Download Physics B AP Review Packet: Mechanics Name

Physics B
AP Review Packet: Mechanics
Acceleration (a) (unit: m/s2)
Any change in velocity, including speeding up,
slowing down, or turning.
If the sign of the velocity and the sign of the
acceleration is the same, the object speeds up.
If the sign of the velocity and the sign of the
acceleration are different, the object slows down.
Position (x) (unit: m)
Location of a particle in space.
Distance (unit: m)
The total length of the path traveled by an object.
Does not depend upon direction.
Displacement (∆x) (unit: m)
Change in position. Depends only on the initial
and final positions, not on path.
Includes direction.
Uniformly Accelerated Motion
aave = v/ t
3. Acceleration (A-182 #1)
In which of the following situations would an
object be accelerated?
I. It moves in a straight line at constant speed.
II. It moves with uniform circular motion.
III. It travels as a projectile in a gravitational
field with negligible air resistance.
(A) I only (B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III
Explain your answer:
1.
Distance vs Displacement (PAB)
A hiker hikes 25 miles due north and then all the
way back to the starting point.
a) How far does the hiker hike? Show your
work:
b) What is the hiker’s displacement? Show
your work:
Kinematic Equations
v = vo + at
x = xo + vot + 1/2 at2
v2 = vo2 + 2a( x)
Average Velocity (unit: m/s)
vave = x/ t
Average speed (unit: m/s)
save = d / t
For motion in a straight line, average speed is
the magnitude (abs. value) of the average
velocity.
2.
Name:________________
4. Kinematic Equations (A-195 #65)
A body moving in the positive x direction passes
the origin at time t = 0. Between t = 0 and t = 1
second, the body has a constant speed of 24
meters per second. At t = 1 second, the body is
given a constant acceleration of 6 meters per
second squared in the negative x direction. The
position x of the body at t = 11 seconds is
(A) +99 m
(B) +36 m
(C) -36 m
(D) -75 m
(E) -99 m
Show your work:
Average Speed/Velocity (S-113 #11)
The graph above represents position x versus
time t for an object being acted on by a constant
force. The average speed during the interval
between 1 s and 2 s is most nearly
a. 2 m/s
b. 4 m/s
c. 5 m/s
d. 6 m/s
e. 8 m/s
Show your work:
3/30/2010
1
Bertrand
Physics B
5.
AP Review Packet: Mechanics
Kinematic Graphs for 1-D motion
Show your work:
Stationary particle
x vs t
v vs t
Particle moving with constant non-zero velocity
x vs t
v vs t
Particle moving with constant non-zero acceleration
x vs t
6.
v vs t
a vs t
Free Fall
An object falls accelerated by gravity
g = 9.8 m/s2 downward.
a = -g if up is positive.
acceleration is down when ball is thrown up
EVERYWHERE in the balls flight.
a vs t
8. Free Fall (A-182 #5)
An object is released from rest on a planet that
has no atmosphere. The object falls freely for 3.0
meters in the first second. What is the magnitude
of the acceleration due to gravity on the planet?
(A) l .5 m/s2 (B) 3.0 m/s2 (C) 6.0 m/s2
(D) 10.0 m/s2 (E) 12.0 m/s2
Show your work:
a vs t
Kinematic Graphs (S-199 #1)
The displacement x of an object moving along
the x-axis is shown above as a function of time t.
The acceleration of this object must be
(A) zero
(B) constant but not zero
(C) increasing
(D) decreasing
(E) equal to g
Explain your answer:
Projectile Motion
Something is fired, thrown, shot, or hurled near
the earth’s surface.
Horizontal velocity is constant.
Vertical velocity is accelerated.
Air resistance is ignored.
Trajectory of Projectile
Parabolic path of a projectile
RANGE is how far it travels horizontally.
MAXIMUM HEIGHT occurs halfway through
range, if fired over level ground.
Acceleration is DOWN at 9.8 m/s2 everywhere.
Instantaneous velocity is tangent to the path.
The vertical velocity changes while the
horizontal velocity remains constant.
7. Kinematic Graphs (S-195 #3)
The graph shows the velocity versus time for an
object moving in a straight line. At what time
after time = 0 does the abject again pass through
its initial position?
(A) Between O and 1 s
(B) at 1 s
(C) Between 1 and 2 s
(D) at 2 s
(E) Between 2 and 3 s
3/30/2010
9.
Kinematic Graphs for 2D Projectiles
x-component of motion
x vs t
2
v vs t
a vs t
Bertrand
Physics B
AP Review Packet: Mechanics
Explain your
reasoning:
y-component of motion
x vs t
v vs t
a vs t
10. Projectile Motion (A-182 #64, #65)
a) How do the speeds of the ball at the three
points compare?
