PHYSICS
Dimensions (length or distance, time)

One dimension = magnitude of length or distance only.

Two dimensions = length or distance on a 2D plane (xy coordinates).

Three dimensions = length or distance in 3D space (xyz coordinates).

Four dimensions = length or distance in 3D space at a given time (xyzt
coordinates).
Vectors, components

Scalar: without direction. For example, length, time, mass.

Vector: with direction. For example, displacement, acceleration, force.

Components: the portion of the vector in a given direction.




Trigonometric rules:
o
SOH CAH TOA = silly old Harry, caught a herring, trolling off
Anglesea.
o
SOH: sinθ = opposite / hypotenus.
o
CAH: cosθ = adjacent / hypotenus.
o
TOA: tanθ = opposite / adjacent.
Vector addition

You can only directly add vectors if they are in the same direction.

To add vectors in different directions, you must add their x, y and z
components. The resulting components make up the added vector.

The vector sum of all components of a vector equal to the vector itself.

Operation involving a vector and a vector may or may not result in a
vector (kinetic energy from the square of vector velocity results in
scalar energy).

Operation involving a vector and a scalar always results in a vector.

Operation involving a scalar and a scalar always results in a scalar.
Speed, velocity (average and instantaneous)

Speed: scalar, no direction, rate of change in distance.

Velocity: vector, has direction, rate of change in displacement.

Average speed:

Average velocity:

Instantaneous speed is the speed at an instant (infinitesimal time
interval).

Instantaneous velocity is the velocity at an instant (infinitesimal time
interval).

Instantaneous speed equals instantaneous velocity in magnitude.

Instantaneous velocity has a direction, instantaneous speed does not.

The direction of instantaneous velocity is tangent to the path at that
point.
Acceleration

Acceleration is the rate of change in velocity

Average acceleration:
o
Uniformly accelerated motion along a straight line
o
If acceleration is constant and there is no change in direction, all
the following applies:
o
The value of speed/velocity, distance/displacement are
interchangeable in this case, just keep a mental note of the
direction.
o
o
o
o
o
o
o
You need to memorize those, be able to rearrange them,
combine them, and how to use them.
o
You need to assign one direction as + and the opposite as -, and
then keep this scheme for all your calculations.
o
For Cartesian coordinates, take upward and rightward motion as
positive; down and left as negative.
o
For free falls, take downward as positive.
o
You can assign in what ever fashion you want, as long as the
opposite direction is opposite in sign.
Freely falling bodies

Free falling objects move toward the ground at constant acceleration.

On Earth, the rate of acceleration is g, which is 9.8 m/s2.

Whenever something is in the air, it's in a free fall, even when it is
being tossed upwards, downwards or at an angle.

For things being dropped, it's easier if you take down as positive, since
that will make g positive.

For things being tossed downwards, it's easier if you take down as
positive, since that will make both initial velocity and g positive.

For things being tossed upwards, the initial velocity will have opposite
sign as g. You can take either up or down as positive depending on the
question and what's convenient, but either way, initial velocity will
have opposite signs as g.

The acceleration due to gravity is constant because the force (weight)
and mass of the object is constant.

The net acceleration is a constant g if you don't take air resistance into
consideration. Usually questions ignore air resistance. But if the
question gives you air resistance, then the acceleration is no longer
constant - it will decrease with time until it gets to zero at terminal
velocity.

When there's air resistance, the acceleration will decrease because the
force (weight - resistance) is decreasing due to increasing resistance
or friction at higher speeds.

At terminal velocity, weight = friction, so the net force is 0. Thus, the
acceleration is 0. So, the speed stays constant at terminal velocity.
Projectiles

Projectiles are free falling bodies.

The vertical component of the projectile velocity is always accelerating
toward the Earth at a rate of g.

The vertical acceleration of g toward the Earth holds true at all times,
even when the projectile is traveling up (it's decelerating on its way
up, which is the same thing as accelerating down).

There is no acceleration in the horizontal component. The horizontal
component of velocity is constant.

What is the time the projectile is in the air? Ans: use the vertical
component only- calculate the time it takes for the projectile to hit the
ground.

How far did the projectile travel? Ans: first get the time in the air by
the vertical component. Then use the horizontal component's speed x
time of flight. (Don't even think about over-analyzing and try to
calculate the parabolic path).

When you toss something straight up and it comes down to where it
started, the displacement, s, for the entire trip is 0. Initial velocity and
acceleration are opposite in sign.

When you toss something straight up and it comes down to where it
started, there is symmetry. Initial velocity and final velocity are equal
and opposite. Time spent going up = time spent coming down.
Orbiting in space

Satellites orbiting the Earth are in free fall.

Their centripetal acceleration equals the acceleration from the Earth's
gravity.

Even though they are accelerating toward the Earth, they never crash
into the Earth's surface because the Earth is round (the surface curves
away from the satellite at the same rate as the satellite falls).
Below are old AAMC topics that has been deprecated or changed
Units and dimensions

A unit is a label for a quantity.

unit + unit = unit

unit - unit = unit

unit x unit = unit2

unit / unit = no unit

Dimensions are powers of units.

unit = one dimension.

unit2 = two dimension.

unit3 = three dimension.

Common SI units

Quantity
SI unit Name
Length
m
meter
2
Area
m
meter squared
Volume
m3
meter cubed
Mass
kg
kilogram
Density
kg/m3 kilogram per meter cubed
Time
s
second
Speed
m/s
meter per second
Acceleration
m/s2
meter per second squared
Force
N
Newton
Pressure
Pa
Pascal
Temperature
K
Kelvin
Energy
J
Joule
Power
W
Watt
Charge
C
Coulomb
Potential
V
Volt
Current
A
Ampere
Resistance
Ω
Ohm
Magnetic field T
Tesla
The product of operations involving all SI units is also in SI units.

Prefixes for units
Prefix Abbreviation Multiplier
exa
E
1018
peta
P
1015
tera
T
1012
giga
G
109
mega M
106
kilo
k
103
hecto h
102
deka da
101
deci
10-1
d
centi c
10-2
milli m
10-3
micro μ
10-6
nano n
10-9
pico
p
10-12
femto f
10-15
atto
10-18
a
Center of mass
The center of mass is the average distance, weighted by mass

In a Cartesian coordinate, the center of mass is the point obtained by
doing a weighted average for all the positions by their respective
masses.

The center of mass of the Earth and a chicken in space is going to be
almost at the center of the Earth, because the chicken is tiny, and its
coordinate is weighted so.

The center of mass between two chickens in space is going to be right
in the middle of the two chickens, because their positions are weighted
equally.

You do not have to obtain the absolute coordinates when calculating
the center of mass. You can set the point of reference anywhere and
use relative coordinates.

The center of mass for a sphere is at the center of the sphere.

The center of mass of a donut is at the center of the donut (the hole).
Newton's first law, inertia
The law of inertia basically states the following: without an external
force acting on an object, nothing will change about that object in
terms of speed and direction.
In the absence of an external force:

Something at rest will remain at rest

Something in motion will remain in motion with the same speed and
direction.

Objects are "inert" to changes in speed and direction.
Newton's second law (F = ma)
A net force acting on an object will cause that object to accelerate in
the direction of the net force.

The unit for force is the Newton. N =
kg·m/ 2
s

Both force and acceleration are vectors because they have a direction.

Many MCAT questions omit the direction attribute because it is so
obvious. For example, when an apple falls to the ground (or on
Newton), we all know that the force of gravity acts downwards, and
the apple of course, falls downwards. Questions in this scenario are
just simple cases of plugging in the formula

However, more difficult questions have directional attributes
associated with them. For example, when a bar of soap slides down an
inclined plane, the force of gravity acts downwards, but the
acceleration is not completely downwards, but is "slanted". Therefore,
you need to do vector analysis (simple ones only. The MCAT is too
short for complex, time-consuming ones that appear in your physics
midterm).
Newton's third law, forces equal and opposite
Every action has an equal and opposite reaction

for the MCAT, you need to know that this law applies to propulsion.
This is why rockets work even in the vacuum of space.
Concept of a field

For the purposes of the MCAT, fields are lines.

When lines are close together, that's shows a strong field.

When lines are far apart, that shows a weak field.

Lines / fields have direction too, and that means they are vectors.

Things travel parallel, perpendicular, or spiral to the field line.
Law of gravitation (F = Gm1m2/r^2)

Gravity decreases with the square of the distance.

If the distance increases two fold, gravity decreases by a factor of
four.

The "distance" is the distance from the center of mass between the
two objects.

Gravity is the weakest of the four universal forces.

This weakness is reflected in the universal gravitational constant, G,
which is orders of magnitude smaller than the Coulomb's constant.
Uniform circular motion
Memorize the equations

acceleration:

force:

circumference:

arc:

area:

sector:

note that theta is always in radians. 2pi radians = 360 degrees.

