MCAT Physics Equation Guide

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MCAT Physics Equation Guide
Summer 2011
Kinematics,Dynamics,Forces,Torques,Work,Energy,and momentum:
Equations
Conditions
v
f
= v
i
+at
x
f
= x
i
+v
i
t +
1
2
at
2
v
2
f
= v
2
i
+2a(x
f
x
i
)
v
avg
=
x
t
=
v
i
+v
f
2
 One dimensional motion
 Constant Acceleration (keywords)
{ Freefall
{ Inclined plane
{ Constant force
 Time is often an important parameter in these problems
v
xf
= v
xi
-x
f
= x
i
+v
xi
t
v
yf
= v
yi
gt
y
f
= y
i
+v
yi
t 
1
2
gt
2
v
2
yf
= v
2
yi
2g (y
f
y
i
)
Projectile Motion (a
x
= 0 and a
y
= g)
These can solve any number of 2D motion problems where the
only force on an object is the gravitational force.All you need to do is
determine the parameters that you have and solve for all of the others
that you might need.
R =
v
2
i
sin(2)
g
H
max
=
v
2
i
sin
2

2g
Special Case:Range equation.Requires y
f
= y
i
Gives range (R) and maximum height (H
max
).
~
F
net
= m~a
Units (Newtons - N):1N =
1kgm
s
2
This allows us to calculate acceleration given dierent forces.Break all
vectors into x and y-components.If this is an inclined plane problem,
use a tilted coordinate system.
F
f
= N
Two kinds of friction:Static (
s
) and Kinetic (
k
)
F
c
=
mv
2
R
= mR!
2
Circular motion.This is the value for the centripetal force.Positive
direction is toward the center of the circle.
1
Equations
Conditions
!
f
=!
i
+t

f
= 
i
+!
i
t +
1
2
t
2
!
2
f
=!
2
i
+2(
f

i
)
!
avg
=

t
=
!
i
+!
f
2
Units: = rad,!=
rad
s
,and  =
rad
s
2
Rotational motion.Analogous to linear motion.x!,v!!,and
a!.
~ =~r 
~
F
~
net
= I
Denition of Torque,and Newton's law of rotation.Use just like New-
ton's laws,separate\clockwise"and\counterclockwise"components.
~
F
net
= 0
~
net
= 0
Statics problems.Use for objects that do not move or roll.
~p = m~v Units:
kgm
s
.
Denition of momentum
m
1
v
1i
+m
2
v
2i
= m
1
v
1f
+m
2
v
2f
Conservation of Momentum (works for all collisions)
v
f
=
m
1
v
1i
+m
2
v
2i
m
1
+m
2
For Totally Inelastic collision (only when objects stick together).Look
for words like\embedded"or\stick"or\totally inelastic."WARNING:
Objects may y apart in ordinary inelastic collisions.
v
1f
=
m
1
m
2
m
1
+m
2
v
1i
+
2m
2
m
1
+m
2
v
2i
v
2f
=
2m
1
m
1
+m
2
v
1i
+
m
2
m
1
m
1
+m
2
v
2i
For elastic collisions only.
~
J =
R
~
Fdt =
~
F
ave
t = ~p
Impulse-Momentum relationship
W =
~
F  ~x
Units (Joules - J):1J = 1Nm=
1kgm
2
s
2
Work | For forces that depend on position (like gravity,spring forces,
electromagnetic forces) the work done does not depend on the path
taken,only the endpoints.For these,so called,conservative forces,we
can dene a potential energy.
W = K
K =
1
2
mv
2
Work-Energy Theorem (K is Kinetic Energy)
P =
dW
dt
=
W
t
Units (Watts - W):W =
J
s
=
kgm
2
s
3
Power
P =
~
F  ~v
For a constant force only.
Potential Energy
U = mgh
Gravitational potential (near earth's surface)
1
U = 
GMm
r
Generalized Gravitational potential (far from planet surface OR for two
general objects)
U =
1
2
kx
2
Spring potential
Oscillations and Waves:
Equations
Conditions
x(t) = Asin(!t +)
General Oscillation motion for displacement (x) as a function of time
(t).A is amplitude,!is angular frequency,and  is phase shift.We
almost universally use radians for these calculations.
~
F = k~x
Hooke's law (Spring law).k is spring constant and has units of
N
m
.
!=
q
k
m
=
2
T
!= 2f f =
1
T
Angular frequency dependance on spring constant and mass.Also rela-
tionship between angular frequency (!measured in
rad
s
) and frequency
(f measured in Hz =
1
s
).Don't forget about the period (T = time for
1 oscillation).
(t) = Asin(!t +)
!=
p
g
l
=
2
T
Pendulum motion (l is length of pendulum).All relations involving!,
f,and T remain the same.
c = f
Wave Characteristics:c wave speed, wavelength,f frequency (regular
frequency measured in Hz),I intensity,A amplitude.
I = A
2
!
r
= Natural frequency (!) of oscilla-
tor.
Resonance{Idea that the amplitude of oscillations is larger at certain
driving frequencies than others.
 Electrons in a LC circuit { Radio tuner {
1
p
LC
 Children on the playground { Swing set (pendulum) {
p
g
L
Total Wave = Wave#1 + Wave#2
Superposition Principle:\Things"add.
 Forces
 Waves
 Fields (electric and magnetic)
Constructive Interference { Crests meet crests and troughs meet troughs,in the end amplitude of total wave increases.
Destructive Interference { Crests meet troughs,in the end amplitude of total wave decreases.May lead to nodes.
Nodes { Places where the medium is not changed even though there are waves all around.
Wave superposition leads to BEATS.(See Figure 1)
-10
-5
0
5
10
-1.0
0.0
0.5
1.0
t
cos(t)
-10
-5
0
5
10
-1.0
0.0
0.5
1.0
t
cos(1.1 t)
-100
-50
0
50
100
-2
-1
0
1
2
t
cos(t) + cos(1.1 t)
-100
-50
0
50
100
-2
-1
0
1
2
-100
-50
0
50
100
-2
-1
0
1
2
Figure 1:Explaining the beat phenomenon:Beats are produced when two waves are interfering.The top gure shows
a wave oscillating in time with a frequency f
1
,the middle gure shows another wave with a slightly dierent frequency f
2
.
The bottom gure shows the sum of these two waves.Note the regular appearance of minima.The blue envelope curve
has a frequency that is f
beat
=
jf
1
f
2
j
2
.The fast oscillation occurs at the average frequency
f
1
+f
2
2
.
Equations
Conditions
Standing Waves:

n
=
2L
n+1
Both ends closed or both ends open.Also works with guitar strings.
Harmonic = n +1
Overtone = n

n
=
2L
2n+1
Half open pipe.
Harmonic = 2n +1
Overtone = n
c =
q
T

Speed of waves on a string.
T = Tension,and  = mass per unit length of the string.
c =
q
P

Speed of sound waves.
P = Pressure (or compressibility of material),and  = density of the material.
For sound waves:
 The speed of sound in a gas < The speed of sound in a liquid < The speed of sound in a solid.
 Pitch increases as frequency increases.
 Pitch increases as wavelength decreases.
Equations
Conditions
Doppler Eect
f
(obs)
= f
(source)

1 
v
o
c
1 
v
s
c

Remember:c is speed of sound,v
o
is speed of observer,and v
s
is speed
of the source.
Which sign to use?Consider each motion separately (source object and
observer).
 In numerator:(+) when observer moves toward source,() when
observer moves away from source.
 In denominator:() when source moves toward observer,(+)
when source moves away from observer.
Remember your NASCAR intuition if memorization fails.(What makes
the frequency increase?What makes it decrease?)
Other sound equations:
I/A
2
Intensity (I),Wave Amplitude (A)
I/
1
r
2
For a point source sound intensity follows the inverse square law.
(dB) = 10 log
I
I
0
Sound intensity level (measured in Decibels{dB).I
0
= 10
12 W
m
2
is the
threshhold of hearing.
I
pain
= 1 10
W
m
2
(I have seen this both ways,and wouldn't suggest memorizing it,other
than having an idea of scale.If you get intensities much larger than
this...it probably means that you should re-calculate unless the sound
is supposed to be really loud.)
 = 10log
I
2
I
1
Relative intensity level. = 
2

1
.
I
tot
= I
1
+I
2

tot
= 
1
+10 log

1 +10

2

1
10

Adding sound waves.Note,you should add the intensities directly,not
the decibel levels.
Fluids and Solids:
Equations
Conditions
 =
m
V
Density = mass/Volume SI units:
kg
m
3
specic gravity =


water
P =
F
A
Pressure = Force/Area SI units:
N
m
2
= Pa
P
gauge
= P P
atm
Gauge Pressure,pressure above atmospheric pressure.
P = P
top
+gd
Pressure as a function of depth (d).P
top
is pressure at top and  is the
density of the uid.
F
B
= 
fluid
V
displaced
g
Archemedes'principle:Buoyant force is equal to the weight of displaced
liquid.

object
 
fluid
Condition for oating.