(B) vR < vQ < vP
(A) vP < vQ< vR
(C) vQ < vR < vP
(D) vQ < vP = vR
(E) vP = vR < vQ
Explain your choice:
Working 2-D Motion Problems
Resolve vectors into components.
Work as one-dimensional problems.
Horizontal Component of Velocity
Not accelerated by gravity (or anything)
Follows equation x = Vo,xt
b) Which of the following
diagrams best shows the
direction of the acceleration of
the ball at point P ?
Explain your choice:
12. Horizontal Component (A-177 #9)
A diver initially moving horizontally with speed
v dives off the edge of a vertical cliff and lands
in the water a distance d from the base of the
cliff. How far from the base of the cliff would
the diver have landed if the diver initially had
been moving horizontally with speed 2v ?
(A) d
(B) 2d
(C) 2d (D) 4d
(E) It cannot be determined unless the height
of the cliff is known.
Show your work or explain your reasoning:
11. Graphs of Projectiles (A-177 #63)
A projectile is fired with initial velocity
v0 at an
angle θ with the horizontal and follows the
trajectory shown above. Which of the following
pairs of graphs best represents the vertical
components of the velocity and acceleration, v
and a, respectively, of the projectile as functions
of time t ?
3/30/2010
Vertical Component of Velocity
Accelerated by gravity (9.8 m/s2 down)
Use kinematic equations for accelerated motion.
3
Bertrand
Physics B
AP Review Packet: Mechanics
13. Vertical Component (S-199 #5)
A 2-kilogram block rests at the edge of a
platform that is 10 meters above level ground.
The block is launched horizontally from the edge
of the platform with an initial speed of 3 meters
per second. Air resistance is negligible. The time
it will take for the block to reach the ground is
(A) 0.3 s
(B) 1.0 s
(C) 1.4 s
(D) 2.0 s
(E) 3.0 s
Show your work:
Equilibrium
A body with no net force on it is in equilibrium.
Bodies in static equilibrium are stationary;
bodies in translational equilibrium are moving.
15. Newton’s 1st Law (A-187 #7 - mod)
Three forces act on an object. If the object is in
translational equilibrium, which of the following
must be true?
I.
The vector sum of the three forces must equal
zero.
II. The magnitudes of the three forces must be equal.
III. One force must be the equilibrant of the other two.
(A) I only
(C) I and III only
(E) I, II, and III
Explain your reasoning
14. Vertical Component (A-187 #59)
(B) II only
(D) II and III only
16. Newton’s 1st Law (A-187 #44 - mod)
A rock of mass m is thrown horizontally off a
building from a height h, as shown above. The
speed of the rock as it leaves the thrower'
s hand
at the edge of the building is υ 0 . How much
time does it take the rock to travel from the edge
of the building to the ground?
(A)
hυ 0
The sum of the forces on the object is zero in
which of the cases?
(A) II only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III
Explain your reasoning
(B) h υ 0
(C) hυ 0 g
(D) 2h g
(E)
2h g
Show your work:
17. Newton’s 1st Law (S-195 #5)
Force (F) (unit: N)
A force is a push or pull on an object.
Forces cause an objects to accelerate.
A ball of mass m is suspended from two strings
of unequal length as shown above. The tensions
T1 and T2 in the strings must satisfy which of the
following relations?
(A) T1= T2
(B) T1>T2
(C) T1<T2
(D) T1+T2=mg
(E) T1-T2 = mg
Newton’s First Law (Law of Inertia)
A body in motion stays in motion at constant
velocity and a body at rest stays at rest unless
acted upon by an external force.
3/30/2010
4
Bertrand
Physics B
AP Review Packet: Mechanics
Show your work or explain your reasoning:
20. Second Law (A-182 #2)
A ball falls straight down through the air under
the influence of gravity. There is a retarding
force F on the ball with magnitude given by F =
bv, where v is the speed of the ball and b is a
positive constant. The magnitude of the
acceleration a of the ball at any time is equal to
which of the following?
(A) g − b
Show your work:
(B) g − bv
18. Newton’s 1st Law (A-177 #58)
When an object of weight
W is suspended from the
center of a massless string
as shown above, the
tension at any point in the
string is
(A) 2W cosθ
(B) W cosθ
2
m
(C) g + bv
(D) g
m
b
bv
(E)
m
Show your work:
(C) W cosθ
(D) W
21. Second Law (A-182 #45)
2 cosθ
(E) W
cosθ
A block of mass 3m can move without friction
on a horizontal table. This block is attached to
another block of mass m by a cord that passes
over a frictionless pulley, as shown above. If the
masses of the cord and the pulley are negligible,
what is the magnitude of the acceleration of the
descending block?
