Translational Motion
I.
II.
III.
Vectors vs Scalars
a. Scalars: length, time, mass, distance, speed, (kind of: energy)
b. Vectors: displacement, velocity, acceleration, force
Motion Formulas:
a. vav = Δx / Δt
b. x = vavt
c. vav = (vf + vi)/2
d. a = Δv / Δt = (vf - vi)/t
e. vf2 = vi2 + 2aΔx
f. Δx = vit + ½at2
Types of problems:
a. Trajectory of ball
b. Trajectory off cliff
c. Free fall
Forces
I.
II.
III.
Newton’s 3 Laws
a. 1 – law of inertia – object in motion stays in motion, object at rest stays at rest unless
acted upon by external force
b. 2 – F = ma
c. 3 – every action has an equal an opposite reaction
i. Understand that things can move because action reaction pairs are acting on
different objects
ii. Rockets work in vacuum because if push off
Gravity
a. F = GMm/r2
b. Weakest of the fundamental forces (‘four’ces because there are four). Only accounts for
attractive force between bodies.
c. Normal force is actually electrostatic force pushing back
d. Terminal velocity
i. When the force of air resistance is equal to the weight of the object in free fall,
the object is said to reach terminal velocity because it stops accelerating. That
object is now in equilibrium and no forces are acting on it. This is the highest
velocity it will go.
Uniform circular motion
a. Diameter = dπ = 2πr
b. Radian -> 2π = 360 degrees
c. a = v2/r
d. velocity is the tangential velocity at any given point
e. Centripetal force: ma = mv2/r
i. When thinking about centripetal force, think of it as an outward centrifugal
force for calculations when you are balancing forces. Centripetal force isn’t
actually a new force, it’s just the sum of the inwardly facing forces of a circular
path.
ii. Ball on string in horizontal direction: T = mv2/r ; there is no other force acting on
the ball so all of the inward force comes from the centripetal force. But if it isn’t
completely horizontal and it is at an angle with the horizontal – then the vertical
component of the Tension = mg and the horizontal component of the Tension =
mv2/r
iii. Ball on vertical string: At what point is the tension the highest? When you are
the top of the loop, the sum of the inward forces is the Tension + mg. So mv2/r =
FT + mg or FT = mv2/r – mg. When at the bottom of the loop, Tension points up
while mg points down, so they oppose each other (and since you are still
moving inward toward the circle, T is bigger) so mv2/r = FT – mg or FT = mv2/r +
mg. Tension is highest at the bottom.
1. It may help to picture the centripetal force as a centrifugal force from
the reference point of the object. When at the top of the circle, mv2/r is
the outward force and opposes the two inward forces (T and mg). When
at the bottom of the circle, mv2/r and mg are outward forces opposing
T. That’s why T is bigger
iv. Normal force: Where is your weight the highest? Equator or North Pole? At the
North pole, the only force acting on you is mg, so the normal force (and your
weight) is mg [N = mg]. At the equator, since you are spinning with r = the radius
of the earth, the sum of forces, mv2/r = mg – N because technically you are
accelerating toward the center along the circular path. As such, your weight is
actually [N = mg - mv2/r] or slightly smaller than mg. So you weigh less. To
conceptualize, think about the centrifugal force pushing you away from the
earth slightly so you are slightly less heavy
f. Frequency = amount of revolutions in 1 second
g. Period = Time it takes for one revolution
h. Angular velocity. If you travel inward toward the center of the circle, your r decreases so
your tangential velocity decreases (why you don’t get thrown out of a merry go round
on the inside), but your angular velocity (the amount of angle you traverse in an amount
of time) does not decrease. If a 90 degrees = π/2 in radians and represents ¼ of the
arclength of a circle (1/4 * 2πr = r* π/2), then S arclength = r*θ or θ = S/r.
i. ω = Δθ / Δt = the angular velocity, so ΔS/r * 1/t = ω. ΔS/t = v, so
ω = v/r
ii. This explains why you feel less acceleration at the center of a merry go round.
a = v2/r would make it seem like a goes up when r goes down, but in reality,
your tangential velocity is changing quite a bit but your angular velocity isn’t.
Plugging in for a non-changing value, a = (ωr)2/r, a = ω2r, so a ∝ r.
IV.
V.
Friction
a. Friction is the force that opposes movement
i. Two coefficients of friction – static and kinetic. Static is the larger μ that first
opposes motion. Once you overcome the max static friction force and the object
is in motion, you now only overcome the kinetic friction force which is smaller.
ii. Counterintuitive – static friction is also the force that moves a resting object.
When you push off the ground when you start running or move forward from
being still in a car, the static friction with the floor gives you the push to start
moving. Kinetic friction always opposes movement
iii. Counterintuitive – the acceleration you feel going uphill is always greater in
magnitude than the acceleration going downhill. That is because when you are
moving uphill, the component of gravity + friction are pointing in the same
direction, so they combine to increase acceleration (ma = mgsin θ + μkN)
whereas when you go downhill they oppose each other so (ma = mgsin θ - μkN).
iv. Counterintuitive – if you think about it, when you lift an object first, you make it
easier to move. So when you pull a rope upward on an object to pull it, you
reduce the normal force so you reduce friction. Even though you only get a
component of force moving in the x direction, the reduced normal force in y
direction still makes the thing easier to push.
b. Force of friction formulas
i. F = μsN Static friction: This is a maximal value that you must overcome.
1. Conceptualize: When an object is at rest (say on a flat surface) it does
not have this max frictional force acting on it. It has no frictional force.
Once you start moving the object (pulling it, increasing the incline on
the plane) then the friction force grows to match the opposing force
until the opposing force overcomes the maximum force of static
friction, in which case the object will move.
ii. F = μkN Kinetic friction: This force always opposes movement and does work
against you, dissipating energy as heat.
Tension
i. Tension is the force that opposes the other forces in a rope. Remember to treat
moving tension (such as in a pendulum) different from static tension. Static
tension is easy, just balance the components of the forces.