The simple harmonic laws of frequency and period applies here also.
Get the concepts

Distinguish between velocity and speed: Velocity is displacement over
time. Speed is the distance over time.

Displacement is the shortest, straight-line distance between two points
on the perimeter of a circle (technically, this is called the chord).
Distance is circumference and arc.

Some typical cases:
o
For displacements and distances that approach zero, the
instantaneous velocity equals the speed.
o
For a quarter around the circle (pi/2 radians or 90 degrees), the
displacement is the hypotenuse of a right-angled triangle with
the radius as the other two sides. Using Pythagoras, the
displacement is square root of 2r^2. The distance is the arc of
1/4 circumference.
o
For half around the circle, the displacement is the diameter and
the distance is half the circumference.
o
For three quarters around the circle, the displacement is again
obtained by Pythagoras. The magnitude of the displacement here
is the same as that at a quarter of a circle, but the direction is
different. The distance, is 3/4 of the circumference.
o
Complete around the circle, the displacement is zero, which
makes the velocity also zero. The distance is the circumference.

The velocity is always less or equal to the speed.

The displacement is always less or equal to the distance.

Displacement and velocity are vectors. Distance and speed are not.

Moving around a circle at constant speed is also simple harmonic
motion.

frequency = how many times the object goes around the circle in one
second.

period = time it takes to move around the entire circle.
Centripetal Force (F=-mv2/r)
Centripetal force is due to centripetal acceleration. Centripetal
acceleration is due to changes in velocity when going around a
circle. The change in velocity is due to a constant change in
direction.

Centripetal force:
o
Sometimes a negative sign is used for centripetal force to
indicate that the direction of the force is toward the center of
circle.

Centripetal acceleration:

The direction of both the acceleration and the force is toward the
center of the circle.

The tension force in the string (attached to the object going in circles)
is the same as the centripetal force.

When the centripetal force is taken away (Such as when the string
snaps), the object will fly off in a path tangent to the circle at the point
of snap.
Weight
Weight is the force that acts on a mass

Weight is a force. It has a magnitude and a direction. It is a vector.

Because it is a force, F=ma holds true.

Your weight on the surface of the Earth: F=mg, where g is the
acceleration due to Earth, which is just under 10.

You weigh more on an elevator accelerating up because F=mg + ma,
where a is the acceleration of the elevator.

An elevator accelerating up is the same thing as an elevator
decelerating on its way down, in terms of the acceleration in F=mg +
ma.

You weigh less on an elevator accelerating down because F=mg - ma,
where a is the acceleration of the elevator.

An elevator accelerating down is the same thing as an elevator
decelerating on its way up, in terms of the acceleration in F=mg - ma.

You weigh less when you are further away from the Earth because the
force of gravity decreases with distance.

However, you are not truly "weightless" when orbiting the Earth in
space. You are simply falling toward the Earth at the same rate as
your space craft.

You gain weight as you fall from space to the surface of the earth.

For a given mass, its weight on Earth is different from its weight on
the Moon.

When something is laying still on a horizontal surface, the normal force
is equal and opposite to the weight.

When something is laying still on an inclined plane, the normal force
and friction force adds up in a vector fashion to equal the weight.
Friction, static and kinetic
Friction is a force that is always in the direction to impede the
sliding of surfaces.
Static friction:
Kinetic friction:
u is the coefficient of friction and N is the normal force.

Like any other force, friction is a vector. However, its direction is easy
because it's always opposite to the motion of the surface involved.

Static friction pertains to objects sitting still. An object can sit still on
an inclined plane because of static friction.

Kinetic friction pertains to objects in motion. A key sliding across the
table eventually comes to a stop because of kinetic friction.

Static friction is always larger than kinetic friction.

The coefficient static friction is always larger than the coefficient of
kinetic friction.

The coefficient of friction is intrinsic to the material properties of the
surface and the object, and is determined empirically.

The normal force at a horizontal surface is equal to the weight

The normal force at an inclined plane is equal to the weight times the
cosine of the incline angle (see inclined planes).

We can walk and cars can run because of friction.

Lubricants reduce friction because they change surface properties and
reduce the coefficient of friction.

Every time there is friction, heat is produced as a by-product.
Motion on an inclined plane

Gravity is divided into two components on an inclined plane.
o
One component is normal (perpendicular) to the plane surface:
FN = mg·cosθ
o
The other component is parallel to the plane surface: F|| =
mg·sinθ

To prevent the object from crashing through the surface of the inclined
plane, the surface provides a normal force that is equal and opposite
to the normal component of gravity.

Friction acts parallel to the plane surface and opposite to the direction
of motion.

In a non-moving object on an inclined plane: normal component of
gravity = normal force; parallel component of gravity = static friction.

Unless the object levitates or crashes through the inclined plane, the
normal force always equals the normal component of gravity.

In an object going down the inclined plane at constant velocity:
parallel component of gravity = kinetic friction (yes, they're equal,
don't make the mistake of thinking it's larger. Constant velocity = no
acceleration = no net force).

In an object that begins to slip on the inclined plane: parallel
component of gravity > static friction.

In an object that accelerates down the inclined plane: parallel
component of gravity > kinetic friction.

When you push an object up an inclined plane, you need to overcome
both the parallel component of gravity and friction.

When you push or pull an object up an inclined plane, make sure you
divide that force into its components. Only the component parallel to
the plane contributes to the motion.
Analysis of pulley systems
Pulleys reduce the force you need to lift an object. The catch - it
increases the required pulling distance.

For the purpose of the MCAT, just memorize the simple pulley systems
below.

Rule of thumb: The ropes on either side of a moving pulley contributes
to pulling the load.

The MCAT will most probably give you simple pulleys where only the
above rule is applicable.

Complex pulleys will have additional ropes that contribute to the
pulling of the load (most likely not tested on the MCAT).

The distance of pulling increases by the same factor that the effort
decreases.
There are no moving pulleys here. If the weight of the box is 100 N, you
have to pull with a force of 100 N. For every 1 meter you pull, the box goes
up 1 meter.
When there is one moving pulley, the force needed to pull is halved because
strings on both side of the pulley contribute equally. You supply 50 N (which
is transmitted to the right-hand rope) while the left-hand rope contributes
the other 50 N. Because effort here is halved, the distance required to pull
the box is doubled.
There are two moving pulleys here. Counting the ropes reveal that when we
tug on one rope, it gets transmitted to a system where 4 ropes pull on the
load. Thus, you can pull the 100 N box with only 25 N. However, for every 4
m you pull, the box only goes up 1 m.
This is a complex pulley. Just like the simple pulleys, the ropes on both sides
of the moving pulley contribute. Here, the left-most rope contributes also.
This makes 3 contributing ropes, which makes the effort required to be
reduced by a factor of 3. The distance you need to pull here is 3 times the
distance the box will travel.
Force

There are 4 universal four-ces... get it?

Universal forces are also called fundamental forces.

The four forces are:
o
The strong force: also called the nuclear force. It is the strongest
of all four forces, but it only acts at subatomic distances. It binds
nucleons together.
o
Electromagnetic force: about one order of magnitude weaker
than the strong force, but it can act at observable distances.
Binds atoms together. Allows magnets to stick to your
refrigerators. It is responsible for the fact that you are not falling
through your chair right now (MCAT people love to throw you
quirky examples like this one).
o
Weak force: roughly 10 orders of magnitude weaker than the
strong force. Responsible for radioactive decay.
o
Gravity: roughly 50 orders of magnitude weaker than the strong
force. Responsible for weight (not mass!). Also, responsible for
planet orbits.
Equilibrium

When something is in equilibrium, the vector sum of all forces acting
on it = 0.

Another way to put it: when something is in equilibrium, it is either at
rest or moving at constant velocity.

Yet another way to put it: when something is in equilibrium, there is
no overall acceleration.
Concept of force, units

Force makes things accelerate, change velocity or change direction.

In the MCAT, a force is indicated by an arrow.

The direction of the arrow is the direction of the force.

The magnitude of the force is often labeled beside the arrow.

F=ma, so the unit for the force is kg·m/s2
Translational equilibrium (Sum of Fi = 0)


When things are at translational equilibrium, the vector sum of all
forces = 0.

Things at translational equilibrium either don't move, or is moving at a
constant velocity.

If an object is accelerating, it's not in equilibrium.

Deceleration is acceleration in the opposite direction.

At translational equilibrium:

o
An apple sitting still.
o
A car moving at constant velocity.
o
A skydiver at falling at terminal velocity.
NOT at translational equilibrium:
o
An apple falling toward the Earth with an acceleration of g.
o
A car either accelerating or decelerating.
o
A skydiver before he or she reaches terminal velocity.
Rotational equilibrium (Sum of Torque = 0)

When things are at rotational equilibrium, the sum of all torques = 0.

