Physics

d, vector (Δx=xf-xi)

slope is velocity

v = d/t, vector

slope is accel, area under curve is displacement

a = v/t, vector

area under curve is velocity

g = 10 m/s^2

d = (1/2)(vo + vf)t, missing a
v = vo + at, missing d
d = vot + (1/2)at^2, missing vf
d = vft - (1/2)at^2, missing vo
vf^2 = vo^2 + 2ad, missing t

place the vectors tip to tip, addition of the vectors is the new shortcut vector

break into component x and y then add up separately

vector magnitude = sqrt(x^2 + y^2)

constant a, constant v_x, v_y = 0 at top

break into component v_x and v_y

sin(0) = 0, sin(30) = .5, sin(45) = .7, sin(60) = .86, sin(90) = 1

90 = (0, 1)
60 = (1/2, sqrt(3)/2)
45 = (sqrt(2)/2, sqrt(2)/2)
30 = (sqrt(3)/2, 1/2)
0 = (1,0)

1. no force means no acceleration, concept of inertia
2. F_net = ma, concerned with 1 object
3. objects exert the same force on each other in opposite directions, concerned with 2 objects

units: N = kg*m/s^2

solving force problems:
1. draw force diagram on the object
2. break into x and y components, F_net = ma for both
3. solve equations

kinetic friction- F = u_kN, while moving/slipping
static friction- F < u_sN, either rolling or not moving

F = Gm_1m_2/ r^2

mass is scalar (measures inertia)
weight is vector (measures gravitational force)

turning

F = mv^2/ r

a = v^2/ r

force diagram on the object moving involves centripetal force, usually *normal force* (someone running will need to contact ground at an angle to turn)

at the top of a loop, the absolute minimum velocity would involve no normal force, so centripetal force equals gravitational force

spring:
F = -kx
U = 1/2 k x^2
KE = 1/2 m v^2
angular frequency (w) = sqrt(k/m)
frequency (f) = 1/T = w/2pi

pendulum:
F = -mgsin(theta)
U = mgh
KE = 1/2 m v^2
angular frequency (w) = sqrt(g/L)
frequency (f) = 1/T = w/2pi

max U at ends of swing
max KE at middle of swing

torque = rFsin(theta)

units are N*m

the greater the distance from the pivot point, the greater the torque

constant force acting on object over a distance

W = F d cos(theta)

units: J = N*m

force perpendicular to displacement does no work

W = P dV , area under a PV curve is work done by the system

when volume expands, work is done by the system
when volume compresses, work is done on the system

P = work/time = dKE/time

units: watts = J/s, J = ft*lb

KE = (1/2) m v^2
PE = m g h

energy is conserved, unless outside forces act on the system

W = dKE

vector quantity

p = mv

units: kg*m/s

elastic collision- conservation of momentum, conservation of kinetic energy

inelastic collision- conservation of momentum, but *kinetic energy is lost*

impulse = change in momentum (dp) = Ft

linear expansion: dL = a L dT

volume expansion: dV = b V dT

conduction- direct transfer of energy via molecular collisions

convection- transfer of energy via physical motion of heated gas or liquid

radiation- transfer of energy via EM waves

specific heat: c , q = m c dT
q is heat gained, does not apply during phase change

units of q: J or calories
units of c: J/ g*C

heat of transformation: q = m L
units of L: J/ g

dU or dPE = q - W
potential energy = heat gained - work done by system (by system= -, on system= +)

no heat transfer: dPE = -W
no change in volume: dPE = q
no change in energy: q = W

shear modulus: S = (F/A) / (x/h)
young's modulus: Y = (F/A) / (dL/L)

density: p = m/V
units: kg/m^3

weight of fluid: F_g = pVg
units: N

buoyant force (*equals weight of displaced fluid*)
F_b = p_fluid V_displaced g
units: N

archimedes principle:
object weight in water = object weight in air - weight of water displaced
weight of water displaced = p_fluid V_displaced g
object weight in air = p_object V_displaced g

pressure: P = F/A
gauge pressure: P = pgh
absolute pressure: P = P_atm + pgh
units: pascal (P) = N/m^2

density of object/density of water

used in archimedes principle, where volume of object equals volume of water displaced

flow rate = ΔP(πr^4 / 8Lη)
*Q = dP/R*

flow rate depends on pressure difference, and radius, length, and viscosity of liquid

increased resistance can be caused by increased length, increased viscosity, or decreased radius