1. For moving tension: If you have two sides playing tug of war, on side
pulling with 15N and the other pulling with 20N, the tension on the rope
is 15N, not 35N or 5N! Consider if you were pulling a rope with a weight
at the end upward using 100N and the weight is 50N. The tension is just
going to be the 50N weight.
VI.
VII.
2. For a pendulum, remember that tension is always highest at the bottom
because Tension has to match centrifugal force AND gravity. Treat the
tension as the axis and the gravity breaks into components.
Mechanical Advantage
a. Mechanical advantage is reducing the amount of force but not reducing the amount of
work (like a gear turning). You increase the distance so you have to apply less force to
get it done. If you had to lift 100kg up 5m, it would require 1000N of force and 5000J.
But if you push it up an incline plane, you could apply a force of 100N over 50m and get
to the same place.
b. Mechanical advantage = weight of object/applied force
c. Machines you will have to understand: Lever and fulcrum, incline plane, pulley,
hydraulic lift. For this section, only incline plane and pulley are relevant.
Incline plane problems
a.
Just remember that gravity component = mgsinθ and N = mgcosθ.
b.
VIII.
If you get a pulley system question, split the force diagram and remember that a1 = -a2
and Tension is equal in both equations. In a lot of these questions, plug in extremes
using limiting cases (such as even weight or m=0) to try to solve for the equation if
possible.
Pulley problems
a. The key to understanding pulleys is to count the number of upward forces of tension are
supporting the weight being lifted.
b.
a single upward tension here means that there is no mechanical advantage
c.
in comparing the two pulleys, B has two upward
tensions so it requires half the force (but twice pull length) the move the weight
d.
count the number of upward tensions. This should be
(6) and so you would only need 500N of force to pull this up.
Momentum, Torque, Equilibrium
I.
Center of mass
a. Usually treat things as point masses, but if the problem has torque then the center of
mass becomes relevant. Center of mass for simple objects are at the geometric center
(such as a sphere), but for a donut it is not in the shape at all (in the hole).
b. For problems that have center of mass being relevant, the density or shape is
nonuniform. Simple problems will likely include hanging ruler with weights or seesaw.
Massless objects do not matter, but you may need to take into account the center of
mass of a massed object. That will generally be at the geometric center for these types
of problems.
c. Center of mass formula:
xcenter = (m1x1 + m2x2 + … ) / (m1 + m2 + …)
II.
Equilibrium
a. Something is in equilibrium when no net forces acting on it. For translational motion,
that means sum of the forces = 0, a = 0. The object can be at rest or constant velocity
b. Note that if something is moving at a constant velocity with a frictional force of 50N
opposing the motion, the constant velocity still means net a = 0, so there is no net force
acting on it.
c. 3 types of equilibrium > static: no velocity/movement; translational: Fnet = 0; rotational =
torque = 0
i. Translational equilibrium > when the object’s center of mass is moving at a
constant or zero velocity, it is in translational equilibrium. If you had a fulcrum
NOT at the center of mass for an object and it rotates, then the center of mass is
accelerating and is NOT in translational equilibrium
III.
Torque
a. Torque is the measure of how much a force rotates an object about a fixed point. Since
there is a force making it rotate, note that this force will make the object accelerate
rotationally. It is measured in Newton meters Nm
b. Torque equation: τ = rFsinϴ
i. Usually the equation is r x F (cross product) since you are really only considering
the Force that’s perpendicular that would cause it to rotate. If the Force were
applied parallel, no rotation.
c. Equation can also be given by τ = F * l where l = the lever arm. Lever arm is the shortest
distance between the pivot point and the line of action. If they give you the lever arm
distance, then use that instead.
d. Notice that the further you apply the force, the less force you need (or the greater the
force exerted if using the same force) to make rotation (should be intuitive), but you
need to move the point further.
IV.
Momentum
a. Important concept is that when things collide, the momentum is conserved. These are
vector quantities so make sure you don’t mix up your signs. Otherwise intuition serves
well in this area.
b. Momentum equation: p = mv
c. Impulse measures the change of momentum of an object. It has the same units as
momentum (kg m/s). Since Force = kgm/s2, you can multiply by t to get
J (impulse) = Δp = FΔt = mΔv = mvf - mvi
d. Conceptualize for MCAT: Injuries occur when you experience a massive force. If a two
runners with the same mass and velocity fall and one tumbles but the other hits the
ground, they both have the same impulse (mv = mv). While FΔt is the same for both
parties, since Δt is much smaller for the second runner, he experiences a much higher
force.
e. Conservation of momentum – for any action reaction pair, the total momentum is
conserved, so change in momentum = 0. Imagine radioactive decay where He is ejected.
The parent nucleus must recoil with a momentum equal to the ejected particle.
Momentum is only conserved in the absence of other forces (for instance it is not
conserved if friction is taken into account)
f. Collisions: there are three important types of collisions – note that momentum is always
conserved in collisions – only way to have no conservation is if you lose force to friction
i. Elastic collision: Kinetic energy is also conserved. The energy transfer is perfect
and lossless. Think of two rubber balls bouncing off each other. Never assume
this is the case unless it is specified (doesn’t occur in real world)
ii. Inelastic collision: There is some loss of energy from deformation/heat loss.
Think of a car hitting a bike and denting both (work done to dent is energy lost).
iii. Perfectly inelastic: loss of energy and objects are stuck together afterward and
move together. Think of a truck crushing a car and they keep driving
g. Conservation of momentum: m1v1i + m2v2i = m1v1f + m2v2f > note that for perfectly
inelastic, the right half is (m1+ m2)vf
Work and Energy
I.
II.
III.
IV.
V.
Work: W = Fdcosϴ
a. Work is only done when there is a displacement. Work is measured as the force
applied in the direction of displacement. If you push a box downward on the floor
with 100N, it doesn’t go anywhere (direction is perpendicular to movement
direction) so no work is done.
b. WORK IS PATH INDEPENDENT
c. Measured in Joules, or kg*m2/s2
Kinetic Energy: KE = ½mv2
Potential Energy (gravity): PE = mgh
a. Space/orbiting: U = -GMm/R
Potential Energy (spring): PE = ½kx2
Power – measured in watts where 1 Watt = 1 J/s
a. P = ΔW / Δt = change in work per change in time
Waves and Periodic Motion
I.