(A) Zero
(B) g/4 (C) g/3
(D) 2g/3
(E) g
Show your work:
Newton’s Second Law
ΣF = ma
Procedure for Second Law Problems
Step 1: Draw the problem
Step 2: Free Body Diagram
Step 3: Set up equations
ΣF = ma ΣFx = max ΣFy = may
Step 4: Substitute known values
Step 5: Solve
19. Second Law (A-173 #11)
Newton’s Third Law
For every action there exists an equal and
opposite reaction.
If A exerts a force F on B, then B exerts a force
of -F on A.
When the frictionless system shown above is
accelerated by an applied force of magnitude F,
the tension in the string between the blocks is
(A) 2F
(B) F
Show your work:
2
F
(C)
3
1
(D)
F
2
1
(E)
F
3
3/30/2010
Weight (W) (N)
W = mg (near the surface of the earth)
Normal Force
Force that prevents objects from penetrating each
other
Reaction to other forces
Commonly a reaction to gravity
5
Bertrand
Physics B
AP Review Packet: Mechanics
22. Normal Force Flat (A-177 #4)
Show your work
A block of weight W is pulled along a horizontal
surface at constant speed v by a force F. which
acts at an angle of θ with the horizontal, as
shown above. The normal force exerted on the
block by the surface has magnitude
(A) W - Fcos θ
(B) W - Fsin θ
(C) W
(D) W + Fsin θ
(E) W + Fcos θ
Show your work
Friction (f) (unit: N)
The force that opposes a sliding motion.
Static friction exists before sliding occurs.
Kinetic friction exists after sliding occurs.
In general Kinetic friction <= Static friction
fs ≤ µsN (for static friction)
Static friction increases as the force trying to
push an object increases, until it reaches its
maximum value.
fk = µkN
25. Friction on Flat Surface (S-195 #61)
23. Normal Force Ramp (A-271 #62)
A push broom of mass m is pushed across a
rough horizontal floor by a force of magnitude T
directed at angle θ as shown above. The
coefficient of friction between the broom and the
floor is µ. The frictional force on the broom has
magnitude
(B) µ(mg -Tsinθ)
(A) µ(mg +Tsinθ)
(C) µ(mg +Tcosθ)
(D) µ(mg -Tcosθ)
(E) µmg
Show your work
A plane 5 meters in length is inclined at an angle
of 37°, as shown above. A block of weight 20
newtons is placed at the top of the plane and
allowed to slide down. The magnitude of the
normal force exerted on the block by the plane is
most nearly
(A) l0 N
(B) 12N
(C) l6 N
(D) 20 N
(E) 33 N
Show your work
26. Friction on Ramp(S-195 #6-#7)
Questions 6-7
A 2-kilogram block slides
down a 30° incline as shown
above with an acceleration of
2 meters per second squared.
24. Elevators and Normal Force (PAB)
A 50-kg middle school student stands on a scale
in an elevator that is moving downward, but
slowing with an acceleration of magnitude 2.0
m/s2. What does the scale read (in N)?
(A) 300
(B) 400
(C) 500
(D) 600
(E) 700
3/30/2010
(for kinetic friction)
(a) Which of the following diagrams best
represents the gravitational force W, the
frictional force f, and the normal force N that act
on the block?
6
Bertrand
Physics B
AP Review Packet: Mechanics
27. Centripetal Force (A-184 #46)
A car initially travels north and then turns to the
left along a circular curve. This causes a package
on the seat of the car to slide toward the right
side of the car. Which of the following is true of
the net force on the package while it is sliding?
(A) The force is directed away from the center of the circle.
(B) The force is directed north.
(C) There is not enough force directed north to keep the
package from sliding.
(D) There is not enough force tangential to the car'
s path to
keep the package from sliding.
(E) There is not enough force directed toward the center of
the circle to keep the package from sliding.
Explain your reasoning:
Explain your Reasoning:
28. Centripetal Force (A-355 #2)
The horizontal turntable shown below rotates at a
constant rate. As viewed from above, a coin on
the turntable moves counterclockwise in a circle
as shown. Which of the following vectors best
represents the direction of the frictional force
exerted on the coin by the turntable when the
coin is in the position shown?
(b) The magnitude of the frictional force along
the plane is most nearly
(A) 2.5 N (B) 5N (C) 6 N (D) 10 N (E) 16 N
Show your work:
Uniform Circular Motion
An object moves at uniform speed in a circle of
constant radius.
Explain your reasoning:
Acceleration in Uniform Circular Motion
Turns object; doesn’t speed it up or slow it down.
Acceleration points toward center of the circle.
Called centripetal acceleration.
Universal Law of Gravity
Fg = Gm1m2/r2
Most orbit problems can be solved by setting the
gravitational force equal to the centripetal force.
Gm1m2 / r2 = m1v2 / r
Centripetal Acceleration
ac = v2/r
29. Orbit (A-266 #61)
A satellite of mass M moves in a circular orbit of
radius R at a constant speed v. Which of the
following must be true?
