Download AP Physics Review Sheet 1

A.P. Physics Review Sheet 1
Chapters 1-13,15-18 in Walker, First Semester, 1/10/08
MECHANICS
 Review types of zeros and the rules for significant digits
 Vectors have both magnitude and direction whereas scalars have magnitude but no direction.
Velocity
x x f  xi

t t f  ti
1
(v  vo )
2

Two equations for average velocity:

Instantaneous velocity can be determined from the slope of a line tangent to the curve at a
particular point on a position-versus-time graph.
vave 
and
vave 
Acceleration
aave 
v v f  vi

t t f  ti

One equation for average acceleration:

Instantaneous acceleration can be determined from the slope of a line tangent to the curve at
a particular point on a velocity versus time graph.
Acceleration due to gravity = 9.81 m/s2
In a velocity-versus-time graph for constant acceleration, the slope of the line gives
acceleration and the area under the line gives displacement. See Table 1.


Table 1: Graphing Changes in Position, Velocity, and Acceleration
Constant
Constant
Constant
Ball Thrown
Position
Velocity
Acceleration
Upward
Position
Versus
Time:
Slope = vave
x
t
Velocity
Versus
Time:
x
t
t
Slope = aave
v
v
t
Acceleration Versus
Time:
x
x
v
t
t
Slope = -9.81 m/s2
v
t
t
a = -9.81m/s2
a
a
t
a
t
a
t
t
A.P. Physics Review Sheet 1, Page 2
Table 2: Relationship Between the Kinematic Equations and Projectile Motion Equations
Kinematic Equations
Missing
Projectile Motion,
Projectile Motion,
Variable
Zero Launch Angle
General Launch Angle
Assumptions made:
Assumptions made:
a   g and vo , y  0
a   g , vo , x  vo cos  ,
and vo , y  vo sin 
x  xo  vavet
a
x  vxt
where vx  const.
x  (vo cos  )t
where vx  const.
v  vo  at
x
v y   gt
v y  vo sin   gt
1
x  xo  vot  at 2
2
v final
1
y   gt 2
2
1
y  (vo sin  )t  gt 2
2
v 2  vo 2  2ax
t
vy 2  2 g y
vy 2  vo 2 sin 2   2 g y
Projectile Motion
 The kinematic equations involve one-dimensional motion whereas the projectile motion
equations involve two-dimensional motion. See Table 2 to better understand how the
projectile motion equations can be derived from the kinematic equations.
 Recall that velocity is constant and acceleration is zero in the horizontal direction.
 Recall that acceleration is g = 9.81 m/s2 in the vertical direction.

For an object in free fall, the object stops accelerating when the force of air resistance, F Air ,

equals the weight, W . The object has reached its maximum velocity, the terminal velocity.
 g 
Projectiles follow a parabolic pathway governed by y  h   2  x 2
 2vo 




When a quarterback throws a football, the angle for a high, lob pass is related to the angle for
a low, bullet pass when both footballs land in the same place. The sum of the angles is 90o.
In distance contests for pumpkins launched by cannons, catapults, trebuchets, and similar
devices, pumpkins achieve the farthest distance when launched at a 45o angle.
v 2 
The range of a projectile launched at initial velocity vo and angle  is R   o  sin 2
 g 
The maximum height of a projectile above its launch site is
ymax 
vo 2 sin 2 
2g
A.P. Physics Review Sheet 1, Page 3
Vectors
 Vectors have both magnitude and direction whereas scalars have magnitude but no direction.
 Examples of vectors are position, displacement, velocity, linear acceleration, tangential
acceleration, centripetal acceleration, applied force, weight, normal force, frictional force,
tension, spring force, momentum, gravitational force, and electrostatic force.
 Vectors can be moved parallel to themselves in a diagram.
 Vectors can be added in any order. See Table 3 for vector addition.

For vector A at angle  to the x-axis, the x- and y-components for A can be calculated from
and Ay  A sin  .
Ax  A cos 

The magnitude of vector A is
the x-axis is
 Ay
 Ax
  tan 1 
Vector Orientation
Vectors are parallel:
Vectors are perpendicular:
r
y

x
Vectors are neither parallel
nor perpendicular:
B
a
c

C
b
A
A  Ax 2  Ay 2
and the direction angle for A relative to

.

Table 3: Vector Addition
Calculational Strategy Used
Add or subtract the magnitudes (values) to get the resultant.
Determine the direction by inspection.
Use the Pythagorean Theorem, x 2  y 2  r 2 , to get the
resultant, r , where x is parallel to the x-axis and y is
parallel to the y-axis.
 y 
Use   tan 1   to get the angle,  , made with the x-axis.
 x 
Adding 2 Vectors
Limited usefulness
(1) Use the law of cosines to
determine the resultant:
c 2  a 2  b 2  2ab cos 
(2) Use the law of sines to
help determine direction:
a
b
c


