List of Principles and Equations
Principle
Equations
vectors - find components
vx = v cos(θ)
vy = v sin(θ)
vectors - find magnitude
and direction
v = sqrt(vx2 + vy2)
θ = tan-1(vy / vx)
add 180° to θ if vx < 0
vectors - addition
If c = a + b, then
cx = ax + bx
cy = ay + by
average velocity and
acceleration
v̅ x = (x - x0) / t
a̅ x = (vx - v0x) / t
constant acceleration
x = x0 + v0x t + ½ ax t2
vx = v0x + ax t
vx2 - v0x2 = 2 ax (x - x0)
Projectile motion
x = x0 + v0x t
y = y0 + v0y t - 4.9 m/s2 t2
vy = v0y - 9.8 m/s2 t
vy2 - v0y2 = -19.6 m/s2 (y - y0)
Newton's 2nd law
(Fnet)x = m ax
(Fnet)y = m ay
Object traveling along an
inclined plane
n = m g cos(θ)
Friction force
fk = μk n ← kinetic friction
fs ≤ μs n ← static friction
Spring force
Fx = - k x
Work - constant force
W = Fx (x - x0)
W = F d cos(θ)
Wg = - m g (y - y0)
Kinetic energy
KE = ½ m v2
Single object, with work
done on it
Wnet = KEf - KEi
PEg = m g y ← gravitational potential energy of one object
System of interacting
objects (energy)
PEs = ½ k x2 ← potential energy of one spring
E = Σ PE + Σ KE ← mechanical energy is the sum of all
potential energies and kinetic energies
Ef - Ei = Wnc ← change in mechanical energy is the net nonconservative work done on the system
Principle
Equations
Momentum of a single object
p = m v ← vector
px = m vx ← component
Impulse
I = F Δt ← impulse due to a single force
pf - pf = Inet
Collision
p1i + p2i = p1f + p2f ← vector
p1ix + p2ix = p1fx + p2fx ← component
1d elastic collision
v1i - v2i = v2f - v1f ← use in conjunction with
collision equation
1d totally inelastic collision
v1fx = v2fx ← use in conjunction with collision
equation
Rotating rigid object, OR
object traveling on a circular path
θ=s/r
ω = vt / r
α = at / r
Uniform circular motion of an object
ac = v2 / r ← ac points from the object towards
the center of the path
Non-uniform circular motion of an object
a = sqrt(at2 + ac2)
Gravitational force between any two
massive objects
F G = G m1 m 2 / r 2
PEG = - G m1 m2 / r
Torque
τ = r F sin(θ)
τ > 0 → counter clockwise change in rotation
Equilibrium
Fnet = 0
τnet = 0
Rolling or spinning object - moment of
inertia
Point mass → I = m r2
ring → I = m r2
disk → I = ½ m r2
sphere → I = 2 m r2 / 5
Non-zero net torque
τnet = I α
Rolling or spinning object - kinetic
energy
KEr = ½ I ω2
Rotating object with 0 net torque
Ii ωi = If ωf
Fluid
ρ=M/V
P=n/A
Pressure in a stationary fluid
P2 = P1 + ρ g h
Buoyant force on a solid object in contact
with a fluid or gas
B = ρfluid g Vdisplaced ← direction is ALWAYS
upwards
Principle
Flowing fluid
Equations
v 1 A1 = v 2 A2
P1 + ½ ρ v12 + ρ g y1 = P2 + ½ ρ v22 + ρ g y2
Length and volume of a heated solid
object
ΔL = α L0 ΔT ← change in length
ΔV = β V0 ΔT ← change in volume,
β=3α
Ideal gas
PV=nRT
molar mass = sum of atomic numbers of atoms in
molecule, in g/mol
n = number of moles = mass of gas / molar mass
Change in temperature of a heated
object
Q = m c ΔT
Change in phase of a heated object
Q = ± m L ← use + for an object that is melting or
evaporating
Isolated system of stationary objects
in thermal contact
ΣQ=0
System which experiences work, and
transfers heat
ΔE + Q = W
Q = heat of the system, W = work done on the
system (not by the system)
Mass on a spring
ω = sqrt(k / m)
Periodic motion of an object
T = 2 π / ω ← period
f = 1 / T = ω / (2 π) ← frequency
x = A cos(ω t)
vx = -A ω sin(ω t)
ax = -ω2 x
Pendulum
ω = sqrt(g / L)
Periodic wave motion
OR standing wave pattern
v=fλ
Speed of a wave on a string
v = sqrt(FT / μ)
μ = m / L ← mass per unit length
Standing wave on a string
fn = f1 / n ← nth harmonic frequency
f1 = v / (2 L) ← fundamental frequency
n = 1, 2, 3, ...
Standing sound wave in an open-open
ended pipe
fn = f1 / n
f1 = v / (2 L)
n = 1, 2, 3, ...
Standing sound wave in an openclosed ended pipe
fn = f1 / n
f1 = v / (4 L)
n = 1, 3, 5, ...
Interference of sound waves from two
sources
constructive if r1 - r2 = n λ
destructive if r1 - r2 = (n + ½) λ
n = 0, ±1, ±2, ...