Physics formulas: Mechanics

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Linear kinematics

Displacement projection on the X axis

sx=x−x0s_x = x - x_0
Symbols
  • sxs_x — displacement projection on the X axis, m (meter)
  • xx — coordinate at the final instant, m (meter)
  • x0x_0 — coordinate at the initial instant, m (meter)
Linear kinematics

Speed of uniform rectilinear motion

v=stv = \frac{s}{t}
Symbols
  • vv — speed of uniform motion, m/s
  • ss — path (displacement), m (meter)
  • tt — time of motion, s (second)
Linear kinematics

Average speed

vav=stv_{\text{av}} = \frac{s}{t}
Symbols
  • vavv_{\text{av}} — average speed, m/s
  • ss — path (displacement), m (meter)
  • tt — time of motion, s (second)
Linear kinematics

Equation of uniform rectilinear motion

x=x0+vtx = x_0 + vt
Symbols
  • xx — coordinate at time t, m (meter)
  • x0x_0 — initial coordinate, m (meter)
  • vv — speed (constant), m/s
  • tt — time from the start, s (second)
Linear kinematics

Acceleration in uniformly accelerated motion

a=v−v0ta = \frac{v - v_0}{t}
Symbols
  • aa — acceleration, m/s²
  • vv — speed at the final instant, m/s
  • v0v_0 — speed at the initial instant, m/s
  • tt — time interval of the speed change, s (second)
Linear kinematics

Velocity in uniformly accelerated motion

v=v0+atv = v_0 + at
Symbols
  • vv — speed at time t, m/s
  • v0v_0 — initial speed, m/s
  • aa — acceleration (constant), m/s²
  • tt — time from the start, s (second)
Linear kinematics

Displacement in uniformly accelerated motion

s=v0t+at22s = v_0 t + \frac{at^2}{2}
Symbols
  • ss — displacement (path), m (meter)
  • v0v_0 — initial speed, m/s
  • aa — acceleration, m/s²
  • tt — time of motion, s (second)
Linear kinematics

Displacement vs. time in uniformly accelerated motion

x=x0+v0t+at22x = x_0 + v_0 t + \frac{at^2}{2}
Symbols
  • xx — coordinate at time t, m (meter)
  • x0x_0 — initial coordinate, m (meter)
  • v0v_0 — initial speed, m/s
  • aa — acceleration, m/s²
  • tt — time from the start, s (second)
Linear kinematics

Displacement projection without time (UAM)

vx2−v0x2=2axsxv_x^2 - v_{0x}^2 = 2a_x s_x
Symbols
  • vxv_x — final speed projection on the X axis, m/s
  • v0xv_{0x} — initial speed projection on the X axis, m/s
  • axa_x — acceleration projection on the X axis, m/s²
  • sxs_x — displacement projection on the X axis, m (meter)
Linear kinematics

Equation of uniformly accelerated motion

x=x0+v0t+at22x = x_0 + v_0 t + \frac{at^2}{2}
Symbols
  • xx — coordinate at time t, m (meter)
  • x0x_0 — initial coordinate, m (meter)
  • v0v_0 — initial speed, m/s
  • aa — acceleration, m/s²
  • tt — time from the start, s (second)
Curvilinear kinematics

Rotation frequency

ν=Nt\nu = \frac{N}{t}
Symbols
  • ν\nu — rotation frequency, Hz (hertz)
  • NN — number of full revolutions, dimensionless quantity
  • tt — time over which the revolutions occur, s (second)
Curvilinear kinematics

Rotation period

T=tNT = \frac{t}{N}
Symbols
  • TT — rotation period, s (second)
  • tt — time of motion, s (second)
  • NN — number of full revolutions, dimensionless quantity
Curvilinear kinematics

Relation between period and frequency

ν=1T\nu = \frac{1}{T}
T=1νT = \frac{1}{\nu}
Symbols
  • ν\nu — rotation frequency, Hz (hertz)
  • TT — rotation period, s (second)
Curvilinear kinematics

Linear speed

v=ltv = \frac{l}{t}
Symbols
  • vv — linear speed (speed magnitude), m/s
  • ll — arc length (path) in time t, m (meter)
  • tt — time of motion, s (second)
Curvilinear kinematics