II.
III.
Period, Frequency, Amplitude
a. Period: Symbol T, measured in s. A period is the time it takes to complete one full
cycle
b. Frequency: Symbol f, measured in Hz. A Hz is 1/T and is the number of vibrations per
second
i. Angular frequency ω = 2πf (or v/r). tells you the rate in radians per second
c. Amplitude: symbol A, measured in m. Magnitude of the displacement from
equilibrium. Measure of the total energy of the system
i. Since A measures energy, Intensity is related to A. Note that energy of the
whole wave increases with A, but energy per photon is dictated by f.
d. Concept: if a system is losing energy, that means the amplitude is decreasing, but
the period and frequency remain the same. Think about a pendulum that slows
down. It travels less far, but always the same period.
Springs
a. F = -kx
b. U = ½kx2
c. Concept: At max amplitude, Force (kx) is at maximum so acceleration is at
maximum. Also, potential energy is at maximum. At x = 0, KE and velocity is
maximum but PE and acceleration are 0.
d. For simple harmonic motion ->
i. T = 2π √(m/k)
ii. f = 1/T = (1/2π)* √(k/m)
iii. Note that the frequency and period do not depend on A (or x). That means
that if you pull a spring 1cm vs 10cm, the period and frequency will be the
same.
Pendulums
a. Two forces acting on the pendulum -> The Tension and gravity
b. Concepts: Tension is highest at the bottom (think of a swing) because mg points
directly down/opposite tension instead of at an angle
c.
Note that the
restoring force is F = -mgsinϴ
IV.
d.
Wave Properties
a. Transverse wave: direction of propagation of wave is perpendicular to the wave’s
motion. Any single point on the wave is moving up and down, but the wave
propagates to the right. Think of a rope that’s fixed. If you wiggle up and down, the
wave travels down the rope but at any point on the rope it is moving up and down.
i. Important types: Light, emf, string
b. Longitudinal wave: direction of propagation of wave is parallel to the vibrations.
Think of a slinky – if you pull a slinky, any point on the slinky is moving in the same
back and forth direction as the propagation of the wave.
i. Important types: sound, pressure, earthquakes
c. v = fλ
i. Concept: Speed of wave is determined by medium, not frequency. This is
why when you change f, you change λ, but not v. Think about sound – speed
of sound isn’t faster for 20Hz for 20,000Hz.
ii. Concept: When a wave hits a new medium (only thing that affects its
speed), the speed is affected but not the frequency.
iii. v on a string: v = √(Tension/linear density). Probably won’t need to know,
but you can see how the properties affect wave speed.
d. Interference
i. Waves can interfere. When peaks meet peaks, you get constructive
interference. You just add the waves at any given position. Full constructive
is when they are completely in phase. Full destructive is when they are
completely out of phase (180 or pi out of phase).
ii. When frequencies are not identical, they will not fully interfere. But there
will be regions where they have troughs and peaks together. The troughs
are soft and form beats. This is called beat frequency and happens when
you tune something. fbeat = |f2 – f1|
e. Standing waves
i. Standing waves look like they’re standing still and have fixed troughs and
peaks.
ii. Nodes are the A = 0, antinodes are the A = max.
f. Resonance – at specific frequencies (dictated by the object’s natural frequency) if
you apply a vibration to it you can make it oscillate with greater amplitude at those
frequencies. You can impart a lot of vibrational energy (enough to shatter a wine
glass, or heat up a carbonyl bond to show on an IR)
g. Harmonics of standing waves
i. Know first/fundamental harmonic, second harmonic/1st overtone, etc
ii.
Fixed on both ends so you have two nodes. Fundamental harmonic is half of
a full wavelength, so L = ½λ or λ = 2L. Second harmonic is a full one so L = λ.
[Will be the same as both ends open, just with nodes and antinodes
reversed]
iii.
fixed on one end so first harmonic is L = ¼λ, or λ = 4L. Second one is L = ¾λ
(notice it’s never a full wavelength from peak to peak)
Sound
I.
II.
III.
IV.
Properties of sound
a. Sound is produced by vibrations in a medium
b. Sound waves are longitudinal
c. Sound waves must have a medium (cannot be created in or travel through a
vacuum)
d. Vibrations whose frequency is too low is infrasound (below 20Hz), too high is
ultrasound (above 20,000Hz)
Speed of sound
a. Speed of sound is dependent on the medium it travels in
b. Media with greater intermolecular forces have greater restoring force so vibrations
can more rapidly compress for a new wave. As a result vsolid > vliquid > vgas
i. Think of someone banging on a hammer on a train track. If you ear is to the
track, you hear the sound faster than if you are listening from the air
c. Speed of sound greater in stiff than compressible objects
d. **Speed of sound is faster in less dense medium, slower in denser media. Gases are
less dense than solids, but they are so much more compressible that sound travels
slower
e. Speed of sound in hotter objects > colder objects
i. Speed of sound at 0oC = 331m/s.
ii. Speed of sound at 20oC = 340m/s.
iii. Relates to speed of sound based on density
Intensity of sound
a. Intensity of sound is how we distinguish loudness. Since it is a measure of energy, it
is affected by the Amplitude of the wave
b. Intensity = energy per area and time. I = Power / Area or Watts per m2.
c. I = P/A, measured in W/m2
d. We measure sound levels in decibels (β), which is just the log form of Intensity.
e. β = 10 log I/Io
i. I0 is 10-12 W/m2 and is the minimum intensity of sound that can be heard.
ii. I0 would have an intensity of 0 dB (Intensity = 10-12 W/m2)
iii. 100* I0 would be a whisper. 10log 100 = 10*2 = 20dB
iv. Every factor of 10 times the intensity of sound, the decibel raises by 10dB.