Force in Uniform Circular Motion
Any force responsible for uniform circular
motion is called a centripetal force. Centripetal
force can arise from one force, or a combination
of sources.
ΣF = mac = m v2 / r
Since speed of object remains constant, kinetic
energy remains constant, and work is zero.
Friction, tension, normal force, gravity and the
magnetic force are common forces that can act
centripetally to cause uniform circular motion.
3/30/2010
I. The net force on the satellite is equal to mv2/R and
is directed toward the center of the orbit.
II. The net work done on the satellite by gravity in one
revolution is zero.
III. The angular momentum of the satellite is a
constant.
(A) I only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III
7
Bertrand
Physics B
AP Review Packet: Mechanics
Explain your reasoning:
33. Torque (A-187 #68)
A rod on a horizontal tabletop is pivoted at one end and is
free to rotate without friction about a vertical axis, as
shown above. A force F is applied at the other end, at an
angle θ to the rod. If F were to be applied perpendicular
to the rod, at what distance from the axis should it be
applied in order to produce the same torque?
(A) L sin θ
Show your work:
(B) L cosθ
(C) L
(D) L tan θ
(E) 2 L
30. Orbit (S-226 #67)
A satellite of mass m and speed v moves in a
stable, circular orbit around a planet of mass M.
What is the radius of the satellite'
s orbit?
a.
GM
mv
GmM
v
Show your work:
d.
b.
e.
Gv
mM
GmM
c.
GM
v2
v2
31. Law of Gravity, Weight (S-226 #10)
A new planet is discovered that has twice the Earth'
s
mass and twice the Earth'
s radius. On the surface of
this new planet, a person who weighs 500 N on Earth
would experience a gravitational force of
a. 125 N
b. 250 N
c. 500 N
d. 1000 N
e. 2000 N
Show your work:
32. Gravity, Acceleration (A-182 #48)
The planet Mars has mass 6.4 x 1023 kilograms and
radius 3.4 x 106 meters. The acceleration of an object
in free-fall near the surface of Mars is most nearly
(A) zero
(B) l.0 m/s2
(C) 1.9 m/s2
(D) 3.7 m/s2
(E) 9.8 m/s2
Show your work:
34. Rotational Equilibrium (A-271 #57)
Two objects, of masses 6 and 8 kilograms, are hung
from the ends of a stick that is 70 centimeters long
and has marks every l0 centimeters, as shown
above. If the mass of the stick is negligible, at which
of the points indicated should a cord be attached if
the stick is to remain horizontal when suspended
from the cord?
(A) A (B) B (C) C (D) D (E) E
Show your work:
35. Rotational Equilibrium (A-173 #13)
A 5-meter uniform plank of mass 100 kilograms
rests on the top of a building with 2 meters
extended over the edge as shown above. How
far can a 50-kilogram person venture past the
edge of the building on the plank before the
plank just begins to tip?
Torque ( ) (Nm)
Torque is a “twist” caused by a force in
combination with a “moment arm”.
τ = F r sin θ
3/30/2010
Rotational Equilibrium
If counterclockwise torques equal the clockwise
torques, the system is balanced and no rotation occurs.
Στcw = Στccw
8
Bertrand
Physics B
1
A)
2
2
m
m
AP Review Packet: Mechanics
perpendicular to the circle and through its
center?
Show your work:
(A) 9 N ⋅ m
B) 1 m
kg
D) 2 m
C) 3
E) It is impossible to make the plank tip since
the person would have to be more than 2 meters
from the edge of the building.
Show your work:
(B)
12
m2
s
kg ⋅ m
(C) 135
.
2
2
s
(D) 18 N ⋅ m
kg
2
(E) 24 kg ⋅ m
s
36. Rotational Equil (S-115 #3-#33)
A horizontal, uniform board of weight 125 N and
length 4 m is supported by vertical chains at each
end. A person weighing 500 N is sitting on the
board. The tension in the right chain is 250 N.
a) What is the tension in the left chain?
a. 250 N
b. 375 N
c. 500 N
d. 625 N
e. 875 N
Show your work:
38. Angular Momentum (A-173 #14)
An asteroid moves in an elliptic orbit with the
Sun at one focus as shown above. Which of the
following quantities increases as the asteroid
moves from point P in its orbit to point Q?
(A) Speed
(B) Angular momentum
(C) Total energy
(D) Kinetic energy
(E) Potential energy
Show your work or explain your reasoning:
b) How far from the left end of the board is the
person sitting?
a. 0.4 m
b. 1.5 m
c. 2 m
d. 2.5 m
e. 3 m
Show your work:
Periodic Motion
Repeats itself over a fixed and reproducible
period of time.
Oscillators undergo periodic motion.