Linear speed via rotation period

v=2πRTv = \frac{2\pi R}{T}
Symbols
  • vv — linear speed, m/s
  • RR — radius of the circle, m (meter)
  • TT — rotation period, s (second)
Curvilinear kinematics

Linear speed via rotation frequency

v=2πRνv = 2\pi R \nu
Symbols
  • vv — linear speed, m/s
  • RR — radius of the circle, m (meter)
  • ν\nu — rotation frequency, Hz (hertz)
Curvilinear kinematics

Angular speed

ω=φt\omega = \frac{\varphi}{t}
Symbols
  • ω\omega — angular speed, rad/s
  • φ\varphi — angle of rotation (in radians), rad
  • tt — time of rotation, s (second)
Curvilinear kinematics

Angular speed via rotation period

ω=2πT\omega = \frac{2\pi}{T}
Symbols
  • ω\omega — angular speed, rad/s
  • TT — rotation period, s (second)
Curvilinear kinematics

Angular speed via rotation frequency

ω=2πν\omega = 2\pi \nu
Symbols
  • ω\omega — angular speed, rad/s
  • ν\nu — rotation frequency, Hz (hertz)
Curvilinear kinematics

Relation between linear and angular speed

v=ωRv = \omega R
Symbols
  • vv — linear speed, m/s
  • ω\omega — angular speed, rad/s
  • RR — radius of the circle, m (meter)
Curvilinear kinematics

Centripetal acceleration via linear speed

ac=v2Ra_{\text{c}} = \frac{v^2}{R}
Symbols
  • aca_{\text{c}} — centripetal acceleration, m/s²
  • vv — linear speed, m/s
  • RR — radius of the circle, m (meter)
Curvilinear kinematics

Centripetal acceleration via angular speed

ac=ω2Ra_{\text{c}} = \omega^2 R
Symbols
  • aca_{\text{c}} — centripetal acceleration, m/s²
  • ω\omega — angular speed, rad/s
  • RR — radius of the circle, m (meter)
Dynamics

Newton’s second law

F=maF = ma
Symbols
  • FF — net force (sum of forces), N (newton)
  • mm — mass of the body, kg (kilogram)
  • aa — acceleration of the body, m/s²
Dynamics

Newton’s third law

F12=−F21F_{12} = -F_{21}
Symbols
  • F12F_{12} — force with which body 1 acts on body 2, N (newton)
  • F21F_{21} — force with which body 2 acts on body 1, N (newton)
Dynamics

Magnitude of friction force

Ff=μNF_{\text{f}} = \mu N
Symbols
  • FfF_{\text{f}} — magnitude of the friction force, N (newton)
  • μ\mu — coefficient of friction, dimensionless quantity
  • NN — magnitude of the normal reaction force, N (newton)
Dynamics

Projection of elastic force

Fx=−kxF_x = -kx
Symbols
  • FxF_x — projection of the elastic force on the X axis, N (newton)
  • kk — spring stiffness (spring constant), N/m
  • xx — extension (compression) from equilibrium, m (meter)
The minus sign reflects the direction of the elastic force.
Dynamics

Gravitational force (weight)

Fg=mgF_{\text{g}} = mg
Symbols
  • FgF_{\text{g}} — gravitational force (weight), N (newton)
  • mm — mass of the body, kg (kilogram)
  • gg — acceleration of free fall, m/s²
Dynamics

Weight on a stationary or uniformly moving support

P=mgP = mg
Symbols
  • PP — weight (force pressing on the support), N (newton)
  • mm — mass of the body, kg (kilogram)
  • gg — acceleration of free fall, m/s²
Dynamics

Weight on an accelerated support

P=m(g±a)P = m(g \pm a)
Symbols
  • PP — weight on an accelerating support, N (newton)
  • mm — mass of the body, kg (kilogram)
  • gg — acceleration of free fall, m/s²
  • aa — acceleration of the support (suspension), m/s²
Use «+» when accelerating upward, «−» when accelerating downward.
Dynamics