So 10,000 times louder, or 104 times louder, you have an increase of 40dB.
f. Intensity of sound is inversely proportional to r2, so it drops off pretty fast at larger
distances. That’s because the area the wave travels through is so much larger and
Intensity is inversely proportional to Area.
g. Attenuation is the gradual loss of intensity as it moves through a medium
i. Attenuation is greatest for soft, elastic, viscous, less dense
Pitch
a. Pitch is the human perception of the frequency of sound.
V.
b. Higher frequency = lower wavelength = higher pitch (measured in Hz)
c. Humans can hear between 20Hz and 20,000Hz
Doppler Effect
a. Movement of the speed and listener will impact perception of frequency.
Movement toward each other will always result in higher perceived frequency
b. Moving Observer -> Top formula. You move toward the sound at speed vobserver,
while sound wave is traveling toward you at speed vsource so you perceive the
wavelengths to be shorter.
i. fo = fs (v + vo)/v
c. Moving source -> bottom formula. Even though the waves are the same frequency
apart, because the source is moving toward you, when they get to you the starting
point has been getting closer and closer so they appear to be shorter distances
apart.
i. fo = fs v/(v – vs)
d. Reverse the signs if they are moving away from each other.
i. If only one is moving, towards = higher f, away = lower f
e.
VI.
TOP OBSERVER MOVES, BOTTOM SOURCE MOVES [o comes
before s on the alphabet so it is on top]
f. Red shift (universe is expanding) – because the source of light is moving away from
the observer, the frequency is shifted down (longer wavelength) so we see it light
shift red toward the upper end of the visible spectrum
Ultrasound
a. Sound waves will diffract out until they hit a surface, at which point they will reflect
b. Source emits ultrasound, which will hit an object and reflect and give information to
the detector. Other waves will go around until they hit another surface.
Fluids
I.
Properties
a. ρ = mass/volume = kg/m3 or g/cm3
b. specific gravity = density compared to water
i. ρwater = 1kg/L or 1g/mL
ii. specific gravity is useful because it tells you how much of something will be
submerged in water (ice has a specific gravity of .917 so 91.7% of ice is
submerged in water)
c. Pressure = Force/Area. Pressure = N/m2 = 1 Pa (pascal).
i. 1 atm = 100kPa
ii. Force: F = mg; Density ρ = m/V
1. F = ρVg
2. Pressure = F/A = ρVg/A; V/A = length (or depth in a fluid) so the
pressure a liquid exerts on an object is Pgauge = ρgD
iii. Gauge pressure vs total pressure
1. Gauge pressure will tell you the pressure of the liquid (which is ρgD)
2. Ptotal = P(system) + Pgauge
a. If the system is exposed to air (like a sea diver) then the
total Ptotal = Patm + Pgauge
b. If the system is closed, the pressure of the gas above the
liquid provides the P(system) which could be 0 if there is no air
above the tank or if there is a vacuum.
iv. Note that pressure for an object submerged in a liquid depend only on the
density of the liquid (ρliquid) and depth – it doesn’t matter what shape the
container is or how much total water is above your head.
d. Surface tension – attraction between molecules of a fluid/solvent create surface
tension. Because the molecules are pulling on each other, higher surface tension
means lower surface area.
II.
Buoyancy
a. Archimedes principle – “the magnitude of the buoyant force is equal to the weight
of the fluid displaced by the object”
i. In other words, the buoyant force is given by the weight of the liquid that is
displaced, not the weight of the object.
ii. FB = ρliquidVsubmergedg
iii. For Buoyancy, remember that mg (or ρobjectVobject) is your downward force
while ρliqVsubdg = is your upward force. If mg > ρliqVsubdg then the object will
sink.
iv. Apparent weight: weight – FB
v. Float: FB = weight
Sink: weight > FB
Rise: FB > weight
III.
Hydrostatic Pressure
a. Pascal’s Law: Pressure in = Pressure out
b. Since the pressure exerted on the liquid becomes the pressure on the other end,
then F1 = A1/A2 * F2. If A1 < A2 then F2 > F1. Or in other words, a small input force on a
smaller area creates a higher output force on the place with the larger area
c. However, because the total work ends up being the same (F1d1 = F2d2) then d2 < d1 .
That means that you raise the side with the greater area much less (the total volume
moved is the same)
IV.
Hydrodynamics
a. Flow rate = the volume of liquid that passes a particular point per unit time
i. Know the difference between flow rate and flow speed. Flow speed = how
fast the water moves out of the hose when it’s open (slower) vs pinched
(faster) but the flow rate is the same (liquid output per second)
ii. Expressed in m3/s
iii. Flow rate can be calculated by knowing how fast a liquid is travelling and
the cross sectional area of the point it passes through, or
f = Av
iv. Since the flow rate cannot change (the water in a pipe isn’t stopping for the
water in front that’s going through a smaller opening) so a smaller opening
means a faster flow: A1v1 = A2v2
b. Bernoulli’s Equation
i. Bernoulli’s equation describes the conservation of mechanical energy for a
flowing liquid
ii. Bernoulli’s equation only applies to ideal fluid flow! There are four
requirements:
1. Fluid is incompressible
2. Negligible viscosity
3. Laminar (non-turbulent) flow
4. Steady flow rate
iii. Equation: P + ½ρv2 + ρgh = constant or P1 + ½ρv12 + ρgh1 = P2 + ½ρv22 + ρgh2
iv. Essentially KE + PE = KE + PE where gravity is providing the potential energy
and the flow is providing the kinetic energy
c. Bernoulli/Venturi Effect
i. By the Bernoulli equation, if h1 = h2, then the side which is faster much have
lower pressure. Thus, if a fluid has a faster flow speed v, then the pressure is
lower. Think of a shower, when you turn on the water (and the air starts
moving faster) then the pressure drops so the curtain pulls into the shower.
d. Viscosity and Poiseuille’s Principle
i. Viscosity is a measure of the friction within fluids. A highly viscous liquid
(like honey) would oppose the flow of the liquid. Viscosity is higher for
colder fluids
ii. Poiseulle’s Principle – flow rate is affected by viscosity, length of tube, and
radius of tube.
iii. Important takeaway – know that
Q (flow rate) ∝ r4
Q ∝ 1/L
Q ∝ 1/viscosity
And especially for blood flow, that r = radius of blood vessel and L = length
of tract. Narrow vessels mean much less blood flow
REMEMBER – a larger radius gives a much larger flow rate, but much smaller
flow speed. If r goes to ½r, then flow speed increases by 4 but flow rate
decreases by 16 times.
e. Turbulent vs Laminar flow
i. Laminar flow is streamlined flow while turbulent is the chaotic flow
ii. Reynolds number is used to predict turbulent flow
iii. Think of white river rafting for turbulent flow – what causes turbulent flow?