sin A sin B sin C
Adding 2 or More Vectors
(Vector Resolution Method)
Used by most physicists
(1) Construct a vector table
based on your diagram.
(Use vector, x-direction,
and y-direction for the
column headings.)
(2) Determine the sum of the
vectors for each direction,
xtotal and ytotal .
(3) Use the Pythagorean Thm
to get the resultant, r :
xtotal 2  ytotal 2  r 2
(4) To get the angle,  , made
with the x-axis, use:
 y 
  tan 1  total 
 xtotal 
A.P. Physics Review Sheet 1, Page 4
Vectors (continued)
 To subtract a vector, add its opposite.
 Multiplying or dividing vectors by scalars results in vectors.
 In addition to adding vectors mathematically as shown in the last table, vectors can be added
graphically. Vectors can be drawn to scale and moved parallel to their original positions in a
diagram so that they are all positioned head-to-tail. The length and direction angle for the
resultant can be measured with a ruler and protractor, respectively.
Relative Motion
 Relative motion problems are solved by a special type of vector addition.
 For example, the velocity of object 1 relative to object 3 is given by v13  v12  v23 where
object 2 can be anything.
 Subscripts on a velocity can be reversed by changing the vector’s direction: v12  v21
Table 4: Newton’s Laws of Motion
Modern Statement for Law
Translation
If the net force on an object is An object at rest will remain at
Newton’s First Law:
zero, its velocity is constant.
rest. An object in motion will
(Law of Inertia)
Recall that mass is a measure
remain in motion at constant
of inertia.
velocity unless acted upon by
an external force.
Newton’s Second Law:
An object of mass m has an Fnet  ma
acceleration a given by the
net force  F divided by m .
That is a 
Newton’s Third Law:
Recall action-reaction pairs
F
m
For every force that acts on an For every action, there is an
object, there is a reaction force equal but opposite reaction.
acting on a different object
that is equal in magnitude and
opposite in direction.
Survey of Forces
 A force is a push or a pull. The unit of force is the Newton (N); 1 N = 1 kg-m/s2
 See Newton’s laws of motion in Table 4. Common forces on a moving object include an
applied force, a frictional force, a weight, and a normal force.
 Contact forces are action-reaction pairs of forces produced by physical contact of two
objects. Review calculations regarding contact forces between two or more boxes.
 Field forces like gravitational forces, electrostatic forces, and magnetic forces do not require
direct contact. They are studied in later chapters.
A.P. Physics Review Sheet 1, Page 5






Forces on objects are represented in free-body diagrams. They are drawn with the tails of the
vectors originating at an object’s center of mass.
Weight, W , is the gravitational force exerted by Earth on an object whereas mass, m , is a
measure of the quantity of matter in an object ( W  mg ). Mass does not depend on gravity.
Apparent weight, W a , is the force felt from contact with the floor or a scale in an
accelerating system. For example, the sensation of feeling heavier or lighter in an
accelerating elevator.
The normal force, N , is perpendicular to the contact surface along which an object moves
or is capable of moving. Thus, for an object on a level surface, N and W are equal in size
but opposite in direction. However, for an object on a ramp, this statement is not true
because N is perpendicular to the surface of the ramp.
Tension, T , is the force transmitted through a string. The tension is the same throughout the
length of an ideal string.
The force of an ideal spring stretched or compressed by an amount x is given by Hooke’s
Law, F x  kx . Note that if we are only interested in magnitude, we use F  kx where k is
the spring or force constant. Hooke’s Law is also used for rubber bands, bungee cords, etc.
Friction



Coefficient of static friction = S 
FS ,max
where F S ,max is the max. force due to static friction.
N
F
Coefficient of kinetic friction =  K  K where F K is the force due to kinetic friction.
N
A common lab experiment involves finding the angle at which an object just begins to slide
down a ramp. In this case, a simple expression can be derived to determine the coefficient of
static friction: S  tan  . Note that this expression is independent of the mass of the object.
Newton’s Second Law Problems (Includes Ramp Problems)
1. Draw a free-body diagram to represent the problem.
2. If the problem involves a ramp, rotate the x- and y-axes so that the x-axis corresponds to the
surface of the ramp.
3. Construct a vector table including all of the forces in the free-body diagram. For the vector
table’s column headings, use vector, x-direction, and y-direction.
4. Determine the column total in each direction:
a. If the object moves in that direction, the total is ma .
b. If the object does not move in that direction, the total is zero.
c. Since this is a Newton’s Second Law problem, no other choices besides zero and ma are
possible.
5. Write the math equations for the sum of the forces in the x- and y-directions, and solve the
problem. It is often helpful to begin with the y-direction since useful expressions are derived
that are sometimes helpful later in the problem. Recall that the math equations regarding
friction and weight are often substituted into the math equations to help solve the problem.
A.P. Physics Review Sheet 1, Page 6
Equilibrium
 An object is in translational equilibrium if the net force acting on it is zero,



Equivalently, an object is in equilibrium if it has zero acceleration.
If a vector table is needed for an object in equilibrium, then  F x  0 and
F  0.
F
y
0.
Typical problems involve force calculations for objects pressed against walls and tension
calculations for pictures on walls, laundry on a clothesline, hanging baskets, pulley systems,
traction systems, connected objects, etc.
Connected Objects
 Connected objects are linked physically, and thus, they are also linked mathematically. For
example, objects connected by strings have the same magnitude of acceleration.
 When a pulley is involved, the x-y coordinate axes are often rotated around the pulley so that
the objects are connected along the x-axis.
 A classic example of a connected object is an Atwood’s Machine, which consists of two
masses connected by a string that passes over a single pulley. The acceleration for this
 m  m1 
system is given by a   2
g .
 m1  m2 
Work
 A force exerted through a distance performs mechanical work.
 When force and distance are parallel, W  Fd with Joules (J) or Nm as the unit of work.
 When force and distance are at an angle, only the component of force in the direction of
W  ( F cos  )d  Fd cos 
motion is used to compute the work:
 Work is negative if the force opposes the motion (  >90o). Also, 1 J = 1 Nm = 1 kg-m2/s2.

n
If more than one force does the work, then
WTotal   Wi
i 1
WTotal  K  K f  Ki 

The work-kinetic energy theorem states that

See Table 5 for more information about kinetic energy.
 A  F 
In thermodynamics, W  Fd  Fd       Ad   PV
 A  A