ADVANCE TOPIC: ANGULAR
MOMENTUM
The tendency of a spinning body to remain spinning.
L=Iω
L: angular momentum (kg m2/s)
I: rotational inertia (MR2 for a particle)
ω: angular speed (radians/second)
Angular momentum, like linear momentum, is
conserved. Unless there is an external torque on
the system, there can be no change in angular
momentum
Simple Harmonic Motion (SHM)
Periodic motion described by sine or cosine function.
Springs and pendulums are Simple Harmonic
Oscillators (SHOs) that obey Hooke’s Law.
Restoring Force (Hooke’s Law)
F = -kx (negative sign indicates force is restoring)
Restoring force is greatest at maximum
displacement and zero at equilibrium
Equilibrium
The midpoint of the oscillation of a simple
harmonic oscillator.
Position of minimum potential energy and
maximum kinetic energy.
37. Angular Momentum (A-187 #6)
A 2 kg object moves in a circle of radius 4 m at a
constant speed of 3 m/s. A net force of 4.5 N
acts on the object. What is the angular
momentum of the object with respect to an axis
3/30/2010
9
Bertrand
Physics B
AP Review Packet: Mechanics
Amplitude (A) (unit: m)
How far the oscillating mass is from equilibrium
at its maximum displacement.
Period of a pendulum
T = 2π√l/g
Potential Energy of a Pendulum
Ug = mgh
Period (T) (unit: s)
The length of time it takes for one cycle of
periodic motion to complete itself.
41. Period of Pendulum (A-177 #8)
The length of a simple pendulum with a period
on Earth of one second is most nearly
(A) 0.12 m
(B) 0.25 m
(C) 0.50 m
(D) 1.0 m
(E) 10.0 m
Show your work:
Frequency (f) (unit: Hz or s-1)
How fast the oscillation is occurring.
f = 1/T
39. Simple Harmonic Motion (A-271 #43)
A particle oscillates up and down in simple
harmonic motion. Its height y as a function of
time t is shown in the diagram above. At what
time t does the particle achieve its maximum
positive acceleration?
(A) 1s (B) 2s (C) 3s (D) 4s
(E) None of the above; the acceleration is constant
Explain your reasoning:
42. Period: Spring, Pendulum (A-088 #44)
An object swings on the end of a cord as a
simple pendulum with period T. Another object
oscillates up and down on the end of a vertical
spring, also with period T. If the masses of both
objects are doubled, what are the new values for
the periods?
Pendulum
Spring
(A) T
2T
2
40. Hooke’s Law (A-271 #3)
An ideal spring obeys Hooke'
s law, F = - kx. A
mass of 0.50 kilogram hung vertically from this
spring stretches the spring 0.075 meter. The
value or the force constant for the spring is most
nearly
(A) 0.33 N/m
(B) 0.66 N/m
(C) 6.6 N/m
(D) 33 N/m
(E) 66 N/m
Show your work:
(B) T
(C) 2T
(D) 2T
(E) 2T
Explain your reasoning:
2T
T
T
T 2
Work (W) (Unit: Joule, J)
The scalar bridge between force and energy.
W = F ∆x cos θ
Period of a spring
T = 2π√m/k
Counterintuitive Results
There is no work if there is no displacement.
Forces perpendicular to displacement don’t work.
By doing positive work on an object, a force or
collection of forces increases its mechanical
energy in some way.
The two forms of mechanical energy are called
potential and kinetic energy.
Potential Energy of a Spring
Us = ½ k x 2
Pendulum
The pendulum can be thought of as an oscillator.
The displacement needs to be small for it to
work properly.
Pendulum Forces: Gravity and tension
43.
3/30/2010
10
Bertrand
Physics B
AP Review Packet: Mechanics
43. Work (A-088 #6)
A horizontal force F is used to pull a 5-kilogran
block across a floor at a constant speed of 3
meters per second. The frictional force between
the block and the floor is 10 newtons. The work
done by the force F in 1 minute is most nearly
(A) 0 J
(B) 30 J
(C) 600 J
(D) 1,350 J
(E) 1,800 J
Show your work:
45. Power (S-115 #68)
A constant force of 900 N pushes a 100 kg mass
up the inclined plane shown above at a uniform
speed of 4 m/s. The power developed by the 900
N force is most nearly
a. 400 W
b. 800 W
c. 900 W
d. 1000 W
e. 3600 W
Show your work:
Kinetic Energy (K) (unit: Joule, J)
Energy due to motion
K = ½ m v2
44. Kinetic Energy (A-177 #3)
Which of the following quantities is a scalar that
is always positive or zero?
(A) Power
(B) Work
(C) Kinetic energy
(D) Linear momentum (F) Angular momentum
State your reasoning:
How We Buy Energy…
The kilowatt-hour is a commonly used unit by
the electrical power company.