1. Obstruction (rocks/plaques in arteries)
2. Flow speed (faster flow increases the likelihood of turbulence)
a. Av = Av so if you decrease the area, you will increase the
flow speed. So decreased area also increases turbulence
3. Decreased viscosity (honey will have a more laminar flow than
water if at the same speed)
V.
Solids
a. When subjected to various forces, solids can also change shape
b. Stretching forces, Compressing Forces, Bending Forces – these forces can strain an
object until it reaches an elastic limit, at which point it gets permanently deformed.
c. Stress: pressure exerted on an object, which is given by σ = F/A
d. Strain: the change as a result of the stress ε = ΔL/L0
e. Shear: bending force that
f. Young’s modulus is a constant like the spring constant for an object until it
permanently deforms, at which point it no longer applies. Y = Stress/Strain
g. Stress = Modulus * Strain (or shear)
h. F/A = Y * ΔL/L0
i. Think of it in terms of Hooke’s Law. F = kx, F = Y(A/L0)ΔL. Since Y is constant,
Area is constant, L0 is constant
i.
Thermal expansion
i. Things expand when it’s hot and shrink when it’s cold
ii. ΔL = αL0 ΔT where alpha = coefficient of linear expansion (constant)
Electricity and Magnetism
I.
II.
Electric Force
a. Electric Force: F = kq1q2/r2
i. k = 9 x 109 Nm2/C2
ii. Force is attractive if charges are opposite, repulsive if charges are same
b. q – charge is quantized and comes in discrete packets
i. elementary unit of charge (of one proton or electron) = 1.6 x 10-19 C
c. Conductors vs. Insulators – electrons are free to move in conductors while do not
generally flow in insulators. Conductors – metals. Insulators – nonmetals like glass or
plastic
Electric Fields
a. Electric fields are lines that help you visualize what a charge would feel from another
charge. All charge sources will create electric fields. These field lines are vectors that
show you what would happen if you put an imaginary positive test charge in the field.
Closer lines denote a strong field while farther lines show weakening of the field
b. F = qE or E = F/q
i. E = kQ/r2 where Q = source charge
III.
c.
Electric Potential
a. Source charges exert a force on nearby charges. So if you have a source charge that has
electric field pointing outward, and you place a positive charge near it, it will push that
charge. There is a potential energy much like gravity, and an object will move from
higher potential energy to lower potential energy (and pick up kinetic energy)
b. Electric potential: V = kQ/r
i. This is described as the voltage, or the electric potential between two points.
Charges will flow from higher potential to lower potential (or higher voltage to
lower voltage). Volts are given in J/C
c. Electric Potential energy: ΔPE = qΔV
d. V = Ed
e. A dipole in an electric field will align itself with the field (MRI)
IV.
V.
i.
Induction
a.
Gauss’s law – just understand that a point charge will send out electric field lines. If you
choose an arbitrary surface area around that charge and measure the electric field lines
coming out of it (the flux), you can determine the amount of charge in that space.
Magnetism
I.
Properties
a. Magnetic field given by B and measured in Tesla (T)
b. Magnetic field lines go from North Pole to South Pole
c. F = qvB sinϴ = ILB sinϴ
d. The force is always perpendicular to the magnetic field and the velocity
e. For a positive charge, use the right hand rule. Thumb is the velocity of the positive
charge, index is the direction of the magnetic field, rest are in direction of force.
f. If charge in wire, the direction of current is the thumb and the electric field wraps
around it in a circle like your fingers in your right hand.
i. qv becomes IL (or current times length of wire) because I = q/t and v = L/t
g. If charge in a solenoid, then your wrapping fingers are the current and the thumb is the
B field
II.
III.
Practical
a. Velocity selector – if you pass a charged particle through an electric field capacitor, then
the particle (let’s say positively charged) will move in the direction of the electric field.
You can then set up a magnetic field such that the force pushes the opposite way so the
charge is deflected by the magnetic field as well. Since F = qvB = qE, if the forces are
equal, the particle isn’t deflected at all and goes straight through the middle. So you can
select for v = E/B
b. A mass spec first uses a velocity selector so that you know all the particles entering the
mass spec have the same velocity. Since that is constant, then you can deflect them in a
third chamber. If qvB = mv2/r and the initial velocity of these was constant, then a bigger
m means a bigger r and you can separate particles by their mass based on how big the r
was in their deflection path.
Flux
Circuits
I.
II.
III.
Current
a. I = ΔQ/ Δt or amount of charge flowing through a cross section of a conductor
b. Conventions: would be direction of positive flow
i. Even though the actual charges that move are electrons
c. Units: Ampere (A) or C/s
Voltage/emf
a. Not really force, but a potential difference with the units of voltage
b. Positive charges flow from high potential (higher voltage) to lower potential (lower
voltage)
c. Batteries are a source of emf. They have a positive cathode and negative anode (by
convention – this is because positive charge flows from positive to negative while
negative charge flows from the anode to cathode. Because oxidation occurs at the
anode (loss of electrons) it actually gets positively charged and will attract anions – think
of a gel)
d. Batteries have internal resistance such that the true potential difference in a battery is
emf – voltage drop due to internal resistance. Together this is called terminal potential
(potential between the terminals)
Resistance
a. V=IR
b. R = ρL/A
i. ρ is the resistivity of a material, which is a constant based on the material. A
bigger cross sectional area will let me charge through, so it reduces resistance. A
longer wire will be more material to traverse so it increases resistivity
c. Power is dissipated by a resistor and given off as heat
d. Circuit problems
i. RULES:
1. Junctions – The sum of all currents entering a junction must equal the
sum of all currents leaving a junction (currents don’t change unless they
hit a junction)
2. Loops – the sum of potential changes around any closed path must be 0.
Emf raises the potential up and as it goes around the circuit it must end
at 0.
e. Resistors in series
i. RT = R1 + R2 +… Resistors in a series add up.