1 2 1 2
mv f  mvi
2
2
for work done on or by a gas.
Determining Work from a Plot of Force Versus Position
 In a plot of force versus position, work is equal to the area between the force curve and the
displacement on the x-axis. For example, work can be easily computed using W  Fd when
rectangles are present in the diagram.
 For the case of a spring force, the work to stretch or compress a distance x from equilibrium
1
is W  kx 2 . On a plot of force versus position, work is the area of a triangle with base x
2
(displacement) and height kx (magnitude of force using Hooke’s Law, F  kx ).
A.P. Physics Review Sheet 1, Page 7
Kinetic Energy Type
Table 5: Kinetic Energy
Equation
Comments
Used to represent kinetic
energy in most conservation of
mechanical energy problems.
Kinetic Energy as a
Function of Motion:
K
Kinetic Energy as a
Function of Time in
Oscillatory Motion:
1 2 2
kA sin (t ) since
2
k
m  2 and v   A sin(t )
Useful for a mass on a spring.
3
1 2
 mv   K ave  kT
2
2
ave
Kinetic theory relates the
average kinetic energy of the
molecules in a gas to the
Kelvin temperature of the gas.
1 2
mv
2
K

Kinetic Energy as a
Function of Temperature:
Determining Work in a Block and Tackle Lab
 The experimental work done against gravity, WLoad , is the same as the theoretical work done
by the spring scale, WScale .

WOutput  WLoad  Fd Load  Wd Load  mgd Load

WInput  WScale  Fd Scale

where d Load = distance the load is raised.
where F = force read from the spring scale and d Scale = distance the
scaled moved from its original position.
Note that the force read from the scale is ½ of the weight when two strings are used for the
pulley system, and the force read is ¼ of the weight when four strings are used.
Power


W
or
P  Fv with Watts (W) as the unit of Power.
t
1 W = 1 J/s and 746 W = 1 hp where hp is the abbreviation for horsepower.
P
Conservative Forces Versus Nonconservative Forces
1. Conservative Forces
 A conservative force does zero total work on any closed path. In addition, the work done
by a conservative force in going from point A to point B is independent of the path from
A to B. In other words, we can use the conservation of mechanical energy principle to
solve complex problems because the problems only depend on the initial and final states
of the system.
A.P. Physics Review Sheet 1, Page 8

In a conservative system, the total mechanical energy remains constant:
Ei  E f .
Since E  U  K , it follows that U i  Ki  U f  K f . See Table 5 for kinetic energy, K ,
and Table 6 for potential energy, U , for additional information.
 For a ball thrown upwards, describe the shape of the kinetic energy, potential energy, and
total energy curves on a plot of energy versus time.
 Examples of conservative forces are gravity and springs.
2. Nonconservative Forces
 The work done by a nonconservative force on a closed path is not zero. In addition, the
work depends on the path going from point A to point B.
 In a nonconservative system, the total mechanical energy is not constant. The work done
by a nonconservative force is equal to the change in the mechanical energy of a system:
WNonconservative  Wnc  E  E f  Ei .

Examples of nonconservative forces include friction, air resistance, tension in ropes and
cables, and forces exerted by muscles and motors.
Potential Energy Type
Table 6: Potential Energy
Equation
Comments
Gravitational Potential
Energy:
U  mgh
Good approximation for an
object near sea level on the
Earth’s surface.
Gravitational Potential
Energy Between Two Point
Masses:
U  G
m1m2
where
r
G  6.67 x 10-11 Nm2/kg2
= Universal Gravitation
Constant
Works well at any altitude or
distance between objects in
the universe; recall that r is
the distance between the
centers of the objects.
Elastic Potential Energy
Useful for springs, rubber
1 2
kx where k is the bands, bungee cords, and other
2
force (spring) constant and x stretchable materials.
is the distance the spring is
stretched or compressed from
equilibrium.
Potential Energy as a
Function of Time in
Oscillatory Motion:
U
U
1 2
kA cos 2 (t )
2
x  A cos(t )
since
Useful for a mass on a spring.
A.P. Physics Review Sheet 1, Page 9
Momentum
 Linear momentum is given by

p  mv
with kg-m/s as the unit of momentum.
n
In a system having several objects,
pTotal   p i .
i 1




Newton’s second law can be expressed in terms of momentum. The net force acting on an
p
F net 
object is equal to the rate of change in its momentum:
t
 p p f  pi mv f  mvi m(v f  vi ) mv
When mass is constant,
F net 




 ma .
t
t
t
t
t
I  F ave t   p
The impulse-momentum theorem states that
where the quantity
F ave t is called the impulse, I . A practical application of this equation involves auto air
bags. In an automobile collision, the change in momentum remains constant. Thus, an
increase in collision time, t , will result in a decreased force of impact, F ave , reducing
personal injury.
For a system of objects, the conservation of linear momentum principle states that the net
momentum is conserved if the net external force acting on the system is zero. In other
words, pi  p f . Thus, for an elastic collision, m1v1i  m2v2i  m1v1 f  m2v2 f
and for a
perfectly inelastic collision, m1v1i  m2 v2i  (m1  m2 )v f . See the collision types in Table 7.

In an elastic collision in one dimension where mass m1 is moving with an initial velocity vo ,
and mass m2 is initially at rest, the velocities of the masses after the collision are:
 m  m2 
v1, f   1
 vo
m

m
 1
2 
and
 2m1 
v2, f  
 vo .
m

m
 1
2 
Table 7: Collision Types
Collision Type
Do the Objects
Is Momentum
and Example
Stick Together?
Conserved?
Elastic:
No permanent
No
Yes
deformation occurs;
billiard balls collide.
Inelastic:
Permanent deformaNo
Yes
tion occurs; most
automobile collisions.
Perfectly Inelastic:
Permanent deformaYes
Yes
tion occurs and
objects lock together
moving as a single
unit; train cars collide
and lock together.
Is Kinetic Energy
Conserved?
Yes
No
No
A.P. Physics Review Sheet 1, Page 10
Rotational Motion

Angular position,  , in radians is given by
 
s
r
where s is arc length and r is
radius.