Conventionally, positive torques act counterclockwise, negative
torques act clockwise.

When things are at rotational equilibrium, they either don't rotate or
they rotate at a constant rate (angular velocity, frequency).

You cannot have rotational equilibrium if there is angular acceleration.

Deceleration is acceleration in the opposite direction.

At rotational equilibrium:

o
Equal weights on a balance.
o
Propeller spinning at a fixed frequency.
o
Asteroid rotating at a constant pace as it drifts in space.
NOT at rotational equilibrium:
o
Unequal weights in a balance such that the balance begins to tilt.
o
Propeller spinning faster and faster.
o
Propeller slowing down.
Analysis of forces acting on an object

Draw force diagram (force vectors).

Split the forces into x, y and z components (normal and parallel
components for inclined planes).

Add up all the force components.

The resulting x, y and z components make up the net force acting on
the object.

Use Pythagoras theorem to get the magnitude of the net force from its
components.

Use trigonometry to get the angles.

... more on vector components
Newton's first law, inertia

The significance of Newton's first law on equilibrium is: things in
equilibrium will remain in equilibrium unless acted on by an external
force.

The significance of Newton's first law on momentum is: things resist
change in momentum because of inertia (try stopping a truck. It's not
easy because it resists changes to its huge momentum).

... more on Newton's first law
Torques, lever arms

Torque
o

o
Torque is the angular equivalent of force - it makes things
rotate, have angular acceleration, change angular velocity and
direction.
o
The convention is that positive torque makes things rotate
anticlockwise and negative torque makes things rotate clockwise.
Lever
o
The lever arm consists of a lever (rigid rod) and a fulcrum
(where the center of rotation occurs).
o
The torque is the same at all positions of the lever arm (both on
the same side and on the other side of the fulcrum).
o
o
If you apply a force at a long distance from the fulcrum, you
exert a greater force on a position closer to the fulcrum.
o
The catch: you need to move the lever arm through a longer
distance.
Weightlessness

There are two kind of weightlessness - real and apparent.
o
Real weightlessness: when there is no net gravitational force
acting on you. Either you are so far out in space that there's no
objects around you for light-years away, or you are between two
objects with equal gravitational forces that cancel each other
out.
o
Apparent weightlessness: this is what we "weightlessness" really
means when we see astronauts orbiting in space. The astronauts
are falling toward the earth due to gravitational forces (weight),
but they are falling at the same rate as their shuttle, so it
appears that they are "weightless" inside the shuttle.
Momentum

Momentum = mv, where m is mass, v is velocity and the symbol for
momentum is p.

Impulse = Ft, where F is force and t is the time interval that the force
acts.

Impulse = change in momentum:

Conservation of linear momentum

o
Total momentum before = total momentum after.
o
Momentum is a vector, so be sure to assign one direction as
positive and another as negative when adding individual
momenta in calculating the total momentum.
o
The momentum of a bomb at rest = the vector sum of the
momenta of all the shrapnel from the explosion.
o
Total momentum of 2 objects before a collision = total
momentum of 2 objects after a collision.
Elastic collisions
o
Perfectly elastic collisions: conservation of both momentum and
kinetic energy.
o
Conservation of kinetic energy: total kinetic energy before =
total kinetic energy after.
o
Kinetic energy is scalar, so there are no positive / negative signs
to worry about.
o
If you drop a ball and the ball bounces back to its original height
- that's a perfectly elastic collision.
o
If you throw a ball at a wall and your ball bounces back with
exactly the same speed as it was before it hit the wall - that's a
perfectly elastic collision.

Inelastic collisions
o
Conservation of momentum only.
o
Kinetic energy is lost during an inelastic collision.
o
Collisions in everyday life are inelastic to varying extents.
o
When things stick together after a collision, it is said to be a
totally inelastic collision.
Work

W = Fdcosθ

F is force, d is the distance over which the force is applied, and θ is the
angle between the force and distance.

Derived units, sign conventions
o
Work is energy, and the unit is the Joule.
o
Joule = N·m = kg·m/s2·m = kg·m2/s2
o
If the force and the distance applied is in the same direction,
work is positive.
o
For example, pushing a crate across a rough terrain involves you
doing positive work (you are pushing forward and the crate is
moving forward).
o
If the force and the distance applied is in opposite directions,
work is negative.
o
For a non-rotating system, friction always does negative work
because it acts against the direction of motion.
o
If the force is acting in one direction, but the object moves in a
perpendicular direction, then no work is done.
o
The classic example is that no work is done by your arms when
you carry a bucket of water for a mile. Because you are lifting
the bucket vertically while its motion is horizontal.
o
If you like math, then everything you need to know is already
contained in the mathematical formula. Cosine of 90 is zero;
cosine of anything below 90 is positive and between 90-180 is
negative ...so forth.



Amount of work done in gravitational field is path-independent
o
Unlike friction, gravity always acts downwards. Thus, it does not
matter what detour you take because sideward motion
perpendicular to the gravitational force involves no work.
o
Pushing an object at constant speed up a frictionless inclined
plane involves the same amount of work as directly lifting the
same object to the same height at constant speed.
o
Sliding down a frictionless inclined plane involves the same
gravitational work as doing a free fall at the same height.
Mechanical advantage
o
Mechanical advantage = little input force (effort) -> large output
force.
o
Using the lever arm can achieve mechanical advantage.
o
Using pulleys can achieve mechanical advantage.
Work-kinetic energy theorem
o
o

Work on an object can transform into kinetic energy.

When you pushing on an object, it will move: Fd = ½mv2

When gravity does work on an object, it will move: Fweighth
= mgh = ½mv2
Kinetic energy of an object can do work.

A moving object can slide up an inclined plane before
coming to a stop: ½mv2 = mgh

A moving object can slide against friction for a while before
coming to a stop: ½mv2 = Ffrictiond
Power
o
Power is the rate of work, or work over time: P = W/t
o
The unit for power is the Watt, or W (don't confuse this W with
the shorthand of work).
o
Watt = Joule / second
Energy

Work and energy are interchangeable.

All types of energy have the same unit - the Joule.

Kinetic energy: KE = 1/2 mv^2; units

o
KE = ½mv2
o
Unit = Joule = kg·m2/s2
o
At the same speed, the larger mass has the larger kinetic
energy.
o
When you double the mass, you double the kinetic energy.
o
At the same mass, the higher speed has the larger kinetic
energy.
o
When you double the speed, you quadruple the kinetic energy.
o
Speed is more important than mass for the kinetic energy
because speed is squared.
Potential energy
o
o
PE = mgh (gravitational, local)

PE = mgh is local because it only works on the surface of
the Earth.

h is the distance from the Earth's surface.

PE = mgh is derived from a more general formula.

On earth, g is 9.8. g is larger for planets with a higher
mass to radius ratio.
PE = 1/2kx^2 (spring)
o



x is distance of the end of the spring from its equilibrium
position.

k is the spring constant.

Stiff springs have a larger k because they are harder to
stretch (it takes more energy to stretch them).
PE = -GmM/r (gravitational, general)

This is the general formula for gravitational potential
energy.

r is the distance between the center of the two attracting
objects.

G is the universal gravitation constant - it is the same for
everything.

m and M are the mass of the two attracting objects.
Conservation of energy
o
The total amount of energy before = the total amount of energy
after.
o
Gravitational potential energy is converted to kinetic energy as
an object falls, but the total amount of energy stays the same.
o
Kinetic energy is converted to heat and sound energy as a crate
slides to a stop on a rough surface.
Conservative forces
o
If a force doesn't dissipate heat, sound or light, then it is a
conservative force.
o
Work done by conservative forces are path independent.
o
Conservative forces are associated with a potential energy.
o
For example, the force from a spring can be stored as spring
potential energy.
o
Gravitational force can be stored as gravitational potential
energy.

o
Electromagnetic forces are also conservative.
o
non-conservative include frictional forces and human exertion.
When friction acts on an object, the heat and sound released
cannot be recovered. When you flex your arm, you lose heat
that cannot be recovered (you cannot re-absorb the heat you
lost).
Power, units
o
Power is the rate of energy use.
o
The unit for power is the Watt, or Joule per second.
o
Lifting a crate in one minute requires more power than lifting the
same crate in an hour.
Periodic motion

Amplitude, period, frequency
o
o
Amplitude (A): how high the peaks are or how low the troughs
are, in meters.

The displacement is how far the wave vibrates / oscillates
about its equilibrium (center) position.

The amplitude is the maximum displacement.