Law of universal gravitation

F=Gm1m2r2F = G\frac{m_1 m_2}{r^2}
Symbols
  • FF — magnitude of the gravitational force, N (newton)
  • GG — gravitational constant, 6.67×10⁻¹¹ N·m²/kg²
  • m1m_1 — mass of the first body, kg (kilogram)
  • m2m_2 — mass of the second body, kg (kilogram)
  • rr — distance between the centers of the bodies, m (meter)
Dynamics

Acceleration of free fall

g=GMR2g = \frac{GM}{R^2}
Symbols
  • gg — acceleration of free fall at the planet’s surface, m/s²
  • GG — gravitational constant, 6.67×10⁻¹¹ N·m²/kg²
  • MM — mass of the planet, kg (kilogram)
  • RR — radius of the planet, m (meter)
At a planet’s surface: RR is radius, MM is mass.
Dynamics

First cosmic velocity

v1=gRv_1 = \sqrt{gR}
Symbols
  • v1v_1 — first cosmic velocity, m/s
  • gg — acceleration of free fall, m/s²
  • RR — radius of the planet, m (meter)
Dynamics

Newton’s second law (impulse form)

FΔt=mΔvF\Delta t = m\Delta v
Symbols
  • FF — force acting on the body, N (newton)
  • Δt\Delta t — time interval of the force, s (second)
  • mm — mass of the body, kg (kilogram)
  • Δv\Delta v — change in speed, m/s
Also: FΔt=ΔpF\Delta t = \Delta p.
Dynamics

Conservation of momentum (two bodies)

m1v1+m2v2=m1v1′+m2v2′m_1 v_1 + m_2 v_2 = m_1 v_1^{\prime} + m_2 v_2^{\prime}
Symbols
  • m1m_1 — mass of the first body, kg (kilogram)
  • m2m_2 — mass of the second body, kg (kilogram)
  • v1v_1 — speed of body 1 before interaction, m/s
  • v2v_2 — speed of body 2 before interaction, m/s
  • v1′v_1^{\prime} — speed of body 1 after interaction, m/s
  • v2′v_2^{\prime} — speed of body 2 after interaction, m/s
Statics

Moment of force about an axis

M=FlM = Fl
Symbols
  • MM — moment of force (torque), N·m
  • FF — magnitude of the force, N (newton)
  • ll — lever arm, m (meter)
ll is the lever arm (shortest distance from axis to line of action).
Statics

Equilibrium without a rotation axis

∑F=0\sum F = 0
Symbols
  • ∑F\sum F — vector sum of all forces acting on the body, N (newton)
Statics

Equilibrium with a rotation axis

∑F=0\sum F = 0
∑M=0\sum M = 0
Symbols
  • ∑F\sum F — vector sum of all forces, N (newton)
  • ∑M\sum M — algebraic sum of moments of forces about the axis, N·m
Hydrostatics

Density

ρ=mV\rho = \frac{m}{V}
Symbols
  • ρ\rho — density of the substance, kg/m³
  • mm — mass of the substance, kg (kilogram)
  • VV — volume of the substance, m³ (cubic meter)
Hydrostatics

Pressure

p=FSp = \frac{F}{S}
Symbols
  • pp — pressure, Pa (pascal)
  • FF — magnitude of the pressure force perpendicular to the surface, N (newton)
  • SS — area of the surface, m² (square meter)
Hydrostatics

Hydrostatic pressure vs. depth

p=p0+ρghp = p_0 + \rho g h
Symbols
  • pp — pressure at depth h, Pa (pascal)
  • p0p_0 — pressure at the liquid surface, Pa (pascal)
  • ρ\rho — density of the liquid, kg/m³
  • gg — acceleration of free fall, m/s²
  • hh — depth (height of the liquid column), m (meter)
Hydrostatics

Liquid pressure force on the bottom

F=pSF = pS
Symbols
  • FF — liquid pressure force on the bottom, N (newton)
  • pp — pressure on the bottom, Pa (pascal)
  • SS — area of the vessel bottom, m² (square meter)
Hydrostatics

Liquid pressure force on a side wall

F=pSF = pS
Symbols
  • FF — pressure force on the side wall, N (newton)
  • pp — pressure on the wall, Pa (pascal)
  • SS — area of the wall, m² (square meter)
Hydrostatics

