List of equations in classical mechanics

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Classical mechanics is the branch of physics used to describe the motion of macroscopic objects.[1] It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known.[2] The subject is based upon a three-dimensional Euclidean space with fixed axes, called a frame of reference. The point of concurrency of the three axes is known as the origin of the particular space.[3]

Classical mechanics utilises many equations—as well as other mathematical concepts—which relate various physical quantities to one another. These include differential equations, manifolds, Lie groups, and ergodic theory.[4] This article gives a summary of the most important of these.

This article lists equations from Newtonian mechanics, see analytical mechanics for the more general formulation of classical mechanics (which includes Lagrangian and Hamiltonian mechanics).

Classical mechanics

Mass and inertia

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Linear, surface, volumetric mass density λ or μ (especially in acoustics, see below) for Linear, σ for surface, ρ for volume.  

 

 

kg m−n, n = 1, 2, 3 M L−n
Moment of mass[5] m (No common symbol) Point mass:

 

Discrete masses about an axis  :  

Continuum of mass about an axis  :  

kg m M L
Center of mass rcom

(Symbols vary)

i-th moment of mass  

Discrete masses:  

Mass continuum:  

m L
2-Body reduced mass m12, μ Pair of masses = m1 and m2   kg M
Moment of inertia (MOI) I Discrete Masses:

 

Mass continuum:  

kg m2 M L2

Derived kinematic quantities

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Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Velocity v Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} = \frac{\mathrm{d} \mathbf{r}}{\mathrm{d} t} } m s−1 L T−1
Acceleration a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a} = \frac{\mathrm{d} \mathbf{v}}{\mathrm{d} t} = \frac{\mathrm{d}^2 \mathbf{r}}{\mathrm{d} t^2 } } m s−2 L T−2
Jerk j Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j} = \frac{\mathrm{d} \mathbf{a}}{\mathrm{d} t} = \frac{\mathrm{d}^3 \mathbf{r}}{\mathrm{d} t^3} } m s−3 L T−3
Jounce s Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{s} = \frac{\mathrm{d} \mathbf{j}}{\mathrm{d} t} = \frac{\mathrm{d}^4 \mathbf{r}}{\mathrm{d} t^4} } m s−4 L T−4
Angular velocity ω Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\omega} = \mathbf{\hat{n}} \frac{ \mathrm{d} \theta }{\mathrm{d} t} } rad s−1 T−1
Angular Acceleration α Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\alpha} = \frac{\mathrm{d} \boldsymbol{\omega}}{\mathrm{d} t} = \mathbf{\hat{n}} \frac{\mathrm{d}^2 \theta}{\mathrm{d} t^2} } rad s−2 T−2
Angular jerk ζ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\zeta} = \frac{\mathrm{d} \boldsymbol{\alpha}}{\mathrm{d} t} = \mathbf{\hat{n}} \frac{ \mathrm{d}^3 \theta}{\mathrm{d} t^3} } rad s−3 T−3

Derived dynamic quantities

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Angular momenta of a classical object.

Left: intrinsic "spin" angular momentum S is really orbital angular momentum of the object at every point,

right: extrinsic orbital angular momentum L about an axis,

top: the moment of inertia tensor I and angular velocity ω (L is not always parallel to ω)[6]

bottom: momentum p and its radial position r from the axis.

The total angular momentum (spin + orbital) is J.
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Momentum p Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p} = m\mathbf{v} } kg m s−1 M L T−1
Force F Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F} = \mathrm{d} \mathbf{p}/\mathrm{d} t } N = kg m s−2 M L T−2
Impulse J, Δp, I Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{J} = \Delta \mathbf{p} = \int_{t_1}^{t_2} \mathbf{F} \, \mathrm{d} t } kg m s−1 M L T−1
Angular momentum about a position point r0, L, J, S Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L} = \left ( \mathbf{r} - \mathbf{r}_0 \right ) \times \mathbf{p} }

Most of the time we can set r0 = 0 if particles are orbiting about axes intersecting at a common point.

kg m2 s−1 M L2 T−1
Moment of a force about a position point r0,

Torque

τ, M Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\tau} = \left ( \mathbf{r} - \mathbf{r}_0 \right ) \times \mathbf{F} = \frac{\mathrm{d} \mathbf{L}}{\mathrm{d} t} } N m = kg m2 s−2 M L2 T−2
Angular impulse ΔL (no common symbol) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \mathbf{L} = \int_{t_1}^{t_2} \boldsymbol{\tau} \, \mathrm{d} t } kg m2 s−1 M L2 T−1

General energy definitions

Main article: Mechanical energy
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Mechanical work due to a Resultant Force W Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = \int_C \mathbf{F} \cdot \mathrm{d} \mathbf{r} } J = N m = kg m2 s−2 M L2 T−2
Work done ON mechanical system, Work done BY WON, WBY Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta W_\mathrm{ON} = - \Delta W_\mathrm{BY} } J = N m = kg m2 s−2 M L2 T−2
Potential energy φ, Φ, U, V, Ep Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta W = - \Delta V } J = N m = kg m2 s−2 M L2 T−2
Mechanical power P Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \frac{\mathrm{d}E}{\mathrm{d}t} } W = J s−1 M L2 T−3

Every conservative force has a potential energy. By following two principles one can consistently assign a non-relative value to U:

  • Wherever the force is zero, its potential energy is defined to be zero as well.
  • Whenever the force does work, potential energy is lost.