Amplitude is correlated with the total energy of the system
in periodic motion. Larger amplitude = greater energy.
o
Period (T): the time it takes for one cycle, in seconds.

o
o

T = 1/f
Frequency (f): the rate, or how many cycles per second, in Hertz
(cycles per second).

f = 1/T

Sometimes, frequency is in rpm (revolutions per minute).
rpm = cycles per second x 60.
Angular frequency (w): the rate, in how many radians per
second.

w = 2πf

w is also called angular velocity.
Phase
o
o
In phase: the waves are 0 or 2π radians (0 or 360°) apart. The
resulting amplitude (sum of the waves) is twice the original.


o
Completely out of phase: the waves are π radians (180°) apart.
The resulting amplitude is zero.
o
Out of phase: resulting amplitude is between 0 and twice the
original.
Hooke's law, force F= -kx
o
F is the force that acts to restore the spring back to its
equilibrium position, or restoring force.
o
k is the spring constant. Stiffer springs have a higher k value.
o
x is the displacement. The amplitude (A) is the maximum x
value.
o
Potential energy = PE = ½kx2
o
Kinetic energy = KE = ½mv2
o
At the equilibrium position x = 0, PE = 0, KE = maximum.
o
At the maximum displacement (amplitude) x = A, PE =
maximum, KE = 0.
o
At any point, PE + KE = maximum PE = maximum KE =
constant.
o
constant = PEmax = ½kA2
o
constant = KEmax = ½mv2 at x = 0
Simple harmonic motion; displacement as a sinusoidal function of time
o
x = A·sin(wt)
o
x is displacement.
o
A is amplitude.
o
w is angular frequency (also called angular velocity).
o
t is time.
o
Examples of simple harmonic motion

Oscillating spring.


Pendulum.

Things going around a circle at constant speed (when plot
the x axis position against time).
Motion of a spring with mass attached to its end
o

o
T is period, m is the mass of the attached mass, and k is the
spring constant.
o
A simpler way to express this is:
o
w is the angular frequency. The spring vibrates faster if it's
stiffer and if the mass attached to it is smaller.
Motion of a pendulum
o

o
T is period, L is the length of the string, and g is 9.8.
o
A simpler way to express this is:
o
w is the angular frequency. The pendulum oscillates faster when
gravity is large and when the string is short.
General periodic motion: velocity, amplitude
o
At the equilibrium position, PE = 0, KE = maximum.
o
At the maximum displacement (amplitude) x = A, PE =
maximum, KE = 0.
o
At any point, PE + KE = maximum PE = maximum KE =
constant.
o
constant = PEmax

= ½kA2 for a spring.

= mgA for a pendulum, where A is the maximum height
that the pendulum can gain during a swing.
o
constant = KEmax = ½mv2 at the equilibrium position.
o
If you are given the velocity at the equilibrium position, then you
should be able to find out the amplitude by setting maximum KE
= maximum PE.
o
If you are given the amplitude, then you should be able to find
out the velocity at the equilibrium position by setting maximum
PE = maximum KE.
Wave Characteristics

Transverse and longitudinal waves
o
o

Transverse wave: wave displacement is perpendicular to the
direction of motion.

Light.

Electromagnetic radiation.

A standing wave by oscillating a string side ways. The
speed for such a wave = square root of (string tension /
mass per unit length of the string). For the MCAT, just
know that tense, light strings can produce faster
transverse waves.
Longitudinal wave: wave displacement is parallel to the direction
of motion.

Sound.

Pressure wave.

Earth quakes.
Wavelength, frequency, velocity
o
v = fλ
o
v is velocity, f is frequency, and λ is wavelength.
o
Sometimes, frequency is also written as ν.
o


Wavelength is in meters, frequency is in Hertz and velocity is in
meters per second.
Amplitude, intensity
o
Amplitude is correlated with the energy of the wave. Greater
amplitude = greater energy of the wave.
o
Intensity = energy per area per time = power per area.
o
Thus, amplitude and intensity are correlated. Greater amplitude
leads to higher intensity.
o
Special note on electromagnetic waves: amplitude and intensity
increases the overall energy of electromagnetic waves such as
light. However, neither amplitude nor intensity changes the
energy per photon. Energy per photon depends on wavelength.
The shorter the wavelength (also the higher the frequency), the
greater the energy.
Superposition of waves, interference, addition
o
o


o
When waves superimpose on each other, they interfere.
o
Interference results from the addition of waves.
o
When in phase waves add, the resulting wave has a greater
amplitude.
o
When out of phase waves add, the resulting wave has a smaller
amplitude.
o
Constructive interference: addition of waves resulting in greater
amplitude.
o
Destructive interference: addition (cancellation) of waves
resulting in diminished amplitude.
Resonance
o
Resonance is when things oscillate at its maximum amplitude.
o
Resonance occurs at resonance frequencies.
Resonance frequencies
o
Examples of standing waves and the resonance frequencies that
produce them
o
Frequencies can be obtained by f = v/λ
o
Both strings and tubes open at both ends have L = n/2λ
o
Tubes with a closed end have L =
o
L is the length of the string/tube
n_odd/ λ
4



Standing waves, nodes
o
Standing waves vibrate at resonance frequencies.
o
Standing waves do not propagate like other waves (that's why
they're called standing waves).
o
Node: point where there's no oscillation.
o
Antinode: point where there's maximum oscillation.
Beat frequencies
o
Beats occur when two waves coexist at different frequencies.
o
The beat frequency is the difference between the frequencies of
the two waves.
Refraction and diffraction
o
o
Refraction is the bending of waves when it meets a boundary
between one medium to another.

Snell's law: n1sinθ1 = n2sinθ2 , where n is the refractive
index and θ is the angle to the normal.

When light moves to a denser medium (higher refractive
index), it bends toward the normal.

Dispersion, the bending of light through a prism, is a
special case of refraction that separates the colors of light
into a rainbow.

Rainbows are created by refraction by water droplets.
o
Diffraction is the spreading (diffusion) of waves around edges of
apertures and obstacles.

You can hear sounds from the other side of a building
because sound spreads.

Shining light through a hole will not produce a dot of light,
instead, it is a diffuse circle.

Diffraction is the basis for the single and double slit
interference experiments with light.

When you think of diffraction, think "diffuse".
Production of sound

Sound is produced by vibrations in a medium.

Sound can not be produced in a vacuum, nor can sound travel across a
vacuum.

Vibrations whose frequency is too low to hear is called infrasound.

Vibrations whose frequency is too high to hear is called ultrasound.

Vibrations produce pressure waves that oscilate parallel to the
direction of propagation.

Sound is a longitudinal wave.
Relative speed of sound in solids, liquids and gases

Speed of sound in solids > liquids > gases.
o

The reason why sound travels the fastest in solids is because
solids are the most stiff.
With all else being equal...
o
Speed of sound in stiff objects > elastic objects.
o
Speed of sound in less dense objects > more dense objects.
Even though gases are less dense than solids, sound still travels
slower in them because they are too elastic.
o
Speed of sound in hot objects > cold objects.
Intensity of sound (decibel units, log scale)

β = 10 logI/I0

β is sound level in decibels. I is intensity. I0 is 10-12 W/m2

Intensity is power per area, or the rate of energy expenditure per
area. The unit is W/m2


Intensity Decibels
I0
0
10 I0
10
100 I0
20
1000 I0
30

The decibel system is based on human perception. The decibel value
for sound with an intensity of I0 is zero - below this intensity, sound is
not audible. As intensity increases, our perception of its loudness only
increases to a much lesser degree.
Attenuation

Sound attenuation is the gradual loss of intensity as sound travels
through a medium.

Sound attenuation is the greatest for soft, elastic, viscous, less dense
material.
Doppler effect (moving sound source or observer, reflection
of sound from a moving object)



Situations where the observed frequency is higher than the actual:
o
Source moving toward stationary observer: fo = fs v/v - vs
o
Observer moving toward stationary source: fo = fs
o
Source and observer both moving toward each other: fo = fs
v /
o v - vs
v+v
o/v
v+
Situations where the observed frequency is lower than the actual:
o
Source moving away from stationary observer: fo = fs v/v + vs
o
Observer moving away from stationary source: fo = fs
o
Source and observer both moving away from each other: fo = fs
-v /
o v + vs
v-v
o/v
Situations where the observed frequency could be either higher or
lower than the actual:
o
Source moving toward the observer, but the observer is moving
away from the source: fo = fs v - vo/v - vs
o
Source moving away from observer, but the observer is moving
toward the source: fo = fs v + vo/v + vs
v

fo is observed frequency. fs is actual frequency emitted by the source.
v is the speed of sound. vo is the speed at which the observer is
travelling. vs is the speed at which the source is travelling.
Pitch

Pitch is the human perception of the frequency of sound.

Higher frequency = higher pitch.
Resonance in pipes and strings


Frequencies can be obtained by f = v/λ

Both strings and pipes open at both ends have L = n/2λ

Pipes with a closed end have L =
(2n-1)/ λ
4
Harmonics

The fundamental frequency is called the first harmonic (n = 1).