Communicating vessels (different liquids)

ρ1gh1=ρ2gh2\rho_1 g h_1 = \rho_2 g h_2
Symbols
  • ρ1\rho_1 — density of the first liquid, kg/m³
  • ρ2\rho_2 — density of the second liquid, kg/m³
  • h1h_1 — height of the first liquid column, m (meter)
  • h2h_2 — height of the second liquid column, m (meter)
  • gg — acceleration of free fall, m/s²
Hydrostatics

Archimedes’ principle

FA=ρflgVdisF_A = \rho_{\text{fl}} g V_{\text{dis}}
Symbols
  • FAF_A — buoyant force (Archimedes’ force), N (newton)
  • ρfl\rho_{\text{fl}} — density of the liquid (gas), kg/m³
  • gg — acceleration of free fall, m/s²
  • VdisV_{\text{dis}} — volume of liquid displaced by the body, m³ (cubic meter)
Hydrostatics

Hydraulic press force relation

F1S1=F2S2\frac{F_1}{S_1} = \frac{F_2}{S_2}
Symbols
  • F1F_1 — force on the small piston, N (newton)
  • F2F_2 — force on the large piston, N (newton)
  • S1S_1 — area of the small piston, m² (square meter)
  • S2S_2 — area of the large piston, m² (square meter)
Work, energy & power

Work of a constant force

A=Fscos⁡αA = Fs\cos\alpha
Symbols
  • AA — work done by the force, J (joule)
  • FF — magnitude of the constant force, N (newton)
  • ss — displacement, m (meter)
  • α\alpha — angle between the directions of force and displacement, rad
Work, energy & power

Work of friction

Af=−FfsA_{\text{f}} = -F_{\text{f}} s
Symbols
  • AfA_{\text{f}} — work of the friction force, J (joule)
  • FfF_{\text{f}} — magnitude of the friction force, N (newton)
  • ss — displacement, m (meter)
Work, energy & power

Work of gravity

A=mghA = mgh
Symbols
  • AA — work done by gravity, J (joule)
  • mm — mass of the body, kg (kilogram)
  • gg — acceleration of free fall, m/s²
  • hh — change in height, m (meter)
Work, energy & power

Work of elastic force

A=kx22A = \frac{kx^2}{2}
Symbols
  • AA — work done by the elastic force, J (joule)
  • kk — spring stiffness, N/m
  • xx — deformation (extension or compression), m (meter)
Work, energy & power

Power in uniform rectilinear motion

N=FvN = Fv
Symbols
  • NN — power, W (watt)
  • FF — magnitude of the force, N (newton)
  • vv — speed of uniform motion, m/s
Work, energy & power

Power

N=AtN = \frac{A}{t}
Symbols
  • NN — average power, W (watt)
  • AA — work done, J (joule)
  • tt — time over which the work is done, s (second)
Work, energy & power

Kinetic energy

Ek=mv22E_k = \frac{mv^2}{2}
Symbols
  • EkE_k — kinetic energy, J (joule)
  • mm — mass of the body, kg (kilogram)
  • vv — speed of the body, m/s
Work, energy & power

Gravitational potential energy

Ep=mghE_p = mgh
Symbols
  • EpE_p — gravitational potential energy, J (joule)
  • mm — mass of the body, kg (kilogram)
  • gg — acceleration of free fall, m/s²
  • hh — height above the chosen zero level, m (meter)
Work, energy & power

Elastic potential energy

Eel=kx22E_{\text{el}} = \frac{kx^2}{2}
Symbols
  • EelE_{\text{el}} — elastic potential energy, J (joule)
  • kk — spring stiffness, N/m
  • xx — deformation from equilibrium, m (meter)
Work, energy & power

Total mechanical energy (closed system)

Ek+Ep=constE_k + E_p = \text{const}
Symbols
  • EkE_k — kinetic energy, J (joule)
  • EpE_p — potential energy, J (joule)
Work, energy & power

Work–energy theorem

A=ΔEkA = \Delta E_k
Symbols
  • AA — work of all forces acting on the body, J (joule)
  • ΔEk\Delta E_k — change in kinetic energy, J (joule)
Work, energy & power