  
 (deg) .
o 
 180 
Counterclockwise rotations are positive, and clockwise rotations are negative.
In rotational motion, there are two types of velocities: angular velocity and tangential
velocity, which are compared in Table 8.
In rotational motion, there are three types of accelerations: angular acceleration, tangential
acceleration, and centripetal acceleration, which are compared in Table 8.
The total acceleration of a rotating object is the vector sum of its tangential and centripetal
accelerations.
Recall that
 (rad )  
Table 8: Comparing Angular and Tangential Velocity and
Angular, Tangential, and Centripetal Acceleration
Calculation
Equations
Units, Comments

t
Angular Velocity:
ave 
Tangential Velocity:
vt  r
Angular Acceleration:
 ave 
Tangential Acceleration:
Centripetal Acceleration:

t
at  r
ac 
vt 2
 r 2
r


radians/s
Same value for horses A
and B, side-by-side on a
merry-go-round.


m/s
Different values for horses
A and B, side-by-side on a
merry-go-round.

radians/s2
Same value for horses A
and B, side-by-side on a
merry-go-round.





m/s2
Different values for horses
A and B, side-by-side on a
merry-go-round.
m/s2
ac is perpendicular to at
with ac directed toward
the center of the circle and
at tangent to it.
A.P. Physics Review Sheet 1, Page 11







mvt2
 mr 2
r
The period, T , is the time required to complete one full rotation. If the angular velocity is
2
constant, then T 
.

A comparison of linear and angular inertia, velocity, acceleration, Newton’s second law,
work, kinetic energy, and momentum are presented in Table 9.
Rotational kinematics is not covered on the AP Physics test.
The moment of inertia, I , is the rotational analog to mass in linear systems. It depends on
the shape or mass distribution of the object. In particular, an object with a large moment of
inertia is hard to start rotating and hard to stop rotating. See Table 10-1 on p.298 for
moments of inertia for uniform, rigid objects of various shapes and total mass.
The greater the moment of inertia, the greater an object’s rotational kinetic energy.
An object of radius r , rolling without slipping, translates with linear speed v and rotates
v
with angular speed   .
r
Centripetal force, FC , is a force that maintains circular motion: FC  maC 
Table 9: Comparing Equations for Linear Motion and Rotational Motion
Measurement or Calculation
Linear Equations
Angular Equations
I  kmr 2 where k  a constant
Inertia:
Mass, m
Average Velocity:
vave 
x
t
ave 

t
Average Acceleration:
aave 
v
t
 ave 

t
Newton’s Second Law:
Fnet  ma
  I  F r
Work:
W  Fd cos
W  
Kinetic Energy:
K
Momentum:
1 2
mv
2
p  mv
K
1 2
I
2
L  I
A.P. Physics Review Sheet 1, Page 12
Torque
 A tangential force, F , applied at a distance, r , from the axis of rotation produces a torque
  rF in Nm. (Since F is perpendicular to r , this is sometimes written as   F r .)
 A force applied at an angle to the radial direction produces the torque   rF sin 
 Counterclockwise torques are positive, and clockwise torques are negative.
 The rotational analog of force, F  ma , is torque,   I , where I  moment of inertia and
  angular acceleration.
 The conditions for an object to be in static equilibrium are that the total force and the total
torque acting on the object must be zero:
 Fx  0 ,  Fy  0 ,   0 . Related
problems often involve bridges, scaffolds, signs, and rods held by wires.
Angular Momentum
 The rotational analog of momentum, p  mv , is angular momentum, L  I in kg-m2/s,
where I = moment of inertia and   angular velocity.
 If the net external torque acting on a system is zero, its angular momentum is conserved and
Li  L f .
Simple Machines
 All machines are combinations or modifications of six fundamental types of machines called
simple machines.
 Simple machines include the lever, inclined plane, wheel and axle, wedge, pulley(s), and
screw.
F
d
MA  out  in . It is a number describing
 Mechanical advantage, MA , is defined as
Fin d out
how much force or distance is multiplied by a machine.
 Efficiency is a measure of how well a machine works, and % Efficiency is calculated using
W 
% Efficiency   out  100 
 Win 
where Wout is the work output and Win is the work input.
Gravity
 Newton’s Law of Universal Gravitation shows that the force of gravity between two point
mm
F  G 12 2
masses, m1 and m2 , separated by a distance r is
where G is the universal
r
gravitation constant, G  6.67x10-11 Nm2/kg2 . Remember that r is the distance between
the centers of the point masses.
 In Newton’s Law of Universal Gravitation, notice that the force of gravity decreases with
1
distance, r , as 2 . This is referred to as an inverse square dependence.
r
A.P. Physics Review Sheet 1, Page 13







