Chapter 1 Equations: Translational Motion
SOH CAH TOA Masses
must be in kg’s Gravity
2
= –10m/s
Sin(x) Cos(x) 0° 01 30° 1/2 √3 /2 =
0.9 45° √2 /2 = 0.7 √2 /2 = 0.7 60° √3 /2 =
0.9 1/2 90° 10
Common right triangles: 3,4,5; and 100,87,50 (has a 30° and a 60° angle)
Vector multiplication: Vperpendicular = V1V2sinθ (cross product – two vectors producing a vector – the “sinθ”
makes the two vectors perpendicular prior to multiplication; if two vectors are already perpendicular, sinθ is 1 and
drops out of the equation) Sproduct = V1V2cosθ (dot product – two vectors producing a scalar – the “cosθ” makes
the two vectors parallel prior to multiplication; if two vectors are already parallel, cosθ is 1 and drops out of the
equation)
Speed = distance/time Vavg = Δx/time = (Vf +
Vo)/2 Displacement = Xf – Xo = Δx = Vavg *
time
Linear Motion Equations (Requires a constant Acceleration to use these, such as gravity):
2
(1) Δx = Vot + 1/2 aconst t
(2) Vf = Vo + aconst t ; aconst = ΔV/t ; Time = ΔV/aconst
2 2
(3) Vf = Vo + 2aconstΔx
Notice: Mass is not involved; projectile motion is mass independent. The force of gravity increases to
2
accelerate larger masses to the same extent as smaller masses. [ Fg = (G*M1*M2) / r ]
Where “x” is displacement in 1 direction
Chapter 2 Equations: Force
Note: masses must be in kg’s
2
Σ F = ma Note that “weight” is a force, = mg Fg = G m1*m2 / r = mg
3 2 2
(G = 6.67E-11 m /kg*s and g = –10m/s )
fs ≤ µs*FN
fk = µk*FN
2
Centripetal acentrip = V /r
2
Fcentrip = m acentrip = mV /r
2
Note that if Fg is the Fcentrip, then mg = mV /r, which means Vel=√(g*r) to maintain orbit
Hooke’s law for spings/deformable objects F
= k Δx
Chapter 3 Equations: Torque and Energy
Equilibrium is where Vel is constant => a = 0 => ΣF = 0
Torque = τ = F r sinθ = F l where l is the lever arm (which is r sinθ), that is, l is always perpendicular to F (note this
is a cross product)
Work/Energy: Note that 1Joule = 1N*1m
2 2
= 1kg m /s 1Watt = 1Joule/1s
Note: Energy, Work and Power are all scalars
Energy equations are best used to describe how much potential energy has been converted into kinetic energy.
2
K = 1/2mV Gravitational potential energy = Ug = mgh = Fg*h = (G
2 2
m1*m2 / r ) * r Elastic potential energy = Ue = 1/2k Δx
W = F*distance (where Force and distance are aligned vectors => a dot product that gives a scalar) W = ΔK + ΔU + ΔE internal
ΔEinternal = W + q (q is heat); Note: Δtemperature ≠ heat
(these last two equations say energy only comes in or out of a system by work or heat) P = ΔE / t
= Work+q / t P = F*vel (where Force and velocity are aligned vectors => a dot product that gives
a scalar)
Chapter 4 Equations: Momentum, Machines, and Radioactive decay
Momentum – momentum is always always always conserved (even when kinetic energy is not): p =
mV
Elastic collisions have no deformation, and so they conserve kinetic energy as well as momentum. Inelastic
collisions have deformation, and so they do not conserve kinetic energy but still conserve momentum. Note: for an
object to bounce backwards, it must have a mass less than the object it strikes.
Impulse J = Δp = Δm*V = Favg Δt Concept: the same change in momentum can occur with reduced force
(less traumatic), if the change in momentum occurs over a longer period of time Δt.
