm)
B.
(3.00 cm; 2.00 cm)
C.
(3.00 cm; 2.20 cm)
D.
(2.20 cm; 3.00 cm)
12. Find the coordinates of the center of mass of the
uniform rectangular plate shown below.
14. How far is the center of mass of the rod from its
left end?
A. 6.42 m
B. 5.73 m
√ C. 5.33 m
D. 4.21 m
E. 3.80 m
Questions 15 and 16:
A thin rod of length a has a linear density
λ(x) = 2ax+a2 where x is the distance from the left end
of the rod and a is a positive integer. All quantities
are expressed in S.I units.
A.
(0.25 cm; 0.25 cm)
B.
(0.50 cm; 0.50 cm)
C.
(1.0 cm; 1.0 cm)
√ D.
15. What is the mass of the rod?
A. 2a2
(1.5 cm; 0.50 cm)
√
Questions 13 and 14
An 8.00 m long thin rod has a linear density
λ(x) = 4.00x where x is the distance from the left end
of the rod. All quantities are expressed in S.I units.
A.
128 kg
B.
105 kg
C.
84.3 kg
D.
51.0 kg
3a2
C.
2a3
D.
3a3
16. How far is the center of mass of the rod from its
left end?
A. 𝑎/6
B. 𝑞/3
C. 𝑎/2
√ D. 7𝑎/12
13. What is the mass of the rod?
√
B.
31
1. Find the coordinates of the center of mass of the uniform sheet shown in the figure below. Consider the mass of
any segment of the sheet to be proportional to the area of that segment.
𝑚$ 𝑥$ + 𝑚! 𝑥! + 𝑚% 𝑥%
𝑚$ + 𝑚! + 𝑚%
(4𝑚 × 1) + (9𝑚 × 3.5) + (12𝑚 × 6.5)
=
4𝑚 + 9𝑚 + 12𝑚
= 4.54 𝑚
𝑥4> =
𝑥4>
𝑥4>
𝑦4> =
𝑚$ 𝑦$ + 𝑚! 𝑦! + 𝑚% 𝑦%
𝑚$ + 𝑚! + 𝑚%
𝑦4> =
(4𝑚 × 1) + (9𝑚 × 1.5) + (12𝑚 × 2)
4𝑚 + 9𝑚 + 12𝑚
𝑦4> = 1.66 𝑚
2. Find the coordinates of the center of mass of the shaded area shown below, knowing that an area of 1 m2 has a
mass M.
𝑚$ 𝑥$ + 𝑚! 𝑥! + 𝑚% 𝑥%
𝑚$ + 𝑚! + 𝑚%
10𝑀 × 3.0 + 4𝑀 × 6.0 + 10𝑀 × 9.0
8𝑀 × 3.0 + 8𝑀 × 6.0 + 8𝑀 × 9.0
=
𝑂𝑅
10𝑀 + 4𝑀 + 10𝑀
8𝑀 + 8𝑀 + 8𝑀
= 6.0 𝑚
𝑥4> =
𝑥4>
𝑥4>
𝑚$ 𝑦$ + 𝑚! 𝑦! + 𝑚% 𝑦%
𝑚$ + 𝑚! + 𝑚%
10𝑀 × 2.5 + 4𝑀 × 4.5 + 10𝑀 × 2.5
8𝑀 × 2.0 + 8𝑀 × 4.5 + 8𝑀 × 2.0
=
𝑂𝑅
10𝑀 + 4𝑀 + 10𝑀
8𝑀 + 8𝑀 + 8𝑀
= 2.8 𝑚
𝑦4> =
𝑦4>
𝑦4>
32
3. A long, thin rod lies along the x-axis. One end of the rod is located at 𝑥 = 2.00 𝑚 and the other end of the rod is
located at 𝑥 = 6.00 𝑚. The linear mass density of the rod is given by 𝜆 = 𝑎𝑥 % + 𝑏 , where 𝑎 = 0.100 𝑘𝑔/𝑚& and
𝑏 = 0.200 𝑘𝑔/𝑚.
a. Find the mass of the rod.
'#
?
𝑀 = d 𝜆(𝑥)𝑑𝑥 = d (𝑎𝑥 % + 𝑏)𝑑𝑥
'$
𝑀 = ‚𝑎
𝑥&
!
?
+ 𝑏𝑥ƒ
4
!
(0.100 𝑘𝑔/𝑚& )(6 𝑚)&
(0.100 𝑘𝑔/𝑚& )(2 𝑚)&
𝑀=‚
+ (0.200 𝑘𝑔/𝑚)(6 𝑚)ƒ − ‚
+ (0.200 𝑘𝑔/𝑚)(2 𝑚)ƒ
4
4
𝑀 = 32.8 𝑘𝑔
b. What is the location of the center of mass of the rod?
?
?
1 '#
𝑋4> = d 𝜆(𝑥)𝑥. 𝑑𝑥 = d (𝑎𝑥 % + 𝑏)𝑥. 𝑑𝑥 = d (𝑎𝑥 & + 𝑏𝑥)𝑑𝑥
𝑀 '$
!
!
?
𝑋4>
𝑋4>
𝑋4>
1 𝑥 ) 𝑏𝑥 !
= ‚𝑎 +
ƒ
𝑀
5
2 !
(0.1 𝑘𝑔/𝑚& )(6 𝑚)) (0.2 𝑘𝑔/𝑚)(6 𝑚)!
(0.1 𝑘𝑔/𝑚& )(2 𝑚)) (0.2 𝑘𝑔/𝑚)(2 𝑚)!
1
=
‚„
+
…−„
+
…ƒ
32.8 𝑘𝑔
5
2
5
2
= 4.82 𝑚
33