The superposition principle can be applied to gravitational force. If more than one mass
exerts a gravitational force on a given object, the net force is simply the vector sum of each
individual force. (The superposition principle is also used for electrostatic forces, electric
fields, electric potentials, electric potential energies, and wave interference.)
Replacing the Earth with a point mass at its center, the acceleration due to gravity at the
GM Earth
g
surface of the Earth is given by
.
A similar equation is used to calculate
2
REarth
the acceleration due to gravity at the surface of other planets and moons in the Solar System.
M Earth
gh  G
At some altitude, h , above the Earth,
can be used to calculate the
2
 REarth  h 
acceleration due to gravity.
In 1798, Henry Cavendish first determined the value of G, which allowed him to calculate
2
gREarth
the mass of the Earth using
.
M Earth 
G
Using Tycho Brahe’s observations concerning the planets, Johannes Kepler formulated three
laws for orbital motion as shown in Table 10. Newton later showed that each of Kepler’s
laws follows as a direct consequence of the universal law of gravitation.
As previously mentioned in the potential energy table, the gravitational potential energy, U ,
mm
between two point masses m1 and m2 separated by a distance r is U  G 1 2 .
This
r
equation is used in mechanical energy conservation problems for astronomical situations.
Energy conservation considerations allow the escape speed to be calculated for an object
2GM Earth
launched from the surface of the Earth:
ve 
 11,200 m/s  25,000 mi/h.
REarth
Law
st
1 Law:
2nd Law:
3rd Law:
Table 10: Kepler’s Laws of Orbital Motion
Modern Statement for Law
Alternate Description
Planets follow elliptical orbits, with The paths of the planets are ellipses,
the Sun at one focus of the ellipse.
with the center of the Sun at one focus.
As a planet moves in its orbit, it An imaginary line from the Sun to a
sweeps out an equal amount of area in planet sweeps out equal areas in equal
an equal amount of time.
time intervals.
The period, T , of a planet increases as The square of a planet’s period, T 2 , is
its mean distance from the Sun, r , proportional to the cube of its radius, r 3 .
raised to the 3/2 power. That is,
4 2 3
2
r  (constant) r 3
That is, T 
 2  3/ 2
3/ 2
GM S
T 
 r  (constant) r
 GM 
where M S  mass of the Sun.
S 

A.P. Physics Review Sheet 1, Page 14
Simple Harmonic Motion (SHM)
 Periodic motion repeats after a definite length of time. The period is the time required for a
motion to repeat (one “cycle”), and the frequency is the number of oscillations (cycles) per
unit time. Period, T , frequency, f , and angular frequency,  , are related to each other by:
1
2
f 
and   2 f 
. Rapid motion has a short period and a large frequency.
T
T
 Classic examples of simple harmonic motion (SHM) include the oscillation of a mass
attached to a spring and the periodic motion of a pendulum. See Tables 11 and 12 for
additional information like the period for each type of system.
 The position, x , of an object undergoing simple harmonic motion varies with time, t , as
 2 
x  A cos t   A cos 
t
where the amplitude, A , is the maximum displacement
 T 
from equilibrium.
 Simple harmonic motion is the projection of uniform circular motion onto the x  axis.
 The maximum speed of an object in simple harmonic motion is vmax  A and the
maximum acceleration is amax  A 2 .

From
vmax  A
and
amax  A 2 , it can be shown that the amplitude,
A , is
2
2 vmax
vmax
T
and the period, T , is
.
amax
amax
In an ideal oscillatory system, the total energy remains constant. Thus, for a spring-mass
1
system in simple harmonic motion (SHM), E  kA2 which comes from
2
1
1
1
1

 

E  K  U   kA2 sin 2 (t )    kA2 cos 2 (t )   kA2 sin 2 (t )  cos 2 (t )   kA2
2
2
 2
 2
2
2
sin   cos   1 for all  .
since
A

Table 11: Relative Velocity, Acceleration, and Restoring Force
at Various Positions for a Spring-Mass System or Pendulum in SHM
Quantity
Maximum
Equilibrium Position
Maximum
Displacement Left
Displacement Right
Velocity, v
v0
v  vmax
v0
Acceleration, a
a  amax
a0
a  amax
Restoring Force, Fx
F  kx  Fmax
F  kx  0
since x  change in
distance from the
equilibrium position.
F  kx  Fmax
A.P. Physics Review Sheet 1, Page 15
Period Type
Table 12: Periods for Oscillating Objects
Equation
Period of a Mass on a Spring:
T  2
m
k
When a spring obeys Hooke’s
Law, the period is independent
of the amplitude.
For a
vertical spring, Hooke’s Law
gives Fy  ky  mg  W so
k
Period of a Simple Pendulum
with a Small Amplitude:
(<15o, Serway & Faughn)
(<10o, Knight)
Period of a Physical
Pendulum:
Comments
mg
.
y
L
g
For a pendulum with an
amplitude less than 10o, the
period is independent of the
amplitude and the mass. The
pendulum’s mass is assumed
to be a point mass.
L I 
T  2


g  mL2 
A physical pendulum has its
mass distributed over a finite
volume like a hollow, glass
Christmas ornament.
The
T  2
quantity
is a
I / mL2
correction factor that accounts
for the size and shape of the
physical pendulum.
FLUID MECHANICS AND THERMODYNAMICS
 Tables 13 and 14 present key principles and equations for fluid mechanics
Principle
Pascal’s Principle:
Archimedes’ Principle:
Bernoulli’s Principle:
Table 13: Principles of Fluid Mechanics
Description
An external pressure applied to an enclosed fluid is transmitted
unchanged to every point within the fluid.
An object completely immersed in a fluid experiences an
upward buoyant force equal in magnitude to the weight of the
fluid displaced by the object.
The pressure in a fluid decreases as the fluid’s velocity
increases. (Please note that Bernoulli’s Principle refers to a
special case of Bernoulli’s equation.)
A.P. Physics Review Sheet 1, Page 16
Concept
Table 14: Equations for Fluid Mechanics
Equation
Comments
Gauge Pressure:
Pgauge  Pactual  Patm
Pressure read from
gauge in real life.
Pressure with Depth:
P  Po   gh
Assumes the fluid is in
static equilibrium.
Buoyant Force:
FB  mg  Vg
Due to an upward buoyant
force, an object’s weight
in water is less than its
weight in air.
Submerged Volume:

Vsubmerged  Vsolid  solid

 fluid
Continuity Equation:
(1) Compressible Flow
1 A1v1  2 A2v2



(2) Incompressible Flow
A1v1  A2v2
Bernoulli’s Equation:
(1) General Equation
1
1
P1   v12   gy1  P2   v22   gy2
2
2
(2) Change in Speed
1
1
P1   v12  P2   v22
2
2
(3) Change in Height
P1   gy1  P2   gy2
(4) Torricelli’s Law
v  2 gh
Assumes the solid
floating in the fluid.
a
is
Equation 1 is broadly
applicable, but equation 2
is more commonly used in
general physics courses.
Equation 1 is broadly
applicable, but equations 2
and 3 are more commonly
used in general physics
courses.
Bernoulli’s equation is
used to derive Torricelli’s
Law. If a hole is poked in
a container at a depth h
below the surface, the
fluid exits with speed v
given by Torricelli’s Law.
A.P. Physics Review Sheet 1, Page 17
Fluid Mechanics

The density,  , of a material is given by  

The density of distilled water is 1000 kg/m3.

Pressure is defined as force per unit area P 





m
where the unit of density is kg/m3.
V
F
where the unit of pressure is the Pascal (Pa).
A
Note that 1 Pa = 1 N/m2
Patm  1.01 x 105 Pa = 101 kPa = 1 atm = 760 mm Hg = 760 torr = 14.7 lb/in2 = 14.7 psi. In
addition, a common unit for atmospheric pressure in weather forecasting is the bar which is
defined as 1 bar = 105 Pa  1 atm.
Pascal’s Principle, Archimedes’ Principle, and Bernoulli’s Principle are compared in Table
13.
Equations for gauge pressure, pressure with depth, buoyant force, submerged volume,
continuity, and forms of Bernoulli’s equation are presented in Table 14.
Bernoulli’s equation can be thought of as energy conservation per volume for a fluid.
According to Torricelli’s Law, if a hole is poked in a container at a depth h below the
surface, the fluid exits with speed v  2 gh . Note that this is the same speed as an object in
free fall for a distance h .
Heat and Temperature
 Heat is the energy transferred between objects because of a temperature difference. Heat
depends on mass, specific heat, and change in temperature for a substance.
 Temperature is the degree of hotness or coldness for an object. Compare the Fahrenheit,
Celsius, and Kelvin scales and their conversions in Table 15.
Scale
Table 15: Temperature Scales
Melting Point for
Boiling Point for
Water on Scale
Water on Scale
Conversion
Fahrenheit
32oF
212oF
o
Celsius
0oC
100oC
o
Kelvin
273 K
373 K
T  oC  273.15
9
F    oC  32
5
C
5 o
 F  32 
9
A.P. Physics Review Sheet 1, Page 18
Heat and Temperature Continued
 See Table 16 for thermal expansion calculations. The coefficient of linear expansion,  , and
coefficient of volume expansion,  , are given in Table 16-1 on p.521 of the course text.
 Heat is a form of energy. Thus, the mechanical equivalent of heat is 1 cal = 4.186 J.
 1 Food Calorie = 1000 calories = 4186 J.
 The heat capacity, C , of an object is the heat, Q , divided by the associated temperature
Q
C
change, T , as in
. The units of heat capacity are J/K or J/oC. Note that heat
T
capacity depends on the quantity of material present in an object.
Q
 The specific heat is the heat capacity per unit mass as in c 
. The units of specific
m T
heat are J/kgK or J/kgoC. Note that specific heat is independent of the quantity of material in
a given object. Specific heats for several materials are given in Table 16-2 on p.529. Note
that specific heat not only depends on the substance but also on its state of matter.
Expansion Type
Table 16: Thermal Expansion
Equations
Linear Expansion:
L   Lo T where   coefficient of linear expansion
Area Expansion:
A  2 Ao T
Volume Expansion:
V  Vo T  3Vo T where
  coefficient of volume expansion  3
Calculations Involving Specific Heat
 The equation for specific heat is commonly rearranged to give Q  mcP T where cP is the
specific heat for a material at constant pressure (usually atmospheric pressure).
 When heat is gained by the system, Q is positive, and when heat is lost from the system to
the surroundings, Q is negative.
 A common energy conservation (heat transfer) problem involves a hot object added to room
temperature water in a calorimeter whereupon the system is allowed to reach its equilibrium
temperature. An overall equation for the system is QHot Object  QWater  QCalorimeter  0 . Next,
QHot Object is subtracted from both sides of the equation to give QWater  QCalorimeter  QHot Object .
The objects gaining heat are positive, and the object losing heat is negative. Afterward,
Q  mcP T is substituted into the equation for each object, and the problem is solved.
A.P. Physics Review Sheet 1, Page 19
Conduction, Convection, and Radiation
 See Table 17 for three common mechanisms of heat exchange.
 If the thermal conductivity of a material is k , its cross-sectional area is A , and its length L ,
 T 
Q  kA 
the heat exchanged in the time t is
Thermal conductivities for several
t .
 L 
substances are given in Table 16-3 on p.532 of the course text.
Mechanism
Table 17: Mechanisms of Heat Exchange
Description
Conduction
Heat flows through a material with no bulk motion. See the
related equation given above.
Convection
Heat exchange due to the bulk motion of an unevenly heated
fluid.
Radiation
Heat exchange due to electromagnetic radiation, such as
infrared rays and light.
Ideal Gases
 Ideal gas characteristics:
1. Gases are composed of very tiny molecules with lots of empty space between them.
2. Gas molecules move rapidly, move in straight lines, and travel in random directions.
3. Gas molecules do not attract each other.
4. Gas molecules have elastic collisions. In other words, the total kinetic energy of the
molecules before and after a collision is the same.
5. The average kinetic energy of gas molecules is proportional to the absolute (Kelvin)
temperature.
 How are real gases different from ideal gases? They occupy space and can attract each other.
At high pressures and low temperatures, real gases show considerable deviation from ideal
behavior.
 The ideal gas law, combined gas law, Boyle’s law, Charles’ law, Gay-Lussac’s law,
Avogadro’s law, and Dalton’s law are presented in Table 18. Values for the gas law
constant, R , and for Boltzmann’s constant, k , are also given in the table.
 Avogadro’s number is 6.022 x 1023 atoms or molecules/mole.
 The volume occupied by one mole of an ideal gas at standard conditions of temperature and
pressure (STP, 0oC and 1 atm) is 22.4 L/mole.
 Absolute zero on the Kelvin scale is the temperature at which all atomic and molecular
motion stops. It can be determined by extrapolating a line to zero volume on a V-T diagram
(Charles’ law) or by extrapolating a line to zero pressure on a P-T diagram (Gay-Lussac’s
law).
A.P. Physics Review Sheet 1, Page 20
Gas Law Name
Ideal Gas Law
(Equation of State):
(Combines Boyle’s, Charles’,
and Avogadro’s Laws)
Table 18: Gas Laws
Equation
Quantities Held Constant
None
PV  nRT where
R  0.0821 L-atm/(mole-K)
= 8.314 J/(mole-K)
= Gas Law Constant
PV  NkT where
k  1.38 x 10-23 J/K
= Boltzmann’s Constant
Combined Gas Law:
(Combines Boyle’s, Charles’,
and Gay-Lussac’s Laws)
PV
PV
1 1
 2 2
T1
T2
Number of moles or molecules
Boyle’s Law:
( PV  constant gives an
isotherm on a P-V diagram.)
PV  constant or PV
1 1  PV
2 2
Temperature and number of
moles or molecules
Charles’ Law:
(Absolute zero, 0 K, can be
determined by extrapolating a
line to zero volume on a V-T
diagram.)
V V
V
 constant or 1  2
T
T1 T2
Pressure and number of moles
or molecules
Gay-Lussac’s Law:
(Absolute zero, 0 K, can be
determined by extrapolating a
line to zero pressure on a P-T
diagram.)
P P
P
 constant or 1  2
T
T1 T2
Volume and number of moles
or molecules
Avogadro’s Law:
V V
V
 constant or 1  2
n
n1 n2
Temperature and pressure
Dalton’s Law:
Total pressure is equal to the
sum of the partial pressures.
PTotal   Pi
n
i 1
Not applicable
A.P. Physics Review Sheet 1, Page 21
Kinetic Theory
 In kinetic theory, the molecules of a gas are imagined to be a large number of points
bouncing off the walls of a container. Gas pressure is related to the number of collisions that
occur with the walls per unit time.
 The molecules of a gas have a range of speeds. The Maxwell distribution indicates which
speeds are most likely to occur in a given gas.
 Kinetic theory relates the average kinetic energy of the molecules in a gas to the Kelvin
3
1 2
temperature of the gas, T :
 mv   K ave  kT
2
2
ave
 The root mean square (rms) speed of the molecules in a gas at the Kelvin temperature T is
3kT
3RT
where m is the mass in kg and M is the molar mass in kg/mole.
vrms 