Simple Machines – ramp, lever, pulley All based on W=F*d, where distance is increased so
that the exerted force can be reduced.
m
Radioactive Decay – just straight memorization (mid-yield material) a X, where a is the atomic # and
m is the mass #. Alpha decay, Beta decay (3 subtypes including classic beta decay, positron emission,
and electron capture), and Gamma ray (mass is destroyed releasing internal energy)
2
Rest Mass Energy E = mc where m is the mass defect (mass created or destroyed, and E is nucleus
binding energy)
Chapter 5 Equations: Fluids and Solids
ρ = m/Volume Memorize: ρwater =
3 3
1000kg/m = 1g/cm Specific Gravity =
ρsubstance/ρwater
Fluid pressure:
Note, when the term “pressure” is used without further clarification, it means “fluid pressure” Note also that
pressure is analogous to Energy, kinetic energy distributed throughout the total volume (Energy/volume)
Pfluid =the force of molecules against a submerged object (actually exists whether an object is submerged or not) P fluid
2 2
[in Pascals] = F/Area has units of N/m => 1Pascal = 1N/m Pfluid = fluid-weight(aforce)/Area = ρgy (where y is
measured from the top of the liquid down)
Memorize: Patm = 101,000 Pascals = 1 atm Pfluid in a tank that is open to the atmosphere: Pfluid = pgy + Patm Pabsolute =
Pguage + Patm (guage pressure is measured relative to atmospheric pressure, how many Pascals above or below
atmospheric pressure).
Hydraulic Lift (high yield)
W1=W2 => F1*d1 = F2*d2
ΔP1=ΔP2 => F1/A1 = F2/A2
Bouyancy Fbouyant = (ρfluid*Vol)*g = (mass of the fluid)*g Submerged objects: Fbouyant = (mass of displaced
fluid)*g [Fbouyant is less than (mass of object)*g => sinks] Floating objects: Fbouyant = (mass of displaced fluid)*g
= (mass of object)*g [forces cancel]
Fluids in Motion Flux = Q = is constant Q
= A*Vel = volume/t (as A↓, Vel↑)
Mass-Flux = I I = ρ*A*Vel = ρ*volume/t
2
Bernoulli’s Equation (all scalars, no direction) Pfluid + ρgh + 1/2ρVel = K = constant,
meaning conserved energy (=ρgy + Patm) (Ug/vol) (Kinetic energy/vol)
Note that y is measured from the top of fluid down, and h is measured from the bottom of a point of interest up.
2
Written another way: mgy/Vol (i.e. energy of random motion/volume) + mgh/Vol + (1/2mVel )/Vol = K
From Bernoulli’s equation, we can solve for Velocity of fluid coming out of a spigot in a barrel: Vel = √(2gh) Also
to notice from Bernoulli’s equation: As velocity increases (like fluid going through a narrower pipe), the pressure
(i.e. Pfluid) actually decreases.
Non-Ideal Fluids – these have resistance and drag
Laminar Flow concept The narrower a pipe, the
more the drag
ΔPfluid = QR this is Ohm’s law, but with fluids. Ohm’s law is V=IR
4
Resistance = 8ηL/πr (where η is viscosity); note that radius has the biggest effect on resistance.
Cohesive vs. Adhesive Cohesive > Adhesive => convex ∩ (as viewed from
above) (e.g. mercury) Adhesive > Cohesive => concave ∪ (as viewed from
above) (e.g. water)
2
Solids Stress = F/A measured in N/m (not Pascals) Strain = Δx/x0 unitless Modulus = stress/strain = (F/A) /
2
(Δx/x0) units are N/m ; modulus is intrinsic to a material (doesn’t change)
Note: If A↑ 2x, then Δx↓ 2x with the same Force
Thermal Expansion – despite what the Exam Crackers book says, I had MCAT questions on this material ΔL
= L0*α*ΔT ΔV = V0*β*ΔT
Chapter 6 Equations: Waves
λ = wavelenth (in meters) frequency = f = #of waves/sec
(units are 1/sec, not m/sec)
T = period = 1/f
Velocity = λ*f
Very important note: Velocity of a wave is always determined by the medium, and doesn’t change. Only f and λ
change, but the velocity is the same for a given medium. Hence, the speed of light is the same for all λ and f, and
the speed of sound is the same for all λ and f.