P_s = ρgy

y = distance belows the surface

shape and surface area of water does not matter

bernoulli's equation:
P + ρgh + 1/2 ρv^2 = constant

continuity equation (volume flow is constant): vA = constant

pascal's principle (pressure is transmitted to every surface):
V = Ad = constant (V1=V2 or A1d1=A2d2)
P = F/A = constant
W = Fd = constant

venturi effect (constriction in pipe causes increased velocity because of continuity equation, thus causes decreased pressure because of bernoulli's equation):
P + 1/2 ρv^2 = constant

determine velocity of air by measuring pressure differential, one side has stagnant air so cross out KE

P1 = P2 + 1/2 ρv^2

KE of water coming out = PE of water at surface (v=square root 2gh)

due to bernoulli's equation

dP = pg dh

from bernoulli's equation

electrostatic force between two charges (coulomb's law)

F = (k q1 q2) / r^2

units of F = N
units of k: (N m^2) / C^2

electric field (field points from positive to negative charges)

E = F/q = kq/ r^2
E = V/d

*units of E: N/C, V/m*

electric potential energy (work required to move from infinity to this point in the field)

U = q dV = qEd = k q1 q2 / r

electric potential (work required to move positive test charge to this point in the field)

V = U/q = kq/ r
V = Ed (potential difference between two plates)

*units of V: volt = J/C*

magnetic field (due to straight wire)
B= μ I / 2 pi r
units: Tesla (T) = N s / m C

magnetic field (due to wire loop)
B = μ I / 2 r

field strength is related to current over radius of loop

RHR

magnetic force (experienced by moving charge)
F = qvBsin(theta)

magnetic force (experienced by current)
F = ILBsin(theta)

force is 0 when charges move parallel/antiparallel to magnetic field

RHR

current: I = Q/t
units: amp (A) = C/s

voltage: V = IR
units: V = A*ohm
resistance is a property of the wire, so only V and I can realistically be changed

resistance: R = pL/A
R is directly proportional to T
units: ohm

capacitance: Q = VC
C = keA/d
U = 1/2 C V^2 (potential energy stored by capacitor)
units: farad (F) = C/V

power: P = IV
units: watt (W) = AV

power: P = IV
units: watt (W) = AV

power delivered = V^2/R
power lost (as heat) = I^2 R

high voltage, low current maximized power transmission!

R = internal resistance

V_actual = V_emf - IR_internal

voltage in series:
V = V1 + V2

voltage in parallel
V = V1 = V2

current in series:
I = I1 = I2

current in parallel:
I = I1 + I2

resistors in series:
R = R1 + R2

resistors in parallel:
1/R = 1/R1 + 1/R2

capacitors in series:
1/C = 1/C1 +1/C2

capacitors in parallel:
C = C1 + C2

343 m/s

v = sqrt(B/p)

depends on:
1. B (or K) = *bulk modulus* of medium (why sound is faster in solids)
2. p = density (why sound is faster in a dense solid)
3. temperature of medium (why sound is faster in hot air)

every 10 decibels is a factor of 10 (for intensity)

20 Hz to 20 kHz
ultrasound is greater than 20 kHz

tone or character of the sound

4 variables:
frequency of sound, source
frequency of sound, observed
speed of sound
speed of source

f_obs=(v+-v_o/v+-v_s)f_source

if source is moving towards you, frequency observed is higher

if source is moving away from you, frequency observed is lower

used in Doppler ultrasound to calculate speed of blood or fetus, better than standard ultrasound

linear relationship between speed and frequency

v=(wavelength)(frequency)

dwavelength = wavelength*relative velocity/speed of light (also df/f=v/c)
if wave reflects off object (police radar thing), double the doppler effect

