physics formulas

component vectors

X = V cos θ

Y = V sin θ

direction of component vectors

θ = tan -1 Y / X

dot product

A . B = |A| |B| cos θ

cross product

A x B = |A||B|sinθ

instantaneous velocity

v = lim ∆x/∆t

∆x = change in displacement

∆t = time

v = velocity inst

average velocity

v_avg = ∆x/∆t

∆x = change in displacement

∆t = time

Universal gravitation equation

F_g= G m₁m₂ / r²

^G= 6.674×10^-11 N*m²/kg²

r= distance between centers

m = masses

static friction

0≤f_s≤µ_sN

Kinetic friction

f_k=µ_kN

Force of gravity/weight

F_g=mg

center of mass

x = m1x1+m2x2+m3x3/m1+m2+m3

avg acceleration

a_avg = ∆v/∆t

v= velocity

t=time

instantaenous accel

a= lim ∆v/∆t

Newton’s first law

F_net = ma = 0

Newton’s second law

F_net = ma

Newton’s third law

F_AB = - F_BA

kinematics (no displacement)

v = v₀+ at

kinematics (no final velocity)

x = v₀t+ at²/2

kinematics (no time)

v² = v₀² + 2ax

kinematics no acceleration

x = v_avgt

components of gravity on an inclined plane

F_g‖ = mg sinθ

F_g⊥=mg cos θ

centripetal force

F_c= mv²/r

Torque

τ = r x F = r *F sinθ

kinetic energy

K = 1/2mv²

gravitation potential energy

U = mgh

elastic potential energy

U = ½ kx² in Joules

k = spring constant N/m

x = displacement of spring (m)

total mechanical energy

E = U + K in joules

U = grav potential energy mgh

K = kinetic energy 1/2mv²

conservation of mechanical energy

∆E = ∆U + ∆K = 0

work done by nonconservative forces

W_{nonconservative} =∆E = ∆U + ∆K

definition of work (mechanical)

W = F*d = F*d*cosθ

F = force N

d = displacement (m)

Joule = N*m

defition of work (isobaric gas-piston system)

W = P∆V in Joules (N*m)

P = constant pressure ;Pa, N/m²

∆V= change in volume m³

Power definition

P = W/t = ∆E/t

P = power

W = work in Joules (N*m)

t = time (s)

∆E = change in mechanical energy (∆U+∆K=∆mgh +∆1/2mv²)

Work-energy theory

W_net = ∆K = K_f-K_i

_{net work is the change in kinetic energy}

mechanical advantage

= F_out/F_in

efficiency

efficiency = W_out/W_in = (load*load distance)/(effort) (effort distance)

Temperature conversions C to F and K

F = 9/5C +32

K = C+273

thermal expansion equation

∆L = αL∆T

∆L = change in length (m)

α= coefficient of linear expansion 1/C° or K

∆T = change in temp

L = initial length

volume expansion equation

∆V = βV∆T

∆V = change in volume m³ or L

β= coefficient of volume expansion 1/C° or K°

V= initial volume in m³ or L

∆T = change in temp F-I

First law of thermodynamics

∆U = Q-W

∆U = change in internal energy in Joules (N*m)

Q = net heat added

W = work done by system on surroundings in Joules

heat gained or lost (w temp change)

q = mc∆T

heat gained or lost (phase change)

q = mL

entropy and heat

∆S = Q_rev/T

2nd law of thermodynamics

∆S _universe = ∆S_sys+∆S_surroundings >0

weight of a volume of fluid

F_g = ρVg

ρ = density (m/V) kg/m³

V= volume of fluid or submerged object in m³

g = gravity

specific gravity

SG = ρ/1 g(cm³)

Pressure

P = F/A

force N over area m²

absolute pressure

P = P₀ +ρgz

P₀ = initial pressure

density gravity z= depth in (m)

gauge pressure

P_gauge = P-P_atm= (P₀+ρgz) - Patm

pascal’s principle

P = F1/A1 = F2/A2

F2= F1(A2/A1)

consider area carefully whether x² or pir²

Buoyant force

F_buoy= ρ_fluidV_{fluid displaced} g = ρ_fluidV_submergedg

Poiseuille’s Law

Q= πr⁴∆P/8ηL

Q = volumetric flow rate m³/s or L/s

r^4 = internal radius of pipe or blood vessel (small change here makes huge difference m)

∆P = P start- P end (P hi - P lo or Pstart-Pend)

η = dynamic viscosity (Pa*s or kg/ m*s)

L = total length (m)

critical speed

v_c = N_Rη/ρD

v_c= critical velocity m/s

N_R = Reynold’s number = given constant

η= dynamic viscosity in Pa*s or kg/m*s (represents fluid’s internal friction or thickness)

row = density

D = diameter in meters

continuity equation

Q = v₁A₁= v₂A₂

Q = volumetric flow rate m³/s

v = m/s

A= πr²

bernouli

P₁ + ½ρv₁² + ρgh₁ = P2 + ½ ρv₂² + ρgh

P1 and P2 = static pressure

dynamic pressure = ½ ρv²

consider constant height

Coulomb’s Law

F_c = kq₁q₂/r²

Fc = coloumb’s force

k = coloumb’s constant (9.0×10^9 N*m²/C²

q₁= net charge of first charge in C (Amperes x s)

q₂ = net charge of second in Coulomb’s

r = distance between two centers of charge

electric field

E = F_e/ q = kQ/r²

E = electric field N/C

F_e = electrostatic force N

q = test charge in (C)

k =columbs constant 9×10^9 N*m²/C^2

Q = source charge C

r = distnace from center of source charge Q to a space where measuring field

electric potential energy

U = kQq/r

U = electric potential energy ( in J)

k = columbs 9×10^9 N*m²/C²

Q = first source charge in C

q = second charge in C

r = distance between centers of two charges

electric potential (from electric potential energy)