m
M
3
3
U  NkT  nRT
 The internal energy of a monatomic ideal gas is
where N is the
2
2
number of atoms or molecules, and n is the number of moles. This equation is the starting
point for Table 21.
Stress and Strain
 Stress is an applied force per unit area, and strain is the resulting deformation due to the
stress.
 A graph of stress versus strain (see p.565 in the course text) begins as a straight line for
Hooke’s law but it eventually curves to the right as strain >> stress.
 At the elastic limit, greater stress results in permanent deformation. This represents a spring
or rubber band that has been stretched too far or a car fender that has been dented.
Phase Changes and Phase Diagrams
 Be able to sketch a phase diagram for water, label the axes (pressure vs. temperature), label
the regions of the diagram (solid, liquid, and gas), label the triple point and critical point, and
show how the melting and boiling points can be determined at atmospheric pressure (1 atm).
 At the triple point in a phase diagram, all three phases are in equilibrium.
 Beyond the critical point in a phase diagram, the liquid and gas phases become
indistinguishable and are referred to as a fluid.
 Be able to sketch a heating curve for water, label the axes (temperature vs. heat), label the
phases in the diagram (solid, liquid, and gas), label the phase changes (melting, vaporizing),
and clearly show where the melting and boiling points are in relation to the curve.
 The latent heat, L , is the amount of heat per unit mass that must be added to or removed
from a substance to convert it from one phase to another. Latent heats of fusion, L f , and