The next-up frequency is called the second harmonic (n = 2).
Ultrasound

Sound has 3 fundamental properties: reflection, refraction, and
diffraction.

Ultrasound imaging is based on the reflection property of sound.

A source emits ultrasound, which reflects off a surface back into the
detector to form an image.

Ultrasound is sound that is too high in frequency for humans to hear.
Fluids

Liquids and gases are fluids.

Density, specific gravity

o
Density: ρ=m/V, where ρ is density, m is mass, and V is volume.
o
The density of water is ρwater = 1 g/mL = 1 g/cm3 = 1 kg/L.
o
Specific gravity is the density of something compared to water.
o
Specific gravity = ρ/ρwater.
o
The specific gravity of water is 1.
Buoyancy, Archimedes' principle
o

o
Archimedes' principle: buoyant force on an object = weight of
the fluid displaced by the object.
o
FB = weightdisplaced = mdisplacedg =ρfluidVsubmergedg
o
The volume of an object that is submerged = the volume of fluid
displaced by the object.
o
Things float when FB = Weight.
o
Things will rise upward when FB > Weight.
o
Things will sink when FB < Weight.
Hydrostatic pressure
o
Pascal's law: if you apply pressure on a liquid, the pressure is
transmitted equally to all parts of the liquid.


F1/A1 =F2/A2

The pressure input at one end is the same as the pressure
output at the other.

You apply a small force over a small area, and the output
force at the end with the larger area will be greater.

A1d1=A2d2, where d is the distance that the end moves.

The work done on one end is the same as the work output
at the other.
o




P=ρgh

P is pressure, ρ is the density of the fluid; g is the
gravitational constant, h is the height from the surface, or
depth that the object is submerged.

Pressure at the surface is 0 because h = 0.

Pressure at a depth of h is ρgh.

ρgh is the gauge pressure because it ignores the
atmospheric pressure above the fluid.

Absolute pressure of something submerged in the ocean =
ρgh + atmospheric pressure.
Viscosity: Poiseuille flow
o

P = pgh (pressure vs. depth)
When a viscous fluid flows through a pipe, the flow has a front
that is shaped like a parabola bulging outward.
Continuity equation (A·v = constant)
o
The volume flow rate of a fluid is constant.
o
dV/dt = constant, where dV/dt is volume flow rate.
o
dV = A·dL
o
A·dL/dt = A·v = constant, where v is linear flow rate (velocity).
Concept of turbulence at high velocities
o
Low velocity -> laminar flow.
o
High velocity -> turbulent flow, forms eddies.
Surface tension
o
Surface tension gives the surface of a liquid the ability to support
things that are very light.
o
For example, insects can walk on water due to surface tension.
o

Surface tension is due to the attraction between the molecules of
the solvent.
Bernoulli's equation
o
P + ½ρv2 + ρgh = constant
Solids

Density: ρ=m/V, where m is mass and V is volume.

Elastic properties (elementary properties)
o
o
Stress: the pressure exerted on an object. σ = stress = F/A.
o
Strain: the deformation of the object in the direction of the
applied force divided by the original length. ε = strain = ΔL/L0.
o
Young's modulus = stress / strain.
o
Young's modulus, the ratio between stress and strain, is
constant until you reach the elastic limit, where things get
permanently deformed.

Elastic limit: The maximum stress something can handle before it
breaks or become permanently deformed.

Thermal expansion coefficient
o
Things expand when temperature rises, and contract when
temperature falls.
o
ΔL = αL0ΔT
o
ΔL is the change in length, L0 is the initial length, ΔT is the
change in temperature, and α is the coefficient of linear
expansion.

o
In the same fashion as linear expansion, the equations for
volume and area expansions are below.
o
ΔV = βV0ΔT
o
ΔA = γA0ΔT
Shear
o

o
Shear = stress / shear ratio.
o
Shear ratio = ΔL/L0.
o
When ΔL is very small compared to L0, Shear ratio is
approximately the same as the shear angle.
o
Shear angle = tan-1ΔL/L.
o
Note: ΔL and L are perpendicular to each other.
Compression: solids and liquids are generally not compressible. Gasses
are compressible.
Electrostatics
o
Charge, conductors, charge conservation

Charges are either positive or negative. Zero charge is
neutral.

Like charges repel, unlike charges attract.

Charge is quantized, and the unit of charge is the
Coulomb.

Conductors are materials in which charges can move
freely. Metals are good conductors.

o
Insulators

o
o
Charge is always conserved. You can't create or destroy
charge, you can only transfer charge from one source to
another.
Insulators are materials in which charges can not move
freely. Nonmetals are good insulators.
Coulomb's law (F = kq1q2/r2, sign conventions)

F = kq1q2/r2

k = 9E9 Nm2/C2

If the charges have the same sign, the force is repulsive.

If the charges have opposite signs, the force is attractive.
Electric field

field lines


Electric field is denoted by the vector E.

Lines that are closer together denote stronger fields
than lines that are farther apart.


Electric fields come out of positive charges, and goes
into negative charges.


The unit for electric field is N/C, or Newtons per
Coulomb.
field due to charge distribution


Field lines come out of the positive end and goes into
the negative end of a dipole.


Field lines for two negative charges are the same as
those for two positive charges except that the
direction of the field lines would be reversed.


The direction and magitude of the field at any point
in space can be calculated as the vector sum of all
the field components there.


Electric field in between a capacitor is uniform until it
reaches the ends of the capacitor.



Electric field for wires runs radially perpendicular to
the wire.

o
Electric field for a cylinder runs radially perpendicular
to the cylinder, and is zero inside the cylinder.
Potential difference, absolute potential at point in space


Absolute potential (V) is the amount of energy per charge
that something possesses.

V = U/q0 = kq/r

V is the electric potential (absolute potential) caused
by q, which is experienced by q0.

q is the charge that is causing the potential, not the
charge that's experiencing the potential.

Traditionally, q0 is the charge experiencing the
potential. The magnitude of q0 is very small.

U is the electrical potential energy possessed by q0.

r is the distance between the potential-causing
charge and the charge that's experiencing the
potential (r is always positive).

if there are multiple charges contributing to the
potential, then calculate the potentials each of them
causes (positive charges cause positive potentials,
and negative charges cause negative potentials), and
sum them together.

The unit for potential is Volts (V) or Joules per
Coulomb (J/C).

o
Potential difference (ΔV) is the difference between two
potentials.

ΔV = VB - VA

Potential difference is used in scenarios such as the
difference in potential between the two plates of a
capacitor, or the positive and negative terminals of a
battery.
Equipotential lines

o

Equipotential lines are places where the potential is the
same.

Equipotential lines are always perpendicular to electric field
lines.
Electric dipole

definition of dipole


dipole = a positive charge and a negative charge
separated by some distance.
behavior in electric field



A dipole in an electric field will want to align itself
with the electric field, such that the positive end of
the dipole is in the direction of the electric field.
potential due to dipole


o
To calculate the exact potential at a given point, just
calculate the individual potential due to the positive
charge and the negative charge, then add them
together.
Electrostatic induction


Induction does not involve any type of conduction.

Electrostatic induction is where a charged object induces
the movement / redistribution of charges in another
object.

The classical example of electrostatic induction is picking
up pieces of paper using a comb rubbed against fur.

It's called electrostatic induction because it's static - the
charged species polarizes non-charged species by simply
being there. This is not the same as electromagnetic
induction, which is how electric generators work. Luckily
electromagnetic induction is not listed as an official AAMC
topic.
o
Gauss' law



ΦE = EA cosθ

ΦE is electric flux.

E is electric field, A is area that the field goes
through, and θ is the angle between the field and the
normal of the area.
ΦE = q/ε0

For an enclosed surface, the electric flux is equal to
q, the charge inside the enclosure, over the
permitivity of free space.

The net electric flux through any enclosed surface is
totally dependent on the charge inside. If there's no
charge inside, then the net electric flux through the
enclosure is zero.
An important application of Gauss's law is the Faraday
cage. Basically, the electric field inside a closed conducting
cage is zero. This is because the charges on the conducting
cage will rearrange to cancel out any external field.
Magnetism
o
o
Definition of the magnetic field B

Magnetic field B exists in a region of space if a moving
charge experiences a force due to its motion in that region.

The unit for magnetic field is the Tesla (T) or
N·s/
m·C
Existence and direction of force on charge moving in magnetic
field


F = qvB sinθ

θ is the angle between the charge velocity and the
magnetic field. Sometimes the sinθ is omitted as θ is
assumed to be 90°.

The force is always perpendicular to both the magnetic
field and to the velocity of the charge.

You can use the right hand rule to predict the direction of
the force. The thumb is the direction of a positive charge,
the middle finger is the direction of the magnetic field, and
the palm faces the direction of the force.