Power companies charge you by the kilowatthour (kWh), but this not power, it is really
energy consumed.
1 kW = 1000 W
1 h = 3600 s
1 kWh = 1000J/s • 3600s = 3.6 x 106J
46. Power (A-187 #5)
Units of power include which of the following?
I.
Watt
II. Joule per second
III. Kilowatt-hour
(A) I only
(B) III only (C) I and II only
(D) II and III only
(E) I, II, and III
State your reasoning:
The Work-Energy Theorem
Wnet = ∆K
Net work – work due to ALL forces -- is used in
this theorem. When net work is positive, the
kinetic energy of the object will increase, when
negative, the kinetic energy will decrease.
Work and graphs
The area under the curve of a graph of force vs
displacement gives the work done by the force in
performing the displacement.
47. Power (A-187 #9)
A child pushes horizontally on a box of mass m
which moves with constant speed υ across a
horizontal floor. The coefficient of friction
between the box and the floor is µ. At what rate
does the child do work on the box?
(Α) µmgυ
(Β) mgυ
(C) υ/µmg
(D) µmg/υ
(E) µmυ2
Show your work:
Power (P) (unit: Watt, W which is a J/s)
The rate of which work is done.
When we run upstairs, t is small so P is big.
When we walk upstairs, t is large so P is small.
P = W/t
P=FV
3/30/2010
11
Bertrand
Physics B
AP Review Packet: Mechanics
Force Types
Conservative forces:
Work is path independent.
Work along a closed path is zero.
Work done against conservative forces
increases potential energy; work done by
them decreases it.
Ex: gravity, springs
Non-conservative forces:
Work is path dependent.
Work along a closed path is NOT zero.
Work may be related to a change in total
energy (including thermal energy).
Ex: friction, drag
Gravitational potential energy changes far from
earth’s surface.
Ug = -GMem/r (close to earth’s surface)
Ug has been defined to be zero when an object is
infinitely far from the earth, and it gets
increasingly negative as an object
approaches the earth.
Note: This literal definition is impractical in
most problems, but this is the equation that
must be used to calculate ∆U when you a
very far from the earth’s surface.
49. Escape Velocity (PAB)
Use conservation of energy to derive an
expression for the escape velocity of a rocket of
mass m from the surface of a planet of mass M
and radius R. Assume the planet has no
atmosphere.
(A)
(B)
(C)
(D)
Gm
Gm
2GM
2GmM
r
r
r
r
(E) None of the above
Show your work:
Potential energy
Energy of position or configuration.
Examples:
Gravitational, Spring, Electrical energies
Potential energy is related to work done by
CONSERVATIVE FORCES only.
∆Ug = -Wg (gravity)
∆Ug = -Ws (spring)
Gravitational potential energy close to earth’s
surface.
Wg = -mgh (close to earth’s surface)
∆U = -Wg = mgh
Note: we calculate changes in potential energy
only using this method. We assign the potential
energy to be zero at some certain point, usually
the surface of the earth.
Spring potential energy
Us = ½ kx2
Us is zero when a spring is in its equilibrium
position (neither compressed nor extended)
Law of Conservation of Mechanical Energy
The total mechanical energy of a system remains
constant, provided only conservative forces act
upon the system.
U + K = Constant
U1 + K1 = U2 + K2
∆U + ∆K = 0
∆U = -∆K
48. Work, gravity (A-182 #63)
A plane 5 meters in length is inclined at an angle
of 37°, as shown above. A block of weight 20
newtons is placed at the top of the plane and
allowed to slide down. The work done on the
block by the gravitational force during the
5-meter slide down the plane is most nearly
(A) 20 J
(B) 60 J
(C) 80
(D) 100 J
(E) l30 J
Show your work:
3/30/2010
50. Springs (A-088 #11,#12)
A block oscillates without friction on the end of
a spring as shown above. The minimum and
maximum lengths of the spring as it oscillates
are, respectively, xmin and xmax The graphs below
can represent quantities associated with the
12
Bertrand
Physics B
AP Review Packet: Mechanics
Show your work:
oscillation as functions of the length x of the
spring.
52. Cons. of Energy (A-098 #60)
A rock of mass m is thrown horizontally off a
building from a height h, as shown above. The
speed of the rock as it leaves the thrower'
s hand
at the edge of the building is υ 0 .What is the
kinetic energy of the rock just before it hits the
ground?
(A) mgh (B) 1 mυ 2 (C) 1 mυ 2 − mgh
(a) Which graph can represent the total
mechanical energy of the block-spring system as
a function of x ?
(A) A (B) B (C) C (D) D
(E) E
Explain your reasoning:
2
(D) 1 mυ 2 + mgh
0
2
(b) Which graph can represent the kinetic
energy of the block as a function of x ?