Conversion between angular frequency (ω) and frequency (f): ω=2πf
3 Must-Know Concepts 1) Velocity is determined by the medium 2) Velocity is proportional to Tension or
Elasticity of the medium, and inversely proportional to density of the medium.
Vel α (Tension or Elasticity)/density 3) For a gas, velocity of a wave increases with
temperature (faster in hot air than cold air).
Memorize: Sound is faster in water than in air
Intensity of sound is proportional to the square of frequency and the square of amplitude. I α
2 2
f and Ampl
Decibels β = 10 x log10(I / I0) where I0 is the softest perceivable sound
Examples: if I↑ 10x, then that’s +10 dβ; if I↑ 100x, then that’s +20 dβ
fbeat = ⏐⏐1-f2⏐⏐
f for there to be a fbeat f1 and f2 must be close
Pitch is proportional to frequency, not wavelength; high pitch => high frequency
When changing mediums, frequency remains the same, but wavelength changes
Know: how waves reflect (they invert if reflecting off a denser medium)
Standing Waves (Nodes & Antinodes) The equation for a system with two closed nodes (e.g. a closed pipe, a flute, etc.)
or two open nodes (e.g. an open pipe). Note: I would memorize the first two wave-forms for each system (λ1 and λ2, or λ1
and λ3)
The equation for a system with 1 node and 1 antinode (e.g. a pipe with one-side closed and one-side open). Note
that I have rearranged the equations from the book’s form… I think they are much easier to use this way.
Simple Harmonic Motion equations (mid-yield material)
For a mass on a spring: Tspring = 2π√(m/k) fspring = 1/2π *
√(k/m)
For a mass on a pendulum: Tpendulum = 2π√(L/g) fpendulum =
1/2π * √(g/L)
NOTE: that mass does not play a part in a pendulum’s period/frequency
Doppler effect – unlike the book’s suggestion, I find the main equation the best to use, and NOT to use their “simplified” equations.
fsource = fobserved * (Velocity-of-sound +/- vel-of-source
) (Veolicty-of-sound +/-
vel-of-observer)
Velocity-of-sound in air is always = 340m/s If the source is moving towards the observer, use the
minus sign in the top part of the equation. If the observer is moving towards the source, use the plus
sign in the bottom part of the equation. If the source or observer are not moving, don’t add or
subtract anything.
Note: fobserved is always close to fsource, (it never changes drastically)
Chapter 7 Equations: Electricity and Magnetism
If a point-charge is creating E For a constant, uniform E (e.g. between the plates of a capacitor) Fc = k ((q1*q2) /
2 2
r ) [N] Fc = E*q [N] E = k ((q1) / r ) = Fc / q [N/C] E = Voltage / d [V/m] U = k ((q1*q2) / r) [J] U = Fc*d = Eqd =
Voltage*q [J] Voltage = k ((q1) / r) [V = J/C] Voltage = E*d [V = J/C]
Ohm’s Law
ΔV = IR
Kirchoff’s first rule: Flow in = Flow out Kirchoff’s second rule:
Voltage around a circuit sums to zero
Battery = an EMF
Capacitors E is proportional to: Q/A C is
proportional to: K(the dielectric) * A / d
C = Q/Voltage U = ½Q*Voltage =
2 2
½C*Voltage = ½Q /C
“Effective” resistance/capacitance
Rseries = R1 + R2 …
1/Rparallel = 1/R1 + 1/R2 …
1/Cseries = 1/C1 + 1/C2 …
Cparallel = C1 + C2 …
Way to solve circuits 1) combine all resistors 2) calculate current (i) using ohm’s
law and the combined resistance value
Note: Overall resistance is decreased when a resistor is added in parallel Note: At a branch, current (i) is only
the same in each branch if the resistance is the same in each branch
2 2
P = iV = i R = V /R - reflects power lost as heat through a resistance.