latent heats of vaporization, LV , for various substances are given in Table 17-4 on p.571 of
the course text.
A common heat transfer problem is calculating the heat needed to raise the temperature of ice
from -15oC to steam at 115oC. Using the heating curve for water, the heat needed to increase
A.P. Physics Review Sheet 1, Page 22
the temperature of ice, liquid water, and steam can be calculated using Q  mcP T and the
heat needed for the phase transfers can be calculated using Q  mL f and Q  mLV . Add the
heats together, QTotal  QIce  QMelting  QLiquid Water  QVaporizing  QSteam , to get the final answer.
Conversion of Gravitational Potential Energy to Thermal Energy
 A common problem is to calculate the specific heat for an object that has fallen. Ideally, the
gravitational potential energy lost, U  mgh , is equal to the thermal energy gained,
gh
.
Q  mcP T . This allows us set these expressions equal to each other. Thus, cP 
T
Laws of Thermodynamics
 The laws of thermodynamics are described in Table 19.
 U , P, V , T , S and H are state functions and depend only on the initial and final
states of the system. Q and W are not state functions and depend on the pathway between
the initial and final states..
0th
Law
Law:
1st Law:
2nd Law:
3rd Law:
Table 19: Laws of Thermodynamics
Physics Description (Walker)
Chemistry Description (Chang)
When two objects have the same Not described.
temperature, they are in thermal
equilibrium.
Energy can be converted from one form
U  Q  W
to another, but it cannot be created or
 Q pos.  system gains heat.
destroyed.
 Q neg.  system loses heat.
E  q  w
 W pos.  work done by system.
 q pos.  system gains heat.
 W neg.  work done on system.
(endothermic process).
 Q and W are not state functions.
 q neg.  system loses heat.
(exothermic process).
 w pos.  work done on system.
 w neg.  work done by system.
 q and w are not state functions.
When
objects
of
different The entropy of the universe increases in
temperatures are brought into thermal a spontaneous process and remains
contact, the spontaneous flow of heat unchanged in an equilibrium process.
that results is always from the high
temperature object to the low
temperature object.
It is impossible to lower the The entropy of each element in a
temperature of an object to absolute perfect crystalline state is zero at
zero in a finite number of steps.
absolute zero, 0 K.
A.P. Physics Review Sheet 1, Page 23
Thermal Processes
 Isobaric, isochoric, isothermal, and adiabatic processes and their characteristics are described
in Table 20.
 In general, the work done during a process is equal to the area under the process curve in a
PV plot. Recall that W  PV .
 In a complete cycle on a PV plot, the system returns to its original state, which means that the
internal energy, U , must return to its original value. Therefore, the net change in internal
energy must be zero, U net  UTotal  0 .
3
 Using U  nRT and the thermodynamic processes in Table 20, the results in Table 21 are
2
obtained for n moles of a monatomic ideal gas.
 Vf 
 Don’t forget that the work done in an isothermal expansion is
W  nRT ln   .
 Vi 




Specific heats have different values depending on whether they apply to a process at constant
pressure or a process at constant volume.
The molar specific heat, C , is defined by Q  nC T where n is the number of moles.
Table 22 compares the specific heats for a monatomic ideal gas at constant pressure or
constant volume.
Table 23 compares isotherms and adiabats for monatomic ideal gases on a PV diagram.
Table 20: Thermodynamic Processes and Their Characteristics
Process
Assumption(s)
Work, W , = ?
Heat, Q , = ?
Constant Pressure:
(Isobaric)
W  PV and
U  Q  W
W  PV
Q  U  PV
Constant Volume:
(Isochoric)
V  0 so
W  PV  0
W 0
Q  U
Constant
Temperature:
(Isothermal)
T  0 so
3
U  nRT  0
2
W Q
Q  W since U  0
Adiabatic:
(No heat flow)
Q0
W  U
Q0
A.P. Physics Review Sheet 1, Page 24
Table 21: Results for Various Thermodynamic Processes Involving
n Moles of a Monatomic Ideal Gas
Process
Heat, Q , = ?
Work, W , = ?
U = ?
Constant Pressure:
(Isobaric)
Q
5
nRT
2
W  PV  nRT
Constant Volume:
(Isochoric, V  0 )
Q
3
nRT
2
W  0 since V  0
Constant
Temperature:
(Isothermal, T  0 )
V 
Q  nRT ln  f 
 Vi 
 Vf 
W  nRT ln  
 Vi 
from a P-V diagram
where
 Vf 
W  PV ln  
 Vi 
U  0 since T  0
Adiabatic:
(No heat flow, Q  0 )
Q0
3
W  U   nRT
2
U 
U 
3
nRT
2
U 
3
nRT
2
3
nRT
2
Table 22: Specific Heats for an Ideal Gas: Constant Pressure, Constant Volume
(Recall that we are still considering monatomic ideal gases)
Process
Heat, Q , = ?
Heat, Q , = ?
Specific Heat, C , = ?
Constant Pressure:
(Isobaric)
QP 
5
nRT
2
QP  nCP T
Constant Volume:
(Isochoric, V  0 )
QV 
3
nRT
2
QV  nCV T
CP 
5
R
2
CV 
3
R
2
A.P. Physics Review Sheet 1, Page 25
Table 23: Comparing Isotherms and Adiabats on a P-V Diagram
Process
Comparison
Isotherms:
Adiabats:
PV  constant
5
R
C
5
PV   constant where   P  2  for monatomic ideal gases
CV 3 R 3
2

(The value of is different for gases that are diatomic, triatomic, etc.)
Heat Engines and Heat Pumps
 A heat engine is a device that converts heat into work; for example, a steam engine.
 Be able to draw a schematic diagram for a heat engine and label the reservoirs, engine, Th ,





TC , Qh , QC , and W .
The efficiency e of a heat engine that takes in the heat Qh from a hot reservoir, exhausts a
Q
W Qh  QC
e

 1 C .
heat QC to a cold reservoir, and does the work W is
Qh
Qh
Qh
Carnot’s theorem states that if an engine operating between two constant-temperature
reservoirs is to have maximum efficiency, it must be an engine in which all processes are
reversible. In addition, all reversible engines operating between the same two temperatures
have the same efficiency.
The maximum efficiency of a heat engine operating between the Kelvin temperatures Th and
T
emax  1  C .
TC is
Th
Refrigerators, air conditioners, and heat pumps are devices that use work to make heat flow
from a cold region to a hot region.
Be able to draw a schematic diagram for a heat pump and label the reservoirs, engine, Th ,
TC , Qh , QC , and W .
Entropy
 Entropy is a measure of the disorder or randomness of a system. As entropy increases, a
system becomes more disordered.
 The total entropy of the universe increases whenever an irreversible process occurs.
 The change in entropy during a reversible exchange of heat Q at the Kelvin temperature T is
Q
S 
where the unit of entropy is J/K.
T