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PHY121 Formula Sheet
Equations of Motion
v=
One Dimension
x f − xi
Δ x dx
=
Δt → 0 Δ t
dt
v = lim
t f − ti
v = v 0 + at
x − x0 = v0t +
1 2
at
2
v x = v 0 cos θ
ar =
v2
r
From Newton's 2nd Law
Variables
xi = initial position
x0 = initial position
xf = final position
v 2 − v 02
2a
y max
v 02 sin 2θ
=
g
2
v sin 2 θ
= 0
2g
a = ar + at
Σ Fc = m
v2
r
vi = initial velocity
v0 = initial velocity
vf = final velocity
x = displacement in the x-direction
y = displacement in the y-direction
ti = initial time
tf = final time
t = time
θ = angle between the positive x-axis & velocity vector
g = gravity (9.8 m/s² or 32 ft/s²)
m = mass
Laws of Motion
x max
y = (v0 sin θ) t – ½ gt2
Rotational
Circular
x − x0 =
x = (v0 cos θ) t
v y = v 0 sin θ − gt
dv
d ⎛ dx ⎞ d 2 x
=
⎜ ⎟=
dt
dt ⎝ dt ⎠ dt 2
v f2 = vi2 + 2a(xf - xi)
x f = xi + vit + ½at²
Projectile Motion
a=
v = velocity
vx = velocity in the x-direction
vy = velocity in the y-direction
a = acceleration
ar rotational acceleration
at = tangential acceleration
r = radius
Fc = centripetal force
Newton's Laws
(a = 0 → Δ v = 0 )
An object in motion tends to stay in motion
For every action there is an equal and opposite reaction
Force equals mass times acceleration
Friction
force = coefficient of friction times mass times the normal force
Kinetic Friction: F fr =
Static Friction: F fr ≤
μ k FN
μ s FN
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PHY121 Formula Sheet
μs
fr
N
Work and Energy Equations
v v
W = F ⋅ s = Fs cos θ =
Work
xf
yf
∫ F dx + ∫ F dy + ∫ F dz
x
y
xi
xf
W spr =
∫ F spr dx =
xi
0
1
∫ ( − kx ) dx = 2 kx
z
yi
zi
2
m
= ½kxi – ½kxf
− xm
Kinetic K = ½mv²
zf
2
Potential U = mgh
2
Total Energy = K + U
Energy
Power
Momentum
P=
W
Δt
p = mv
Collisions
P = lim
Δt → 0
F =
W
dW
=
Δt
dt
dp
dt
P=F⋅
ds
= F ⋅v
dt
For an isolated system, Δp = 0
tf
Impulse
I =
∫ Fdt
= Δp
ti
Elastic (bounce)
Inelastic (stick)
Variables
m 1v1i + m2v2i = m1v1f + m2v2f
m 1v1i + m2v2i = (m1 + m2)vf
W = work
W spr = work done by spring
F = force
Fspr = force of spring
F x = force in x-direction
k = spring constant
s = distance
x = distance spring stretched/compressed
θ = angle
x m = distance spring stretched/compressed
xi, yi, zi = initial position in the x, y, or z direction K = kinetic energy
xf, yf, zf = final position in the x, y, or z direction m = mass
P = power
v = velocity
Δt = change in time
U = potential energy
p = momentum
h = height
Δp = change in momentum
I = impulse
m1 = mass of first object
ti = initial time
m2 = mass of second object
tf = final time
v1i = initial velocity of first object
v1f = final velocity of first object
v2i = initial velocity of second object
v2f = final velocity of second object
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PHY121 Formula Sheet
Rotational Motion
ω =
s = rθ
θ f −θi
t f − ti
General Equations
α =
Equations of Motion
Inertial Moment
ω f −ϖ i
θ = θ 0 + ω0t + ½αt2
∑ mi ri 2
For Rods: dm =
ar =
I = lim
Δm i → 0
M
dx
L
∑r
i
2
Δ m i = ∫ r 2 dm
Work: dW = F ⋅ ds = ( F sin φ ) rd θ
Other
τ = R×F = R×
vCM = rω
2
s = arc length
r = radius
θ = angle
θi = initial angle
θf = final angle
ρ =
dm
dV
τ = Iα
aCM = rα
dp dL
=
dt
dt
K = ½ICMω2+½MvCM = rotational+translational
Variables
2
I = I CM + MD²
τ = (mrα)r = mr2α
τ = r F sin φ = F·D
v2
= rω
r
For Cylinders: dV = dA ⋅ L = (2π rdr ) L )
Parallel Axis Theorem
Torque
ω2 = ω02 + 2α (θ – θ 0)
at = rα
v = rω
I =
Δω
dϖ
=
Δt → 0 Δ t
dt
α = lim
t f − ti
ω = ω0 + αt
Converting Rotational
to Linear
dθ
Δθ
=
Δt → 0 Δ t
dt
ϖ = lim
ω = angular speed