1. Series increase resistance
2. Larger resistances = larger total resistance
ii. All resistors in a series share same current (no junction to split current)
iii. Each resistor dissipates voltage. Since I doesn’t change across resistors, the
voltages must be different
f. Resistors in Parallel
i. 1/RT = 1/R1 + 1/R2 …
1. Parallels decrease resistance
2. Larger resistance on one resistor = smaller total resistance
ii. Junction – IT splits into I1 + I2 + … and then reconverges.
iii. Since V = IR and I changes across resistors, V remains the same. Each closed loop
adds up to 0, so whichever path you took through each resistor must have
dissipated the whole emf.
iv. Current is split according to resistance – bigger resistance means lower current
(think of a branching river)
IV.
Power
a. Power measured in Watts (W)
b. 1 W = 1 J/s so it is a measure of how much work or heat per unit time
c. P = IV = I2R
V.
Capacitance
a. Conceptually: When you connect two plates to battery and let them build up charge,
then there will be an interface where one side a has a ton of positive charge and one
side has a ton of negative charge. If the space between the plates doesn’t let charge
jump to dissipate the difference, you have a capacitor. See the way the charges move to
build up the difference. The purpose of a capacitor is to store charge that you can
discharge in a rapid controlled manner, much faster than the original battery would
discharge.
Capacitors also don’t actually store charge, they store an imbalance of charge. They
keep storing more and more charge until the repulsive forces of the electrons on one
plate no longer allow for any more induced charge of more electron buildup, and that’s
the total capacitance.
b. Note that it would also create a uniform electric field between the plates!
c. Capacitance is measured in Farads (F) where 1F = 1C/V. Since 1C is a huge unit of charge,
1 F is huge. Q = CV
d. Remember that PE = qV for an electric field (qEd). If you are measuring the energy
stored in a capacitor, it’s q*Vavg. Since the capacitor goes from 0 – V, Vavg = Vf – Vi / 2 and
that’s ½V. Combine all of that so PE = QVavg = CV*½V or ½CV2. PE = ½CV2
e. Capacitance of a capacitor without charge C = εA/d
i. Epsilon is the permittivity of free space constant and is absolute, which means
that any conducting metal plates (regardless of material) will have a capacitance
that is directly proportional to Area of the plates (how much charge buildup)
and the inverse of distance (if they are close, the capacitance is better)
f. Dielectrics
i. The medium between the plates that insulates the plates from each other also
directly impacts the capacitance. This nonconducting material is called the
dielectric (such as air)
ii. Cf = kC0; where C0 could be vacuum and Cf be in air, or Cf could be the
capacitance after you inserted a plastic box between the two plates and C0
would be before. K is always positive as the dielectric always increases the
capacitance because the electric field induces a dipole in the dielectric that in
turn increases the amount of charge stored on the plates
iii. Dielectric in action: a defib is a battery hooked up to a capacitor. Once you need
the defib, you let the battery store charge on the capacitor until it is charged,
and then you can discharge it quickly in a burst!
g. Circuit problems
i. Capacitors in series: If you think about it, the battery is going to induce the same
amount of charge on the far left as it is on the far right (the only two it’s directly
acting on). Then those plates will induce the opposite plate to have its charge
separation and down the line. Since the same amount of charge is going to the
left and to the right, the Q for the whole system is the same and the voltage
across each capacitor is split and given by Q/C.
ii. Capacitors in parallel: Remember that for capacitors in parallel, the voltage drop
from any closed loop must equal to 0. Since you can draw separate closed loops
through each capacitor, they must all drop the whole voltage amount
individually. That means that V stays the same (the V of the battery) for all of
them, but the amount of built up charge each has is different based on its
capacitance.
VI.
h. Charge/Discharge of a capacitor
i. When the circuit is first closed, you have electrons flowing from the negative
anode of the battery to the negative plate. This will repel electrons from the
positive plate to move toward the positive cathode of the battery, inducing the
charge buildup on the capacitor. It’s easier at first to build up charge but soon
the negative buildup resists more electrons flowing toward the negative plate
and the movement of electrons slows down.
Using appropriate conventions, it means the positive current going from the
positive battery cathode to the positive plate is fast at first, and ultimately slows
down as it is harder to buildup charge. It experiences exponential decay.
Voltage or charge (however you want to think about it, since building up the
charge induces the voltage potential across the plates) climbs quickly at first and
builds up, but once flow is opposed, it slows down until it hits the max voltage.
ii. When the circuit is open to the resistor that is going to be using the stored
charge of the capacitor, you have a large initial burst of energy as the electrons
rush to the opposite side plate (or current rushes to the negative plate) through
the resistor, but soon the attractive force is reduced because the total charge on
the plates is reduced, so it also decays exponentially.
iii. Current goes through the path of least resistance first, so it will preferentially go
to the capacitor first rather than the resistors. The current will drop off quickly
to the capacitor as it charges and then start going to the resistors. Voltage
across the capacitor increases gradually as charge is built up in the capacitor.
Conductivity
a. Conductivity is the inverse of resistivity. An object or materials conductivity is high when
there is low resistivity (and thus increases with higher area and shorter length).
b. Conductivity is driven by electrolyte concentration. There is an optimal point where ions
are mobile and there are enough charges to elicit conductivity. Too few electrolytes
means low conductivity, too many electrolytes means too much crowding and low
movement
c. Temperature control
VII.