Special scenarios / cases


Charge moving in a circle

F = qvB = mv2/r

You are setting the electromagnetic force equal
to the centripetal force, which maintains the
orbit. Using this equation, you can solve for
whatever the question asks you.
Current carrying wires

F = qvB sinθ = (it)vB sinθ = (it)(L/t)B sinθ =
iLB sinθ

i is current, L is length of wire.

Consider the current in the wire as moving
positive charges (by tradition, the direction of
the current is defined as the direction of
moving positive charges).

You can calculate the direction of the force on
the wire in the same way using the right hand
rule. Just treat the direction of the current the
same as the direction of velocity of a positive
charge.

Two wires will attract each other if the current
is in the same direction.

Two wires will repel each other if the current is
in opposite directions.
Electromagnetic Radiation (Light)
o
Properties of electromagnetic radiation (general properties only)


radiation velocity equals constant c, in vacuo

Electromagnetic radiation travels fastest in a
vacuum, at a velocity equals c, or 3x108m/s

Light slows down when it travels in a medium other
than in vacuo.

n = c/v, where n is the index of refraction for the
medium, and v is the speed of light travelling in that
medium.
radiation consists of oscillating electric and magnetic fields
that are mutually perpendicular to each other and to the
propagation direction

o
Classification of electromagnetic spectrum (radio, infrared, UV,
X-rays, etc.)

Lower frequency, longer wavelength, less energy
Radio
Causes electronic oscillations in the antenna
Microwave Causes molecular rotation
Infrared
Causes molecular vibration
Visible
Can excite electrons to orbits of higher energy. Visible light
ranges from 400-700 nm. 400ish being violet, 700ish being
red.
Ultraviolet
Can break bonds and excite electrons so much as to eject
them, which is why UV is considered ionizing radiation.
X-rays
Ionizing radiation, photoelectric effect
Gamma
rays
Even more energetic than X-rays
Higher frequency, shorter wavelength, more energy
Old AAMC Topics: the topics below have either been removed
or modified from the official AAMC outline.
Magnetism
o
Orbits of charged particles moving in magnetic field

o

Perfect orbit occurs when qvB = mv2/r

When qvB < mv2/r, there isn't enough centripetal force,
and the charged particle flies out of orbit.

When qvB > mv2/r, there's too much centripetal force, and
the charged particle spirals inward.
General concepts of sources of the magnetic field


o
Anything that involves a moving charge creates a magnetic
field

Moving charges.

Current carrying wire.

Solenoids and toroids.

The Earth (electric current in the liquid core).
Atoms with unpaired electrons is the other source of
magnetic fields. This is basically the same deal as moving
charges, since the unpaired electrons orbiting the nuclei is
the same thing as moving charges.

Magnets.

Individual atoms of Ferromagnetic and Paramagnetic
create magnetic fields because they have unpaired
electrons. Ferromagnetic materials have domains of
aligned atoms that make them even more
susceptible to be magnetized. Both Ferro and
paramagnetic material are attracted to magnetic
fields.

Diamagnetic atoms don't create magnetic fields
because the electrons are paired, so their individual
fields cancel out. Diamagnetic fields actually is
repeled by an external magnetic field.
Nature of solenoid, toroid


Solenoid

The solenoid is just a coil of current-carrying wire.

B = μ0nI.

o
n is the number of loops per meter. I is current.

The magnetic field produced by a solenoid is directly
proportional to the number of coils, and to the
current.
Toroid

Toroid is just a solenoid in a circle.

B = μ0NI/circumference

N is the total number of loops, I is the current.

More loops, smaller circle → greater magnetic field.
Ampere's law for magnetic field induced by current in straight
wire and other simple configurations

o

Ampere's law lets you calculate the magnetic field at a
radius r from a current-carrying wire: B = μ0I/2πr
Comparison of E and B relations


force of B on a current

F = qvB sinθ = (it)vB sinθ = (it)(L/t)B sinθ = iLB
sinθ

i is current, L is length of wire.

Consider the current in the wire as moving positive
charges (by tradition, the direction of the current is
defined as the direction of moving positive charges).

You can calculate the direction of the force on the
wire in the same way using the right hand rule. Just
treat the direction of the current the same as the
direction of velocity of a positive charge.

Two wires will attract each other if the current is in
the same direction.

Two wires will repel each other if the current is in
opposite directions.
energy

Oscilations of electric and magnetic fields
(electromagnetic radiation) has energy.

E = hν

E is energy per photon, h is Planck's constant, and ν
is the frequency of the electromagnetic wave.
Circuit elements

Current (I = ΔQ/Δt, sign conventions, units)
o


o
Current is the rate of charge flow through the cross-section of a
conductor (wire).
o
Traditionally, the direction of current is taken as the flow of
positive charges.
o
The unit for current is Coulombs per second, C/s.
Battery, electromotive force, voltage
o
Electromotive force (emf) is really not a force, but a potential
difference, with the unit voltage.
o
A battery is a source of emf.
o
If the battery has no internal resistance, then potential
difference across the battery = EMF.
o
If the battery has internal resistance, then potential difference
across battery = EMF - voltage drop due to internal resistance.
Terminal potential, internal resistance of battery
o

o
Terminal potential is the voltage across the terminals of a
battery.
o
Internal resistance of a battery is like a resistor right next to the
battery connected in series.
o
Terminal potential = EMF - IRinternal
Resistance
o
Ohm's law (I = V/R)
o
resistors in series

o

Iseries = I1 = I2 = I3

All resistors in series share the same current.

Vseries = V1 + V2 + V3

Voltage drop among resistors in series is split according to
the resistance - greater resistance, greater voltage drop (V
= IR).
resistors in parallel

o


Vparallel = V1 = V2 = V3

All resistors in parallel share the same voltage.

Iparallel = I1 + I2 + I3

Current among resistors in parallel is split according to the
resistance - greater resistance, less current (I = V/R).
resistivity (ρ = RA/L)

Resistivity is the inverse of conductivity.

Greater resistivity, greater resistance of the material.

Rearranging the above equation to get R = ρL/A. To make
a wire of low resistance, select a material that has low
resistivity, keep the wire short, and keep the diameter of
the wire large.

Extension cords are made really thick to keep the
resistance down, so it doesn't heat up and cause a fire.
Capacitance
o
concept of parallel-plate capacitor

o
o

C = Q/V = εA/d

Greater capacitance is created by a greater charge on
plates (Q) for a given voltage (V), greater plate area (A),
or smaller distance between plates (d).

V = Ed, where V is voltage across capacitor, E is electric
field between capacitor, and d is the distance between
capacitor plates.
energy of charged capacitor

U=
Q2/
2C
= ½QΔV = ½C(ΔV)2

U is the potential energy of the charged capacitor, Q is
charge stored (magnitude of either +Q or -Q on one of the
plates), C is capacitance.
capacitors in series


o
1/
Ceq
= 1/C1 + 1/C2 + 1/C3
capacitors in parallel


o
Ceq = C1 + C2 + C3
dielectric



Dielectric = nonconducting material.

Inserting a dielectric between the plates of a capacitor
increases the capacitance by either increasing Q (if V is
constant) or decreasing V (if Q is constant).

V = V0/κ

C = κC0
Discharge of a capacitor through a resistor
o
Charge
o
Discharge
o
During the discharge of a capacitor, the capacitor acts as a
battery and drives current flow, which decreases with time as
the capacitor discharges.
Conductivity theory
o
Conductivity is affected by electrolyte concentration:

No electrolyte, no ionization, no conductivity.
o

Optimal concentration of electrolyte, greatest conductivity
due to greatest mobility of ions.

Too much electrolyte, ions are too crowded, less ion
mobility, less conductivity.
Conductivity is affected by temperature:

In metals, conductivity decreases as temperature
increases.

In semiconductors, conductivity increases as temperature
increases.

At extremely low temperatures (below a certain critical
temperature typically a few degrees above absolute zero),
some materials have superconductivity - virtually no
resistance to current flow, a current will loop almost
forever under such conditions.
o
Conductivity (σ) is the inverse of resistivity (ρ).
o
Place a capacitor inside a solution, the solution will conduct a
current between the plates of the capacitor, thus you can
measure the conductivity of a solution using a capacitor.
Circuits

Power in circuits (P = VI, P = I2R)
o
P = IV = I2R
o
P is power, I is current, V is voltage, R is resistance.
o
Power companies try to save the amount of copper needed for
power lines by using thinner wires, which makes R quite high.
o
To minimize P dissipated by the wires, they minimize I by
maximizing V. This is why power lines transfer electricity at high
voltage.
Alternating Currents and Reactive Circuits

Root-mean-square current
o
Irms =
I
max/√2
= 0.7 Imax

Root-mean-square voltage
o
Vrms =
V
max/√2
= 0.7 Vmax

Vrms = IrmsR

Pavg = IrmsVrms = I2rmsR
Light (Electromagnetic Radiation)

Concept of interference, Young double slit experiment
o
Review basic interference concepts here
o
In order for interference to occur, the follow conditions must
hold:

the interfering light sources must be coherent. This means
they must constantly maintain the same phase
relationship. The light coming from the two slits in Young's
double slit experiment are coherent because a single light
source shines through both slits.

the light source must be monochromatic (of single
color/wavelength).

dsinθ = mλ

bright bands occur at m = 0, +/-1, +/-2 ...etc

dark bands occur at m = +/-0.5, +/-1.5, +/-2.5 ...etc
o

Thin films, diffraction grating, single slit diffraction
o
Thin films provide a means for interference to occur.