1
A
1
v
1
= 
2
A
2
v
2
A
1
v
1
= A
2
v
2
Continuity equation for uid ow through a pipe with a variable diame-
ter.As gases are readily compressed,their density changes quite easily.
However liquids are not very compressible,therefore 
1
= 
2
.
P +
1
2
v
2
+gy = constant
Bernoulli's equation,a form of conservation of energy regarding uid
ow.
P = 8
vL
A
Flow rate =
V
t
=
Pr
4
8L
Non-ideal uids deal with frictional losses as they ow so they must be
\pushed"by a change in pressure.
Elastic Properties:Many solids equations look just like Hooke's law.
V
V
= T
Thermal expansion of a solid.
P = B
V
V
Bulk modulus [B] (given a change in pressure,how much does the volume
change?)
F
A
= E
L
L
Young's modulus [E] (if I pull on a wire with a force F,how much does
the wire stretch?)
F
A
= S
L
L
Shear modulus [S] (if I pull on an object to the side with a force F,how
much does the object shear?)
Stress =
F
A
Strain =
L
L
Elastic Limit:At some point the material will not stretch,but deform
(technical term:plastic deformation),and,ultimately,fail.Once an
object has passed the elastic limit,it will not return to its original shape.
Electrostatics and Electromagnetism:
Equations
Conditions
~
E = 0
Inside a conductor.Charges move freely,and move as far apart as
possible.This means that all charge lies on the edge of a conductor.
Insulators do not allow charge to move freely,so they are\stuck"
where you put them.
Charge is not created or destroyed.
F =
kq
1
q
2
r
2
Coulomb's law:for two charges q
1
and q
2
separated by a distance r,
the force is F.(k = 8:99 10
9 Nm
2
C
2
 10
10 Nm
2
C
2
) Force is along the line
connecting the two.Like charges repel,opposites attract.
F =
1
4
0
q
1
q
2
r
2
Alternate formulation:k =
1
4
0
where 
0
= 8:85 10
12 C
2
Nm
2
~
E is electric eld
Property of a point is space that points in the direction that a positive
charge would feel a force.Field lines begin on a positive charge and end
on a negative charge.Or they begin or end at innity.
~
F = q
~
E
Given a charge q at a point with a eld
~
E.The force felt by the charge
is
~
F.
V = 
~
E  
~
l
Denition of potential given an electric eld and a path between two
endpoints
U
electric
= qV
Denition of electric potential.
Electric eld and potential for several dierent charge distri-
butions.N.B.:You should probably memorize the form with k and not

0
E
point
=
kq
r
2
=
1
4
0
q
r
2
Field:Point charge q,r is distance from the point charge.
V
point
=
kq
r
=
1
4
0
q
r
Potential:Point charge.(Potential is zero at r = 1
E
line
=
2k
s
=
1
2
0

s
Field:Innite line charge, =
charge
length
,s is perpendicular distance from
the line.
V
line
= 2kln
s
s
0
=

2
0
ln
s
s
0
Potential:Innite line charge,(Potential is zero at s = s
0
Usually 1m
by convention).
E
sheet
= 2k =

2
0
Field:Innte sheet charge, =
charge
area
.
V
sheet
(z) = V
0
+2kz = V
0
+
z
2
0
Potential:Innte sheet charge,z is distance from sheet (positive one
way,negative the other) and V
0
is the potential of the sheet of charge.

E
=
~
E 
~
A
Electric ux:Counts the\number"of electric eld lines passing through
a surface.Area vector
~
A is perpendicular to the surface that you want
to calculate the ux for.

E
=
Q
enc

0
= 4kQ
enc
Gauss'Law:Electric ux through any closed surface is directly related
to the charge enclosed by that surface.
Electric dipoles are electrically neutral,but have equal positive and negative charges split into the two halves
of the object.If a dipole is in an electric eld,it will align itself so that if an electric eld is going from left to
right,the positive half is on the right and the negative half is on the left.
Equations
Conditions
~p = q
~
d
Denition of Dipole.See hyperphysics for a sketch of a dipole as well as
a sketch of the electric eld of a dipole.
V
dipole
=
kpcos 
r
2
Dipole potential:r is distance from dipole, is angle from dipole direc-
tion.