stars have red shift (decreased frequency) since they're moving away from us

for moving target or reflector:

df/f=2v/c

if wave reflects off object (police radar thing), double the doppler effect (2 shifted waves coming back at you now)

n = speed of light/speed in new medium (n=c/v)

light slows down in a medium with greater index of refraction (light is slower in glass than air) (n: air=1, water=1.3, glass=1.5)

sound is faster in glass than air (sound travels faster in denser objects) (helium is a lie)

while wavelength does change, frequency does not change between mediums (due to e=hf where changing f changes energy of photon)

n = c/v

n1sin(theta1) = n2sin(theta2)

always measure angles from the normal lines

when light enters higher index medium, refracts towards normal line

when light enters a lower index medium, refracts away from normal line

total internal reflection- enters lower index medium and refracts a full 90 degrees away from normal, how fiber optics work

f > 0 means concave mirror

object beyond f means image is real (in front of mirror), inverted

object inside f means image is virtual (behind mirror), upright, enlarged

f < 0 means convex mirror

all images are virtual (behind mirror), upright, smaller

1/do + 1/di = 1/f

-di/do = hi/ho

height of image/object is proportional to distance of image/object

use to calculate magnification factor

f > 0 means converging lens

object beyond f means image is real, inverted

object inside f means image is virtual, upright, enlarged

f < 0 means diverging lens

all images are virtual, upright, smaller

do (object distance) is always positive

di (image distance) is positive for real images, negative for virtual images

f (focal length) is positive for concave mirrors, converging lens

m (magnification) is positive for upright image

m = -i/o
m < 1 means image is reduced
m > 1 means image is enlarged

negative m means inverted image
positive m means upright image

refractive power unit (D = 1/m)

reciprocal of focal length of a lens

smaller focal length means more powerful lens

spherical aberration:

assumption of sin(theta) = theta

light at top of lens is refracted more, shorter focal length

so not all rays go through focal length, inherent property with spherical lens

chromatic aberration:

dispersion says greater wavelengths have smaller index of refraction

red refracts less than blue

ciliary muscles adjust lens to focus image on retina

requires diverging lens (concave)
corrects myopia, nearsightedness, focal length of eye is too short

requires converging lens (convex)
corrects hyperopia, farsightedness, focal length of eye is too long

image on retina is upside down, brain flips it

work output/work input

kg
m
s

work function is energy to remove an electron

input energy is greater than output energy

KE = hf - W
W = work function

E = hf
units of E: J
units of h = J s
units of f: 1/s

photon absorbed by electron raises its energy level; when electron falls back down to lower energy level, it releases light (photon) proportional to the energy that was absorbed when electron moved

The work done equals the energy transformed.
$\purple{W= \Delta U}$

$\purple {W_{F} = F_{x} \cdot d = F \cos( \theta ) \cdot d = F_{\parallel} d ) $ or as dot product $ \purple {W_{F} = \vec{F} \cdot \Delta \vec{r}} $
$ \red{ \text{Work is NOT a vector }}$
$\purple { \text{The amount of work done equals the amount of energy transformed. }} $

$\purple { KE = \frac{1}{2} mv^{2} [\text{joules}]} $ KE is always positive.

$ \purple{ \text{The work done by the net force on a single-point object is equal to the change in kinetic energy of than object.} } $
$ \purple{ W_{net} = W_{f net} \Delta KE = KE_{f} - KE_{i} }$
W-E applies in cases where the object is regarded as a point withou deformation or rotation.

$\purple { W_{F} = \int \vec{F} \cdot d \vec{r} }$
Work along a curved path or done by an inconstant force

$ \purple {F_{spring} = -ks}$

$ \purple {PE_{gravitational} = mgh}$
$ \purple { PE_{elastic} = \frac{1}{2}kx^2 }$ k is the spring constant

$ E_{mech} = KE +PE $ (ignoring friction within an isolated system)

$ \purple { \Delta PE_{F} = -W_{F} = +W_{external} }$

The normal force never does work, therefore the potential energy associated with the normal force is zero