Light reflecting off the outer and inner boundary of a thin
film interfere with each other.

A film of oil on water has the appearance of a swirly
rainbow due to this interference.
o
o
o
Diffraction grating

Diffraction = light spreads out after passing through the
slit, instead of going in a straight path.

Diffraction grating = a slab with many slits close together.

The equation for a diffraction grating is the same as the
double-slit experiment.

dsinθ = mλ

d is the distance between the slits, everything else is the
same as the double-slit experiment.

bright bands occur at m = 0, +/-1, +/-2 ...etc

dark bands occur at m = +/-0.5, +/-1.5, +/-2.5 ...etc
Single slit



Light shining through a single slit casts a central bright
band followed by a series of maximas and minimas on
either side.

The equation for a single slit diffraction is different from
the equation for the double slit.

asinθ = mλ

a is the width of the slit.

Maxima occurs for m = 0 (big central maxima), +/-1.5,
+/-2.5 , etc.

Minima occurs for m = +/-1, +/-2, +/-3, etc.
Other diffraction phenomena, X-ray diffraction
o
Light shining through a pin hole will not appear on the screen as
a pin hole. Instead, it will be a diffraction pattern of circular
bright and dark bands, with a central bright band.
o
Light shining past an opaque boundary will not cast a sharp
shadow of the boundary on the screen. Instead, fringes of bright
and dark bands appear above the boundary.
o
Light shining past a penny will not cast a completely black
shadow. Instead, there will be a central bright spot, as well as
patterns of bright and dark rings.
o
X-ray diffraction = X-rays diffracting on a crystal. Patterns of
interference that results from this is used to deduce the
structure of the molecules in the crystal.
Polarization of light
o
Unpolarized light = light with electric field oscilating in many
planes.
o
Polarized light = light with electric field oscilating in only one
plane.
o
Applications of polarization:

Selective absorption: pass light through polarizer that
absorbs all but light with electric field in one plane.



Reflection: at a certain polarizing angle, all reflected light
is polarized.

Double refraction: birefringent materials have two indices
of refraction that splits the incident light into two rays
polarized perpendicular to each other.

Scattering: air molecules scatter light, which becomes
polarized.

Opticaly active molecules either rotate polarized light
clockwise or counterclockwise.
Doppler effect (moving light source or observer)
o
Red shift = frequency decreases = occurs when source and
observer is moving away from each other.
o
Blue shift = frequency increases = occurs when source and
observer is moving toward each other.
o
Observed in astronomy, when stars appear redder/bluer than
they really are because they are moving away/toward us.
o
The equation for the doppler effect for light is the same as
the doppler effect for sound, except instead of using speed of
sound v, you now use the speed of light c. For red shift, use the
equation for source moving away from observer. For blue shift,
use the equation for source moving toward observer.
Visual spectrum, color
o
o
energy

Blue = greatest energy, shortest wavelength, highest
frequency.

Red = least energy, longest wavelength, lowest frequency.

Energy per photon = hν, where h is plank's constant and ν
is frequency.
lasers

Laser = light amplification by stimulated emission of
radiation.

Normal light emission = spontaneous emission.

Laser emission = stimulated emission.

Repeated stimulated emission inside the lasing medium
(by reflecting light back and forth through it) amplifies
light.
Geometrical Optics

Reflection from plane surface (angle of incidence equals angle of
reflection)
o
o
mirrors completely reflect light.
o
going from one medium to another results in partial reflection of
light.

Refraction, refractive index n, Snell's law (n1sinθ1 = n2sinθ2)

Dispersion (change of index of refraction with wavelength)
o
blue light refracts more than red light in a prism.
o


white light passes through a prism and gets split into colors of
the rainbow due to dispersion.
Conditions for total internal reflection
o
Going from a medium of high index of refraction to a medium of
low index of refraction.
o
Angle of incidence > critical angle.
o
Find the critical angle by: n1sinθc = n2sin90°

n1 > n2

θc = critical angle
Spherical mirrors
o
Image height vs. Object distance:
note: this curve only shows the height of the image, not the
position.
o
note: this curve only shows the height of the image, not the
position.
o
o
mirror curvature, radius, focal length

mirror curvature can be concave or convex.

concave mirrors can focus light, so it's converging.

convex mirrors can't focus light, so it's diverging.

The focal length is 1/2 of the radius of curvature.

converging mirrors have positive focal length, while
diverging mirrors have negative focal length.

It's called the focal length because rays parallel to the
principle axis of the mirror will converge at the focal point
(for diverging mirrors, the extrapolated rays will pass
through the focal point).
use of formula (1/p) + (1/q) = 1/f with sign conventions

For the purpose of the MCAT, p is always positive unless
the MCAT explicitly tells you otherwise.

q is positive if the image is real. For mirrors, this is when
the image is in front of the mirror. For lenses, this is when
the image is behind the lens.
o


f is positive when the mirror/lens is converging. For
mirrors, this is when the mirror is concave. For lenses, this
is when the lens is convex.

M = h'/h = -q/p, where M is magnification, h' is height of
image, h is height of object.
real and virtual images

real images are always inverted, and can be cast on a
screen.

virtual images are always erect (noninverted), and can not
be cast on a screen.

For concave mirrors, real images (positive q) are formed in
front of the mirror, where light is reflected by the mirror
and can be cast on a screen. It's impossible for light to be
cast behind the mirror, so anything behind the mirror is
virtual (negative q).

For convex mirrors, images are always virtual (negative q).

Note: diverging mirrors and lenses (convex mirrors and
concave lenses) can never form real images.
Thin lenses
o
You don't have to re-learn everything for lenses, because they
are almost the same as mirrors:
o
Convex lenses are the same as concave mirrors (both are
converging) except for the following:
o

Real images are on the opposite side of the lens as the
object. Because light travels through the lens and can
focus on a screen behind the lens.

Virtual images are on the same side of the lens as the
object. Because light can't focus in front of a lens and be
cast on a screen.
Concave lenses are the same as convex mirrors (both are
diverging) except for the following:

The virtual images formed by the lens is on the same side
of the lens as the object. Because light can't focus in front
of a lens and be cast on a screen.
o
The image height vs. object distance curve is exactly the same
as those of mirrors (convex lenses the same as concave mirrors,
concave lenses the same as convex mirrors). Refer to above.
o
converging and diverging lenses, focal length
o
o
o

Focal length for converging lens is positive.

Converging lens is convex.

Focal length for diverging lens is negative.

Diverging lens is concave.
use of formula (1/p) + (1/q) = 1/f, with sign conventions

same deal as with mirrors.

p always positive.

q positive if real, and negative if virtual.

f positive if converging, and negative if diverging.
real and virtual images

Real images are inverted and can be cast on a screen.

Virtual images are erect and can not be cast on a screen.

For convex lenses, real images (positve q) are formed
behind the lens because light passes through the lens and
focuses there.

For concave lenses, images are always virtual (negative
q), and forms in front of the lens.
lens strength, diopters

Lens strength, or lens power is measured in diopters.