ω i = initial angular speed
ω0 = initial angular speed
ωf = final angular speed
ω = average angular speed
l = R x p = mvR sin φ = Iω
α = angular acceleration
α = average angular acceleration
I = Inertia
mi = mass of ith particle
M = mass of an object
ti = initial time
r i = distance from ith particle to axis of rotation
tf = final time
ar = radial acceleration
D = distance from axis
v = linear velocity
at = tangential acceleration
K = kinetic energy
ρ = density of an object
L = length of rod / cylinder
φ = angle
I cm = Inertia about an axis through the center of mass of an object
τ = torque
l = instantaneous angular momentum
F = force
R = instantaneous position vector
p = instantaneous linear momentum
aCM = acceleration at center of mass
VCM = velocity at center of mass
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PHY121 Formula Sheet
Miscellaneous Formulas
Center of Mass
X CM =
∑m x
∑m
i
rCM =
i
i
Rocketry
(M + Δm)v = M(v + Δv) + Δm(v – ve)
Mdv = -vedM
Gravity
Kepler's Laws
M
vi
Mi
Fg = G
Law of
Gravitation
Fluids
Bernoulli's
Equation
Variables
dM
M
p=
F
A
1
Δr
⎛ M
v f − v i = v e ln ⎜ i
⎜M
⎝ f
Fg = −G ∫
⎞
⎟
⎟
⎠
dm
r
r2
⎛ 4π 2 ⎞ 3
3
T2 =⎜
⎟ r = kr
Gm
⎝
⎠
dA
L
=
= constant
dt 2 m
Energy
Pressure
∫
f
m1 m 2
R2
Δ U = − Gm 1 m 2
∫ rdm
MΔv = Δmve
vf
∫ dv = − v e
1
M
E=
Gm 1 m 2 Gm 1 m 2
mm
−
= −G 1 2
r
2r
2r
Buoyant force = weight of displaced liquid
p + ½ρv2 + ρgy = c
X CM = center of mass in x-direction
v = velocity
th
m i = mass of i particle
M = mass of rocket and fuel
x i = position of ith particle
Δm = mass of fuel loss
r CM = radial center of mass
ve = exhaust speed
M = mass
Δv = change in velocity of rocket
m = mass
M i = initial mass of rocket and fuel
m 1 = mass of 1st object
M f = final mass of rocket and remaining fuel
m 2 = mass of 2nd object
v i = initial velocity
F g = force of gravity
v f = final velocity
G =universal gravitational constant(6.673 x 10-11 N·m²/kg²) R = distance seperating m 1 & m 2
L = angular momentum of planet (constant)
T = period of revolution
k = constant (2.97 x 10 -19 s²/m³)
ΔU = change in gravitational potential energy
E = total energy
ρ = density (mass / volume)
g = gravity constant (9.8 m/s² or 32 ft/s²)
c = constant
= unit vector
r = radius
Δr = change in radius
p = pressure
F = force exerted on the piston
A = surface area of piston
y = height
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PHY121 Formula Sheet
Oscillations
x = A cos (ωt + φ)
A=
Simple Harmonic
Motion
2π
T =
ω
⎛v ⎞
x +⎜ 0⎟
⎝ω ⎠
m
k
= 2π
K = ½ mv2 = ½ mω2A2sin (ωt + φ)
Energy
U = ½ kx2 = ½ kA2cos2 (ωt + φ)
Wave Motion
v=
E = K + U = ½ kA2
dx
= −ω A sin( ω t + φ )
dt
Fr =
Waves on a string
2 Fθ =
v=
mv 2
= 2 F sin θ ≈ 2 F θ
R
2 μ Rθv 2
R
v=
⎡ ⎛ x t ⎞⎤
y = A sin ⎢ 2π ⎜ − ⎟ ⎥
⎣ ⎝ λ T ⎠⎦
ω = angular frequency =
2π
T
m = μΔs = 2μRθ
F
μ
, F = tension , μ =
kg
m
v=
2π
λ
dy
= −ω A cos( kx − ω t )
dt
dv
= −ω 2 A sin( kx − ω t )
dt
ΔE = ½Δmω2A2
Energy
dv
= −ω 2 A cos( ω t + φ )
dt
k = angular wave number =
a=
Variables
2
2
0
P=
dE 1
= μω 2 A 2 v
2
dt
x = position
x0 = initial position
K = kinetic energy
U = potential energy
ω = angular frequency
t = time
A = amplitude
T = period
E = total energy
R = radius
φ = phase constant or phase angle
v = velocity
m = mass
θ = angle
k = spring constant
λ = wavelength
= frequency
t = time
ΔE = change in total energy
a = acceleration
Δs = length of small segment of string
Δm = change in mass
v0 = initial velocity
Fr = total radial force
F = force of tension
μ = mass per unit length
P = power
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