(A) A (B) B (C) C (D) D
(E) E
Explain your reasoning:
0
2
0
(E) mgh − 1 mυ 2
0
2
Show your work:
53. Cons. of Energy (A-182 #47)
A block of mass m slides on a horizontal
frictionless table with an initial speed v 0 . It then
compresses a spring of force constant k and is
brought to rest. How much is the spring
compressed from its natural length?
2
(A) v0
2g
Show your work:
mg
(B)
.
51. Conservation of Energy (A-098 #38)
k
(C) m v
0
A block of mass 3.0 kg is hung from a spring,
causing it to stretch 12 cm at equilibrium, as
shown above. The 3.0 kg block is then replaced
by a 4.0 kg block, and the new block is released
from the position shown above, at which the
spring is unstretched. How far will the 4.0 kg
block fall before its direction is reversed?
(A)
9 cm
(B) 18 cm
(C) 24 cm
(D)
32 cm
(E) 48 cm
3/30/2010
k
(D)
(E)
13
m
v0
k
k
v0
m
Bertrand
Physics B
AP Review Packet: Mechanics
54. Cons. of Energy (A-355 #51,52)
block placed at point P at one end of the track is
released from rest and slides past the bottom of
the track. Which of the following is true of the
height to which the block rises on the other side
of the track?
(A) It is equal to h/2 π . (B) It is equal to h/4.
(C) It is equal to h/2.
(D) It is equal to h.
(E) It is between zero and h; the exact height
depends on how much energy is lost to
friction.
Explain your reasoning:
A ball swings freely back and forth in an arc
from point I to point IV, as shown above. Point
II is the lowest point in the path, III is located 0.5
meter above II, and IV is l meter above II. Air
resistance is negligible.
a) If the potential energy is zero at point II,
where will the kinetic and potential energies
of the ball be equal?
(A) At point II
(B) At some point between II and III
(C) At point III
(D) At some point between III and IV
(E) At point IV
State your reasoning:
Momentum (P) (unit: kg m/s or N s)
How hard it is to stop a moving object.
For one particle
p = mv
For a system of multiple particles
P = Σpi = Σmivi
Momentum is a vector!
56. Momentum (A-098 #43)
b) The speed of the ball at point II is most nearly
(A) 3.0 m/s
(B) 4.5 m/s (C) 9.8 m/s
(D) l4 m/s
(E) 20 m/s
Show your work:
The magnitude of the momentum of the object is
increasing in which of the cases?
(A) II only (B) III only (C) I and II only
(D) I and III only (E) I, II, and III
Explain your reasoning:
Conservation of Energy and Dissipative Forces.
Dissipative forces cause loss of mechanical
energy by producing heat.
Wnc = ∆U + ∆K
55. Cons.of Energy (A-093 #4)
Impulse (J) (units: N s or kg m/s)
The product of an external force and time, which
results in a change in momentum
J=Ft
J = ∆P
The figure above shows a rough semicircular
track whose ends are at a vertical height h. A
3/30/2010
14
Bertrand
Physics B
AP Review Packet: Mechanics
57. Impulse (A-262 #56)
Two planets have the same size, but different
masses, and no atmospheres. Which of the
following would be the same for objects with
equal mass on the surfaces of the two planets?
I. The rate at which each would fall freely
II. The amount of mass each would balance on
an equal-arm balance
III. The amount of momentum each would
acquire when given a certain impulse
(A) I only
(B) III only
(C) I and II only
(D) II and II1 only
(E) I, II, and III
Explain your reasoning:
2.0 meters per second relative to the surface.
Which of the following assessments of this
report is most accurate?
(A) Momentum was not conserved; therefore the
report is false.
(B) If potential energy was released to the objects
during the collision, the report could be true.
(C) If the objects had different masses, the report
could be true.
(D) If the surface was inclined, the report could be
true.
(E) If there was no friction between the objects and the
surface, the report could be true.
Explain your reasoning:
Collisions
Follow Newton’s Third Law (forces exerted on
colliding bodies are equal magnitude)
During a collision, external forces are ignored.
The time frame of the collision is very short.
The forces are impulsive forces (high force, short
duration).
58. Impulse (A-187 #57)
A ball of mass 0.4 kg is initially at rest on the
ground. It is kicked and leaves the kicker'
s foot
with a speed of 5.0 m/s in a direction 60° above
the horizontal. The magnitude of the impulse
imparted by the ball to the foot is most nearly
(A) 1 N ⋅ s
(B)
3 N ⋅s
Collision Types
Elastic: P is conserved, K is conserved
Inelastic: P is conserved, K is NOT conserved
Perfectly Inelastic means the bodies stick together
Show your work:
(C) 2 N ⋅ s
(D)
60. Collisions (A-093 #10)
Which of the following is true when an object of
mass m moving on a horizontal frictionless
surface hits and sticks to an object of mass M >
m, which is initially at rest on the surface?