Efficiency

η=AuseAin⋅100%\eta = \frac{A_{\text{use}}}{A_{\text{in}}} \cdot 100\%
Symbols
  • η\eta — efficiency, % (percent)
  • AuseA_{\text{use}} — useful work, J (joule)
  • AinA_{\text{in}} — total input work, J (joule)
Oscillations & waves

Displacement vs. time (oscillation)

x=xmcos⁡(ωt+φ0)x = x_m \cos(\omega t + \varphi_0)
Symbols
  • xx — coordinate at time t, m (meter)
  • xmx_m — amplitude of oscillation, m (meter)
  • ω\omega — angular (cyclic) frequency, rad/s
  • tt — time, s (second)
  • φ0\varphi_0 — initial phase, rad
Oscillations & waves

Velocity projection vs. time (oscillation)

vx=−ωxmsin⁡(ωt+φ0)v_x = -\omega x_m \sin(\omega t + \varphi_0)
Symbols
  • vxv_x — projection of velocity on the X axis, m/s
  • xmx_m — amplitude of oscillation, m (meter)
  • ω\omega — angular frequency, rad/s
  • tt — time, s (second)
  • φ0\varphi_0 — initial phase, rad
Oscillations & waves

Acceleration projection vs. time (oscillation)

ax=−ω2xmcos⁡(ωt+φ0)=−ω2xa_x = -\omega^2 x_m \cos(\omega t + \varphi_0) = -\omega^2 x
Symbols
  • axa_x — projection of acceleration on the X axis, m/s²
  • xmx_m — amplitude of oscillation, m (meter)
  • ω\omega — angular frequency, rad/s
  • xx — coordinate at time t, m (meter)
  • tt — time, s (second)
  • φ0\varphi_0 — initial phase, rad
Oscillations & waves

Angular (cyclic) frequency

ω=2πT\omega = \frac{2\pi}{T}
Symbols
  • ω\omega — angular (cyclic) frequency, rad/s
  • TT — period of oscillation, s (second)
Oscillations & waves

Relation between period and frequency of oscillations

T=1νT = \frac{1}{\nu}
ν=1T\nu = \frac{1}{T}
Symbols
  • TT — period of oscillation, s (second)
  • ν\nu — frequency of oscillation, Hz (hertz)
Oscillations & waves

Maximum speed in oscillation

vm=ωxmv_m = \omega x_m
Symbols
  • vmv_m — maximum speed of the oscillating body, m/s
  • ω\omega — angular frequency, rad/s
  • xmx_m — amplitude of oscillation, m (meter)
Oscillations & waves

Maximum acceleration in oscillation

am=ω2xma_m = \omega^2 x_m
Symbols
  • ama_m — maximum acceleration, m/s²
  • ω\omega — angular frequency, rad/s
  • xmx_m — amplitude of oscillation, m (meter)
Oscillations & waves

Period of a spring pendulum

T=2πmkT = 2\pi \sqrt{\frac{m}{k}}
Symbols
  • TT — period of the spring pendulum, s (second)
  • mm — mass on the spring, kg (kilogram)
  • kk — spring stiffness, N/m
Oscillations & waves

Period of a simple (mathematical) pendulum

T=2πlgT = 2\pi \sqrt{\frac{l}{g}}
Symbols
  • TT — period of the simple pendulum, s (second)
  • ll — length of the pendulum string, m (meter)
  • gg — acceleration of free fall, m/s²
Oscillations & waves

Total energy of a body on a spring

E=kxm22=mvm22E = \frac{kx_m^2}{2} = \frac{mv_m^2}{2}
Symbols
  • EE — total mechanical energy of oscillation, J (joule)
  • kk — spring stiffness, N/m
  • xmx_m — amplitude of oscillation, m (meter)
  • mm — mass of the body, kg (kilogram)
  • vmv_m — maximum speed, m/s
Oscillations & waves

Wavelength

λ=vT=vν\lambda = vT = \frac{v}{\nu}
Symbols
  • λ\lambda — wavelength, m (meter)
  • vv — wave propagation speed, m/s
  • TT — period of oscillation, s (second)
  • ν\nu — frequency of oscillation, Hz (hertz)