P = 1/f

o


where P is in diopters.
lens aberration

spherical aberrations: not all light will focus at the focal
point.

chromatic abberation: blue light gets refracted more than
red light, so different colors focus differently.
Combination of lenses
o
The real image formed by a lens can be used as the object for
another lens.
o
Magnification by multiple lenses is the product of all the
individual magnifications.
Ray tracing
<li
<li
<li
o
For mirrors:
1. First draw a parallel line from the object, as it bounces off
the mirror, it intersects the focal point. Now, which focal
point to intersect? The left or right? Use common sense:
for concave mirrors, it's going to focus the ray to the left
focal point. For convex mirrors, which can't focus, it's
going to diverge the ray, which means you're going to
have to extrapolate it to the right focal point.
2. Next draw a line that intersects the R point on the principle
axis. Which R? Left or right? Should I extrapolate? Again,
use common sense: The ray drawn should bounce right
back its original path, and not be reflected else where. By
eye-balling the mirror, you should be able to figure this
out.
3. Now, you already have two rays drawn, and that is enough
to make an intersection. Use this intersection as a guide to
drawing the last ray. The last ray should first intersect the
focal point, then bounce off the mirror parallel to the
principle axis. Which focal point to intersect? Should I
extrapolate? There's only one combination for the ray here
to fit the intersection already made by the previous two
rays. The trick to do this is to draw the parallel line first,
and force it to intersect the intersection already made by
the previous two rays.
o
o
o
o
For lenses (similar to the way you draw rays for mirrors):
0. First draw the parallel → focal point ray. It should make
sense which focal point the ray should hit/extrapolate
given the converging/diverging nature of the lens.
1. Next draw a ray intersecting the center of the lens.
2. Lastly, using the intersection already made by the previous
two rays as a guide, draw the focal point → parallel ray.
Again, draw the parallel line first and force it to intersect
the intersection already made by the previous two rays.
</li
</li
</li

Optical instruments
o
Eye = lens focuses real image on retina.
o
Glasses = diverging (concave) lens for near-sightedness,
converging (convex) for far-sightedness.
o
Magnifying glass = virtual, erect, larger image formed when p <
f for a converging lens.
Atomic Structure and Spectra
Emission spectrum of hydrogen (Bohr model)


Bohr model:
o
An electron orbits the positively charged nucleus in the same
way that the earth orbits the Sun.
o
Electrostatic attraction pulls the electron toward the nucleus.
o
The electron orbits at high speed to prevent it from crashing into
the nucleus.
o
The electron can orbit at different energy levels: n=1, n=2, n=3
...etc.
o
The higher the energy level, the larger the radius from the
nucleus.
Emission spectrum of hydrogen:
o
When an electron transitions from a higher energy level to a
lower energy level, it emits electromagnetic radiation.
o
The emission spectrum of hydrogen consists of sharp, distinct
lines.
Atomic energy levels

quantized energy levels for electrons
o
The distinct lines of the emission spectrum prove that electron
energy is quantized into energy levels.
o
If electron energy is not quantized, then a continuous spectrum
would be observed.
o
The energy of the energy levels is governed by:
where E is energy and n is the energy level.

The equation is negative, so all energies are negative.

Negative energies mean that it is energy that contributes
to the "stability" of the system - the electron binding
energy.

The more negative (lower) the energy, the more stable the
orbit, the harder it is to knock out the electron.

The less negative (higher) the energy, the less stable the
orbit, the easier it is to knock out the electron.

At the highest energy, 0 eV, there is no binding energy, so
the electron dissociates.

For atoms other than hydrogen, the shape of the energy
level curve stays the same. However, the numerator is a
constant other than 13.6 eV.

The precise relationship for atoms other than hydrogen
is:


,
, where Z is the atomic number.
Higher Z values give more negative binding energy (more
stable) because the more charge, the more electrostatic
attraction.
calculation of energy emitted or absorbed when an electron changes
energy levels
o
The wavelength of the emitted or absorbed radiation is governed
by the Rydberg formula:
, where lambda is the
wavelength, nf is the final energy level, ni is the initial energy
level, and R is the rydberg constant.
o
The energy of the emitted or absorbed radiation
is:
, where E is energy, f and v both mean
frequency and c is the speed of light.
o
Energy is emitted for transitions to lower energy levels (nf < ni).
o
Energy is absorbed for transitions to higher energy levels (nf >
ni).
Atomic Nucleus
Atomic number, atomic weight



Atomic number = the number of protons.
o
The atomic number is what defines an element.
o
When two things have the same number of protons, they are the
same element.
Atomic weight = the weighted average of atomic mass for all isotopes
of a given atom.
o
Atomic mass = number of protons + neutrons.
o
The atomic mass is used for an isotope.
o
The atomic weight is used for an element.
In standard notation the atomic number is always at the bottom, and
the weight is always on top:

An easy way to remember this is that the atomic number is
"fundamental" to the identity of the element, so it is located at the
fundation.
Neutrons, protons, isotopes

Neutrons = neutral particles that reside in the nucleus.

Protons = positive particles that reside in the nucleus.

Isotopes = things with the same number of protons, but different
number of neutrons.
Atomic particles
Name
Mass (amu) Charge Location
Proton
1
+1
In the nucleus
Neutron 1
0
In the nucleus
Electron 0
-1
Surrounding the nucleus

Nucleons = protons or neutrons.
Isotopes

When two things have the same number of protons but different
number of neutrons, they are isotopes of the same element.

Isotopes often have similar chemical properties, but different stabilities
(some decay and give off radiation, some don't).
Nuclear forces

Two forces are at work in the nucleus: the strong force and the
electromagnetic force.

The strong force binds the nucleons together, and is therefore
contributes to the binding energy.

The electromagnetic force is due to electrostatic repulsion between the
positively charged protons in the nucleus.

The nucleus stays together because the strong force is much stronger
than the electromagnetic repulsion.

The strong force is also called the "nuclear force".
... see forces section
Radioactive decay: alpha, beta, gamma, half-life, exponential
decay, semi-log plots

Alpha decay:
low speed.

Beta decay:
. Ejection of a helium nucleus at relatively
. Ejection of a high speed electron.

Gamma decay:
electromagnetic wave.
. Release of high energy

Name


Notation Information
Alpha
particle
Weakest form of radiation. Can be stopped by a sheet of paper. It is
essentially a relatively low speed helium nucleus.
Beta
particle
More energy than an alpha particle. Can be stopped by aluminum
foil. It is a high speed electron.
Gamma
ray
Strongest form of radiation. It is a high energy electromagnetic
wave. Can be stopped by a thick layer of lead or concrete.
Some notes on α, β, and γ decay
o
Conservation of mass dictates that total atomic weight before
the decay equal the total atomic weight after.
o
Conservation of charge dictates that the total atomic number
before the decay equal the total atomic number after.
o
Don't get thrown off by particles you do not recognize. As long
as they have a weight and a charge, just incorporate these
numbers in your calculations.
o
MCAT problems on identifying decay products are just math
work.
o
Remember: the atomic number (the bottom number) determines
what element it is.
half-life is the time it takes for the amount of something to half due to
decay.
o
After 1 half-life, the amount of the original stuff decreases by
half.
o
After 2 half-lives, the amount of the original stuff decreases by a
factor of 4.
o
After 3 half-lives, the amount of the original stuff decreases by a
factor of 8.
o
The mathematical expression for this
is:
, where N sub t=0 is the
amount the original starting material. N sub t is the amount of
the original material that is still left. Lastly, t is time.
o
Although the above is the official half-life equation, people like to
multiply rather than to divide. Therefore, a more user friendly
equation is:

Stability
o
When something is stable, it doesn't decay.
o
When something is unstable, it decays.
o
The more unstable something is, the shorter the half-life.

Exponential decay:

Semi-log plots: for the purposes of the MCAT, semi-log plots convert
exponential curves into straight lines.
o
Something that curves up becomes a straight line with a positive
slope.
o
Something that curves down becomes a straight line with a
negative slope.
o
For exponential decay, a semi-log plot graphs the log of amount
vs. time.
o
For exponential decay, a semi-log plot is a straight line with a
negative slope.
o
The semi-log plot intercepts the x axis where the original y value
is 1.
o
General nature of fission

Fission = one nuclei splitting apart.

Uranium undergoes fission when struck by a free neutron.

The fission of uranium generates more neutrons, which goes on to split
other Uranium nuclei. This is called a chain reaction.
General nature of fusion

Fusion = two nuclei coming together.

The Sun works by fusion.

Hydrogen in the Sun fuses to form helium.
Mass deficit, energy liberated, binding energy

Mnucleons = Matom + binding energy/c2

Mnucleons > Matom because some of the Mnucleons is converted to binding
energy that holds the nucleons together.
o
Mnucleons = mass of all the nucleons that make up the atom in
their free, unbound state.
o
Matom = mass of the atom.
o
Mnucleons - Matom = mass deficit (also called mass defect) = ΔM.
o
Binding energy = converting ΔM into its equivalent in energy =
ΔM c2.
o
Energy liberated = binding energy.

The conservation of mass and energy: the total mass and energy
before a reaction is always the same as the total mass and energy
after the reaction.

If the total mass before the reaction is different from the total mass
after the reaction, then the difference in mass is made up for by
energy.

The difference in mass before and after a reaction is called the mass
deficit or mass defect.

The energy that makes up for the mass deficit is calculated by:

Energy is liberated when mass is lost during a reaction.

Energy is absorbed with mass is gained during a reaction.

More notes on binding energy:
o
Binding energy most commonly refers to nuclear binding energy
(the energy that binds the nucleons together).
o
Binding energy is due to the strong force. ...more on forces
o
Binding energy per nucleon is strongest for Iron (Fe 56).
o
Binding energy per nucleon is the weakest for Deuterium (the 2nucleon isotope of hydrogen).
o
Less commonly used is the electron binding energy. This is
because electron binding energy is more commonly referred to
as the ionization energy.