AC current imax = √2 *
irms = 1.4*irms
Vmax = √2 * Vrms = 1.4*Vrms Memorize: in
the U.S., Vrms is 120 Volts
Magnetism ΔE creates B, and ΔB creates E Any current (i) creates B –
use the right hand rule; if there is i, there is E
FB = q vel*B sinθ (a cross product) Open hand rule (works for positive charges only, otherwise FB is in the
opposite direction for neg. charges): Thumb in the direction of velocity, open fingers in the direction of the magetic field,
2
your palm is the direction of FB FB also = m*vel /radius-of-curvature (a centripetal force) => radius-of-curvature =
(m*vel) / (q*B)
ΔB creates a current (i) that works to create a B in opposition to the change. Note: whenever ΔB creates E,
some mechanical energy is transferred into internal energy (i.e. heat created)
Chapter 8 Equations: Light and Optics
Visible Light: (UV on this side) 390nm – 700nm (IR on this side) } all less than a micron (1µm) c =
8
speed of light = 3x10 m/s = fλ refractive index = n = c/vel => as n↑, light moves more slowly
Note: n changes with the frequency of the light
Memorize: nair = 1, nwater = 1.3, nglass = 1.5
Refraction Snell’s Law – determines which way the light bends at an
interface nleaving*sinθleaving = nentering*sinθentering
Total internal reflection is when θentering = 90°, so that the equation becomes nleaving*sinθcritical =
-1
nentering*(1), which rearranged is θcritical = sin (small-n / big-n)
Dispersion (through a prism) – think Pink Floyd when you read the word “dispersion”
Longer λ’s (like red) move faster and bend less for any given n > 1.
Diffraction Occurs when light bends around barriers. Can create light and dark spots of
interference.
Mirrors and Lenses (good mid-yield material) Concave Mirrors behave like Convex (i.e. converging) ()
lenses. Convex Mirrors behave like Concave (i.e. diverging) }{ lenses. “Behave like” means with respect
to focal points, and where an object is in relation to the focal point.
A converging lens is any lens that has greatest thickness at the middle vs the edges (a bowed shaped is not
sufficient to call a lens converging or not). A diverging lens is the other way, that is, thickest at the edges.
f = focal point fmirror = 1/(2*radius-of-curvature) flens = is found by using the Lens-maker’s equation… all you
should be aware of about this equation is that flens depends on the n of the lens, the n of the medium its in, and the
radius-of-curvature of both sides of the lens.
Plens = 1/flens [is measured in diopters]
o = object and i = image
Magnification = m = – di / d0 = hi / h0 } the negative sign refers to if the image becomes upright or not (a
positive m means an upright image, and a negative m means an inverted image) 1/f = 1/d o + 1/di To use the
above two equations, you must know if f, do, and di are positive or negative
My 1rule: Assume objects (do) are always positive (in front) can’t get behind unless use multiple
lenses/mirrors; then images (di) and focal points (f) on the left are positive, and on the right are negative,
independent of whether that’s “front” or “behind”. Again, objects (do) are positive when they are in “front”,
even if that means being on the left or right. Images (di) and focal points (f) are positive on the left, and
negative on the right.
Note: The view is someone looking from the left. If a question has the eye on the right, switch the question in your
mind so that the eye is on the left, then follow my rule, then flip it back before answering.
Some common cases: Concave Mirror: o = ∞ then i = +f ; o = +2f = i; o = +f then i = ∞ Convex Mirror: o
= ∞ then i = -f ; as o gets closer the the mirror, i gets even closer on the other side Converging Lens: o =
∞ then i = +f ; o = +2f = i; o = +f then i = ∞ (same as concave mirror) Diverging Lens: o = ∞ then i = -f ;
as o gets closer the the mirror, i gets even closer