Most Used Mechanics Equations & Formulas

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)
= 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 A Δz
 F = η dvx A dz
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 A y
Bulk Modulus
 F = K ΔV A V0
Surface Tension
 γ = F ℓ