V = U/q

V = electric potential volts (J/C)

U = electric potential energy joules

q = charge C

electric potential (from source charge)

V = k Q / r

V = voltage V (J/C)

k = coulumb;’s constant 9×10^9 N*m²/C²

Q = source charge C

r = distance from center of charge to point measuring

Voltage

∆V = V_b- V_a = W_ab/q

Volts = Work (J) done per unit charge(C)

electric potential near a dipole

V = (kqd/r² )cosθ

volts

k = coloum’s 9×10^9 N*m²/C²

q = charge C

d = separation distance between + and - charges (m)

r = distance from center of dipole to a specific point r>d (m)

dipole moment

p = qd

p = dipole moment

q = charge C

d = separation distance between 2 centers (m)

electric field on the perpendicular bisector of a dipole

E = 1/4πε₀ x (p/r³)

E = electric field strength V/m

ε₀= 9×10^-12 C²/N*m²

¼ pε₀ = k = 9×10^9

p = qd (dipole moment)

r = distance between centers

Torque on a dipole in an electric field

τ = pEsinθ

torque = N*M

p = q*d dipole moment

E = strength of electric field V/m

magnetic field from a straight wire

B = µ₀I/2πr

B = magnetic field strength in teslas

µ₀ = 1.3×10^-6

I = current (A)

r = shortest radial distance from the center of the wire to the point in space where you are measuring the magnetic field (m)

Magnetic field from a loop of wire

B = µ₀I/2r

B = magnetic field strength in teslas

µ₀ = 1.3×10^-6

I = current (A)

r = shortest radial distance from the center of the wire to the point in space where you are measuring the magnetic field (m)

Magnetic force on a moving point charge

F_B = qvB sin θ

q = charge C

v = velcotiy of charge m/s

F = magnetic force N

B = magnetic field strength T

magnetic force on a current carrying wire

F_B = ILB sin θ

F = force of mag field N

I = current A

L = length

B = magnetic field strength Teslas

Current

I = Q/∆t

current (A) = Columb / sec

kirchhoff’s junction rule

I_{(into junction)}= I_{(leaving junction)}

Kirchhoff’s loop rule

V_source = V_drop

definition of resistance

R = ρL/A ohms

ρ = resistivity in Ohm*m

L = length of conductor

A = CSA m²

Ohm’s law

V = IR

Voltage and cell emf

V = E_cell - ir_int

_{V = voltage drop (V)}

_{E = electromotive force emf (V)}

_{I = current (A)}

_{r = internal resistance of the cell (ohm)}

definition of power

P = W/t = ∆E/t

P = energy transfer or work done in Watts or J/s

W = energy transferred to or from an object of force along displacement (F*d) in Joules

E = difference in total mechanical energy (Final-Inital) in Joules ∆E=∆U-∆K

t = time in s

electric power

P = IV=I²R = V²/R in Watts (J/s)

I = current A

V = volts

R = ohms

P = power

Voltage drop across circuit elements (series)

V _s= V₁+V₂+V₃

equivalent resistance (series)

R_{s =}R₁+R₂+R₃

voltage drop across circuit elements (parallel)

V_p=V₁=V₂=V3

equivalent resistance parallel

1/R_p=1/R₁+ 1/R₂ 1/R₃ + …

definition of capcitance

C = Q/V in Farads (F)

Q = charge in C

Voltage in V

capacitance based on parallel plate geometry

C= ε₀(A/d) in Farads (F)

ε₀₌ permittivity of rfree space (9×10^-12 F/m)

A = surface area of plate m²

d = distance between plates in m

Electric field in a capacitor

E = V/d

ED = electric field strength in V/m or N/C

V = voltage difference in V

d= gap or separation between two parallel plates in m

POTENTIAL ENERGY OF A CAPACITOR

U= ½ CV²

U = potential energy in J

Capacitance in Farads F

V = voltage in V

capacitance with a dielectric material

C’ = κC

C’ = new capacitance in F

K = dielectric constant

C = original capacitance (F)

equivalent capacitance (series)

1/C_s=1/C₁+1/C₂+1/C₃

equivalent capacitance (parallel)

C_p=C1+C2+C3

pico

1 ×10^-12

1 Farad =

1C/V

wave speed

v = ƒ λ

v= m/s

f = frequency Hz

λ = wavelength

period

T = 1/ƒ

T in seconds

f in Hz

angular frequency

ω = 2πf = 2π/T radians per time

f = frequency Hz

T = period (T) in seconds

speed of sound

v = √(B/p)

v = m/s

B = Bulk modulus = N/m² or Pa (how resistant a substance is to compression)

p= density km/m³

Doppler effect

ƒ’ = ƒ (v±v_D)/(v±v_S)

f’ = new /observed frequency in Hz

f= actual or emitted frequenecy

v = speed of wave m/s

v_D = speed of detector or observer m/s

v_s = speed of source m/s

Intensity

I = P/A

Intensity = Power/A which is W/m²

P = power (W or J/s)

A = area m²

Sound level

β = 10 log I/I₀

in W/m² decibel dB

B = sound intensity level in decibels dB

I = sound intensity W/m²

I₀ = reference sound W/m² usually 10^-12 W/m²

change in sound level

β_f = β_i + 10 log I/I₀

final sound level = initial sound level + log change in intensity/reference intensity

beat frequency

ƒ_beat = |ƒ₁-ƒ₂|

frequency 1 - frequency 2 abs

wavelength of a standing wave (strings and open pipe)

λ = 2L/n

n = nodes

L = length