2
N ⋅s
3
(E) 4 N ⋅ s
(A) The collision is elastic.
(B) All of the initial kinetic energy of the less-massive
object is lost.
(C) The momentum of the objects that are stuck
together has a smaller magnitude than the initial
momentum of the less-massive object.
(D) The speed of the objects that are stuck together
will be less than the initial speed of the less-massive
object.
(E) The direction of motion of the objects that are
stuck together depends on whether the hit is a
head-on collision.
Law of Conservation of Momentum
If the resultant external force on a system is zero,
then the momentum of the system will remain
constant.
The sum of the momentums before a collision is
equal to the sum of the momentums after a
collision.
ΣPb = ΣPa
Explain your reasoning:
59. Cons. of Momentum (A-093 #11)
Two objects having the same mass travel toward
each other on a flat surface. each with a speed of
1.0 meter per second relative to the surface. The
objects collide head-on and are reported to
rebound after the collision. each with a speed of
3/30/2010
15
Bertrand
Physics B
AP Review Packet: Mechanics
61. Collisions (A-093 #11)
Two objects having the same mass travel toward
each other on a flat surface, each with a speed of
10 meter per second relative to the surface. The
objects collide head-on and are reported to
rebound after the collision, each with a speed of
20 meters per second relative to the surface.
Which of the following assessments of this
report is most accurate?
A tennis ball of mass m rebounds from a racquet
with the same speed v as it had initially, as
shown above. The magnitude of the momentum
change of the ball is
(A) 0
(B) mv
(C) 2mv
(D) 2mv sin θ
(E) 2mv cos θ
Show your work:
(A) Momentum was not conserved, therefore the
report is false.
(B) If potential energy was released to the objects
during the collision, the report could be true.
(C) If the objects had different masses, the report
could be true.
(D) If the surface was inclined, the report could be
true.
(E) If there was no friction between the objects and the
surface, the report could be true.
64. Collision (A-276 #41)
Explain your reasoning:
Two objects of mass 0.2 kg and 0.1 kg,
respectively, move parallel to the x-axis, as
shown above. The 0.2 kg object overtakes and
collides with the 0.1 kg object. Immediately after
the collision, the y-component of the velocity of
the 0.2 kg object is 1 m/s upward. What is the
y-component of the velocity of the 0.1 kg object
immediately after the collision?
(A) 2 m/s downward
(B) 0.5 m/s downward
(C) 0 m/s
(D) 0.5 m/s upward
(E) 2 m/s upward
Show your work:
62. Collision (A-004 #63)
The two blocks of masses M and 2M shown
above initially travel at the same speed v but in
opposite directions. They collide and stick
together. How much mechanical energy is lost to
other forms of energy during the collision?
a. Zero
b. 1 Mv 2
2
4
3
Mv 2
e.
Mv 2
3
2
Show your work:
c.
3
Mv 2
4
d.
Explosion
Mathematically, handled just like an ordinary
perfectly inelastic collision.
Momentum is conserved, kinetic energy is not.
65. Explosion (A-098 #67)
63. Momentum Change (S-199 #7)
A stationary object explodes, breaking into
three pieces of masses m, m, and 3m. The two
pieces of mass m move off at right angles to
each other with the same magnitude of
momentum mV, as shown in the diagram
3/30/2010
16
Bertrand
Physics B
AP Review Packet: Mechanics
67. Center of Mass (A-182 #8)
above. What are the magnitude and direction of
the velocity of the piece having mass 3m?
The two spheres pictured above have equal
densities and are subject only to their mutual
gravitational attraction. Which of the following
quantities must have the same magnitude for
both spheres?
(A) Acceleration
(B) Velocity
(C) Kinetic energy
(D) Displacement from the center of mass
(E) Gravitational force
Show your work or explain your reasoning:
Show your work:
<<ADVANCED TOPIC>>
Center of Mass
Where all the mass can be considered to exist
For uniform objects, the center of mass resides
at geometric center.
For collection of points, use these equations
xcm = Σ mixi / Σmi
ycm= Σ miyi / Σmi
zcm= Σ mizi / Σmi
where xcm, ycm, and zcm are the coordinates of the
center of mass, and Σmi is the total mass of the
system.
66. Center of Mass (A-098 #63)
Two people of unequal mass are initially
standing still on ice with negligible friction.
They then simultaneously push each other
horizontally. Afterward, which of the following
is true?
(A) The kinetic energies of the two people are equal.
(B) The speeds of the two people are equal.
(C) The momenta of the two people are of equal
magnitude.
(D) The center of mass of the two-person system
moves in the direction of the less massive person.
(E) The less massive person has a smaller initial
acceleration than the more massive person.
Explain your reasoning:
3/30/2010
17
Bertrand