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Most Used Mechanics Equations & Formulas

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velocity
v̅ =  Δs
Δt
v = ds
dt
Acceleration
a̅ =  Δv
Δt
a = dv
dt
Equations Of Motion
v = v0 + at
x = x0 + v0t + ½at2
v2 = v02 + 2a(x − x0)
v̅ = ½(v + v0)
Newton's 2nd Law
∑ F = m a
∑ F =  dp
dt
Weight
W = m g
Dry Friction
ƒ  μN
Centripetal Accel.
ac =  v2
r
ac = − ω2r
Momentum
p = mv
Impulse
J = Δt
J = F dt
Impulse-Momentum
Δt = mΔv
F dt = Δp
Work
W = Δs cos θ
W = F · ds
Work-Energy
Δs cos θ = ΔE
F · ds = ΔE
Kinetic Energy
K = ½mv2
General Potential Energy
ΔU = − F · ds
F = − ∇U
Gravitational Potential Energy
ΔUg = mgΔh
Efficiency
η =  Wout
Ein
Power
 =  ΔW
Δt
P = dW
dt
Power-Velocity
 = v cos θ
P = F · v
Angular Velocity
ω̅ = Δθ
Δt
ω = dθ
dt
v = ω × r
Angular Acceleration
α̅ = Δω
Δt
α = dω
dt
a = α × r − ω2 r
Equations Of Rotation
ω = ω0 + αt
θ = θ0 + ω0t + ½αt2
ω2 = ω02 + 2α(θ − θ0)
ω̅ = ½(ω + ω0)
Newton's second law of rotational motion
∑ τ = I α
∑ τ = dL
dt
Torque
τ = rF sin θ
τ = r × F
Moment Of Inertia
I = ∑ mr2
I =   r2 dm
Rotational Motion
W = τ̅Δθ
W =   τ · dθ
Rotational power
P = τω cos θ
P = τ · ω
Rotational Kinetic Energy
K = ½Iω2
Angular Momentum
L = mrv sin θ
L = r × p
L = I ω
Newton's Law Of Universal Gravitation
Fg = −  Gm1m2 r̂
r2
Gravitational Field Surrounding a Point Mass
g = − Gm r̂
r2
Potential Energy Surrounding a Point Mass
Ug = − Gm1m2
r
Gravitational Potential Surrounding a Point Mass
Vg = − Gm
r
Orbital Speed
v = √ Gm
r
Escape Speed
v = √ 2Gm
r
Hooke's Law
F = − kΔx
Elastic Potential Energy
Us = ½kΔx2
Period of a Simple Harmonic Oscillator
T = 2π √ m
k
Period of a Simple Pendulum
T = 2π √ ℓ
g
Frequency
ƒ = 1
T
Angular-Linear Frequency Relation
ω = 2pƒ
Mass Density
ρ = m
V
Pressure
P = F
A
Pressure-Depth Relation for a Fluid of Uniform Density
P = P0 + ρgh
Archimedes' Law of Buoyancy
B = ρgVdisplaced
Mass Flow Rate
I = m
t
Volume Flow Rate
φ = V
t
Mass Continuity
ρ1A1v1 = ρ2A2v2
Volume Continuity
A1v1 = A2v2
Bernoulli's Equation
P1 + ρgy1 + ½ρv12 =  P2 + ρgy2 + ½ρv22
Dynamic Viscosity
 = η  Δvx
AΔz
F = η  dvx
Adz
Kinematic Viscosity
v = η
ρ
Drag
R = ½ρCAv2
Mach Number
Ma = v
c
Reynolds Number
Re = ρvD
η
Froude Number
Fr = v
√gℓ
Young's Modulus
F = E Δℓ
A ℓ0
Shear Modulus
F = G  Δx
Ay
Bulk Modulus
F = K  ΔV
AV0
Surface Tension
γ = F
ℓ
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