VIII.
i. Metals – conductivity decreases as temperature increases (cold metals are
better conductors)
ii. Semiconductors – conductivity increases as temperature increases
iii. Superconductivity – some materials have superconductivity at incredibly low
temperatures and have no resistance (current loops forever)
iv. In a solution, a capacitor will discharge because the solution or medium is
conducting some charge. You can measure the conductivity of a solution this
way.
AC vs. DC
a. Every formula prior was for a direct current. When you alternate a current by constantly
switching the direction of the flow, you create alternating current.
b. You need to use root mean squared formulas for current and voltage
i. Irms = I / √2 and Vrms = V / √2 = .7
Ammeter and Voltmeter
a. Ammeter measures the current. Objects in a series experience the same current (I isn’t
split). So you plug it in a series and set the resistance insane low so it doesn’t affect the
resistance of the entire circuit (and thus the current).
b. Voltmeter measures the voltage drop. Resistors drop the voltage (closed loop drops
voltage down to 0) so you don’t want to connect it in a series. You want to connect it in
parallel, since parallel resistors do not drop the voltage, just split the current. So you
connect it in parallel, but since you don’t want much current to be diverted through it,
you put it at very high resistance. This way, the 1/R value is so low it doesn’t change the
resistance of the whole circuit
Light
I.
Electromagnetic Spectrum
a. Changing electric field induces a changing magnetic field which induces a changing
electric field etc… They are always perpendicular to each other and then perpendicular
to the direction of propagation. These together create light
b. Light exists in a wave particle duality so it can act like a quantized photon and also like a
wave
c.
d. Formulas
i. v = fλ
ii. c = 3x108 m/s
iii. E = hf
II.
Polarization
a. Light from sources have waves in all directions that is not polarized
i.
b. Polarizers are filters that only allow beams that are oriented in one direction
i.
c. Based on the angle and material, if light is reflected off something, it becomes polarized.
That’s why polarized sunglasses are good at reducing glare
III.
IV.
Young’s double slit
a. Note that with these types of experiments, you need to make sure you have
monochromatic light and high coherence or else it will be smudgier for the interference
pattern
b. dsinϴ = mλ for constructive points, and for destructive points it’s m/2
c. Remember that you get constructive interference when waves have the same
wavelength (they add up!) but if they have the same wavelength but are not aligned you
can get variations and the extreme case is destructive interference.
i. For the experiment, you use a single light source through a double slit to ensure
that you have the SAME amplitude and wavelength!
d. If the wavelengths and amplitudes are the same and you shift over half a wavelength,
the nodes all line up, but the antinodes all line up oppositely and completely destroy the
wave.
e. For young’s double slit, if you think about it in terms of interference, if the distance
traveled for the light is the same, you will have constructive interference, but if the
distance traveled for the wave is off by λ/2, then they are destructive and you get a dark
spot.
f. If you have the double slit, the spot in between the double slit is equidistant from both
slits so it is going to be a bright spot. The next bright spot over is when they’re in cycle
again, which means that it’s where the wavelength traveled one full wavelength over, so
the distance the light travels is + λ.
i. Keep in mind that λ is not the distance between the spots. It’s the distance
difference for the two waves that had to travel from the slit to the wall.
g.
or y = λL/d for small angles
Single slit interference
a.
b. wsinϴ = mλ but instead of for the constructive points, for single slit it’s for the
destructive points.
i. w/2 is half the slit, and a point from w/2 will interfere destructively with a wave
that λ/2 and that wave interfere destructively (since it travels half a wavelength
more).
c. The various constructive waves don’t really happen at the same place. Only thing you
can know is the width of the middle wide smudge (or other smudges since you can only
find the location of the destructive points)
V.
VI.
Thin film interference
a. With a thin film, it’s like having 3 different media (think air, then thin film of oil, then
water). The thin film of oil on top will have some light reflecting off of it, while other
angles of light will go through refracted, hit the third boundary, and then get reflected
off of that third boundary and come back out of the oil back into the air. Those different
waves that are “coming off the oil” might interfere with each other. You see bright spots
where they are constructive and dark rings where they are destructive
b. This is why you see rainbow rings on pavement after rain.
c. If the distance the light traverses in the middle medium = n λ, then you get constructive,
but if it travels n λ/2, then you get a destructive dark spot.
d. Every time there is a reflection, there may be a pi shift such that the wavelength gets
reversed (think of noise cancelling headphones where they switch the direction of the
wave so it creates the opposite wave). This means that unlike with slit interference, the
new wave may not necessarily start with the same peak/valley. Pi shifts occur when
light crosses into a medium where it would travel slower. So if light goes through oil
first, the first gets pi shifted when it is reflected, but the next goes through water and is
NOT pi shifted because light travels faster in water than oil, it does not get pi shifted
when it is reflected. So the two reflected waves are out of phase, so if the distance is an
integer of λ, they would be destructive while λ/2 would be constructive.
Diffraction grating
a.
b. You get perfect overlap with a diffraction grating because each hole down meets the
bright spot at exactly one wavelength more than the last
c. The path length difference for any spot in the pattern that is dark is because all of the
waves add up perfectly destructively at every point. For example, first wavelength
travels 1.1 λ further, then the next travels 2.2 λ further, then the third travels 3.3 λ
further. When they all add up, the .1, .2, .3, .4 etc all end up adding up destructively so
you get a clean dark spot.
d. The bright spots are further than in the case for double slit with the same d because
with all those waves it’s rarer to get a perfectly constructive point
Optics
I.
II.
III.
Reflection
a. Reflection occurs when a light wave strikes a boundary between two media. There will
be some amount of light that is reflected from the plane surface such that the angle of
incidence equals the angle of reflection. Note that the angle of incidence and reflection
are measured from perpendicular to the plane, not to the plane!
i.
b. Mirrors are a special case of reflection where all light is reflected from every angle
Refraction
a. When electromagnetic radiation goes from one medium to another, it is slowed down
(only light in a vacuum travels at c)
b. The index of refraction is given by n where n = c/v and is always greater than 1.
c. Recall that the frequency of a wave does not change when it enters a new medium. The
velocity changes here, and so does the wavelength, but not the frequency
Snell’s law
a. When light strikes the boundary of the medium at an angle, it will be refracted into the
new medium with a different angle. The relationship is governed by Snell’s Law
i. n1 sinϴ1 = n2sinϴ2
ii.
b. Apparent depth – thought exercise
IV.
V.
Dispersion
a. White light through a prism will split the light into a rainbow because each index of
refraction varies by wavelength.
i. If v = fλ such that v is directly proportional to λ, and n = c/v such that v is
inversely proportional to n, then n and λ are inversely proportional. That means
that a bigger λ means a smaller n and a smaller λ is a bigger n. So red has a
smaller n and is refracted less.
Total internal reflection
a. When you go from a medium of higher index of refraction to a lower index of refraction
(such as from water into air), there exists a critical angle where the refracted ray is
incident at 90o and does not appear to leave the water. Any greater angle of incidence
would be reflected internally instead of refracted.
i. If you think about it, the refracted angle cannot be greater than 90 or it will just
go back into the water.
ii.
iii. Remember that Total Internal Reflection can only occur when you go from a
greater index of refraction to a lower one.
1. Critical angle: n1sinϴc = n2sin90o
2. Once you have the critical angle, anything greater than ϴc gets reflected
at the same angle as the incident angle
VI.
General Optics Rules
a. Real vs Virtual image
i. For mirrors, same side as object is real, opposite side is virtual. For lenses,
opposite side is real and same side is virtual.
VII.
ii. UV-IR = upright virtual, inverted real
b. 1/f = 1/o + 1/i
c. m = -i/o
d. Conventions: if m>0, upright, if m<0 inverted
e. If |m|>1 enlarged, if |m|<1 reduce
Mirrors
a. Plane mirrors reflect perfectly, such that i = -o. There is no focal point, the image is
always upright, virtual, and the same size. This is because m = -i/o = 1 (same size,
upright) and since you see the image on the opposite side of the mirror but the light
rays are reflecting back toward you, the image is virtual.
b. Spherical mirrors – there are two types of spherical mirrors you need to know.
i. Concave mirrors – these mirrors are concave and also converging since they
converge a beam. The focal length is always positive since it’s always on the
same side as the object.
ii. Convex mirrors – these mirrors point light rays away and thus are diverging
mirrors. Focal length is always negative as it is always on the opposite side of
the object (or appears to be). These images are ALWAYS upright and virtual and
cannot be projected onto a second image.
iii. f = R/2
iv. For concave mirrors, image depends on location of object. Using 1/f = 1/o + 1/i
in 5 cases: If you had an image at twice the focal length, then 1/f = 1/2f + x, so x
must equal 1/2f (because 2/2f = 1/f). so 1/f = 1/R + 1/R
1. Object beyond R. If object is beyond R, then 1/o got smaller so 1/i must
get bigger. That must mean that i must get smaller (but 1/i still has to be
smaller than 1/f so i>f)
a. Real, Inverted, Smaller; i between f and R
2. Object at R. If object is at R, then i must also be R. Since m = -i/o, the
image must be inverted, real, and same size.
3. Object between R and f. If o gets smaller, then i/o got bigger, which
means that 1/i needs to get smaller. That means that i gets bigger.
a. Real (-i/o is negative), Inverted (-i/o is negative), enlarged and
farther (i>o)
4. Object at f. 1/f = 1/f + 1/i. for this to be true i has to equal infinity, so
the rays converge at infinity/never converge. No image formed.
5. Object inside f. Since o is smaller than f, 1/o is bigger than 1/f, which
means that 1/i must be negative and subtracted from 1/o. If i is
negative, the image forms on the right of the mirror.
a. Virtual (-i/o is positive), upright
c. Lenses
i. Convention is different for lenses. A convex lens will create a converging lens
while a concave lens will create a diverging lens
ii. For lenses, the light goes through the lens instead of being reflected, so the light
rays actually pass through. That means real images are on the opposite side of a
lens and virtual images are on the same side as the object.
iii. A convex/converging lens will have a positive f and concave/diverging lens will
have a negative f. Since we take positive for i to now be on the other side from
the object, we keep m=-i/o. If i is positive, it is real, if i is negative, it is virtual.
That means that a negative magnification still means real.
d. Multi-lens problems
i. Only a real image can be the object that the second lens uses as its image.
Generally speaking, the first lens (objective) is going to invert the image
between it and the second lens (eyepiece) and the eyepiece is going to invert it
again and make it upright.
e. Lens/ray diagrams –
i. Parallel ray goes hits the mirror and reflects/refracts through the focal point
ii. Center is always reflected at same incident angle in a mirror and goes straight
through unimpeded for a lens
iii. Ray through the focal point will be reflected/refracted parallel
f. Diopters and Lens aberrations
i. Lens strength/lens power is measured in diopters. P = 1/f and is in diopters.
ii. An eye will focus a real image on the retina (and your brain inverts it from its
inverted state)
iii. Glasses
1. For near sightedness, the eye lens focuses the light in front of the
retina. To fix this, a diverging or concave lens will diverge the light
before they hit your eye lens, so your eye will converge it properly on
the retina
2. For far sightedness, the eye lens is forming an image behind the retina,
so you want to catch it and converge it (converging, convex lens) before
it hits your eye lens, so your eye converges it on the retina.
iv. Magnifying glass – since p<f (you hold it close to what you are magnifying), you
actually use a converging lens to create the virtual upright image. It is also why
you can converge the sun’s rays to create a small, hot, real light.
v. Aberrations:
1. Spherical aberrations occur when a misshapen lens does not focus all
light properly at the focal point. This is because the lens is not properly
rounded (R changes) The image is clear in the center but distorted
around the edges
2. Astigmatic aberrations occur when the lens is not properly rounded and
is not symmetric on both sides. This causes distortion at all angles
3. Chromatic aberration – since different lights get refracted with different
indices of refraction, you get halos of colors since the different
wavelengths are converging at slightly different foci