Generalized mechanics

Main articles: Analytical mechanics, Lagrangian mechanics, and Hamiltonian mechanics
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Generalized coordinates q, Q varies with choice varies with choice
Generalized velocities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{q},\dot{Q} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{q}\equiv \mathrm{d}q/\mathrm{d}t } varies with choice varies with choice
Generalized momenta p, P Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = \partial L /\partial \dot{q} } varies with choice varies with choice
Lagrangian L Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(\mathbf{q},\mathbf{\dot{q}},t) = T(\mathbf{\dot{q}}) - V(\mathbf{q},\mathbf{\dot{q}},t) }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q} = \mathbf{q}(t) } and p = p(t) are vectors of the generalized coords and momenta, as functions of time

J M L2 T−2
Hamiltonian H Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(\mathbf{p},\mathbf{q},t) = \mathbf{p}\cdot\mathbf{\dot{q}} - L(\mathbf{q},\mathbf{\dot{q}},t) } J M L2 T−2
Action, Hamilton's principal function S, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{\mathcal{S}} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{S} = \int_{t_1}^{t_2} L(\mathbf{q},\mathbf{\dot{q}},t) \mathrm{d}t } J s M L2 T−1

Kinematics

In the following rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to use θ, but this does not have to be the polar angle used in polar coordinate systems. The unit axial vector

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{\hat{n}} = \mathbf{\hat{e}}_r\times\mathbf{\hat{e}}_\theta }

defines the axis of rotation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{\hat{e}}_r } = unit vector in direction of r, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{\hat{e}}_\theta } = unit vector tangential to the angle.

Translation Rotation
Velocity Average:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_{\mathrm{average}} = {\Delta \mathbf{r} \over \Delta t}} Instantaneous:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} = {d\mathbf{r} \over dt}}

Angular velocityFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\omega} = \mathbf{\hat{n}}\frac{{\rm d} \theta}{{\rm d} t}} Rotating rigid body:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} = \boldsymbol{\omega} \times \mathbf{r} }
Acceleration Average:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}_{\mathrm{average}} = \frac{\Delta\mathbf{v}}{\Delta t} }

Instantaneous:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2} }

Angular acceleration

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\alpha} = \frac{{\rm d} \boldsymbol{\omega}}{{\rm d} t} = \mathbf{\hat{n}}\frac{{\rm d}^2 \theta}{{\rm d} t^2} }

Rotating rigid body:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a} = \boldsymbol{\alpha} \times \mathbf{r} + \boldsymbol{\omega} \times \mathbf{v} }

Jerk Average:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j}_{\mathrm{average}} = \frac{\Delta\mathbf{a}}{\Delta t} }

Instantaneous:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j} = \frac{d\mathbf{a}}{dt} = \frac{d^2\mathbf{v}}{dt^2} = \frac{d^3\mathbf{r}}{dt^3} }

Angular jerk

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\zeta} = \frac{{\rm d} \boldsymbol{\alpha}}{{\rm d} t} = \mathbf{\hat{n}}\frac{{\rm d}^2 \omega}{{\rm d} t^2} = \mathbf{\hat{n}}\frac{{\rm d}^3 \theta}{{\rm d} t^3} }

Rotating rigid body:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{j} = \boldsymbol{\zeta} \times \mathbf{r} + \boldsymbol{\alpha} \times \mathbf{a} }

Dynamics

Translation Rotation
Momentum Momentum is the "amount of translation"

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p} = m\mathbf{v}}

For a rotating rigid body:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p} = \boldsymbol{\omega} \times \mathbf{m} }

Angular momentum

Angular momentum is the "amount of rotation":

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{I} \cdot \boldsymbol{\omega} }

and the cross-product is a pseudovector i.e. if r and p are reversed in direction (negative), L is not.

In general I is an order-2 tensor, see above for its components. The dot · indicates tensor contraction.

Force and Newton's 2nd law Resultant force acts on a system at the center of mass, equal to the rate of change of momentum:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{F} & = \frac{d\mathbf{p}}{dt} = \frac{d(m\mathbf{v})}{dt} \\ & = m\mathbf{a} + \mathbf{v}\frac{{\rm d}m}{{\rm d}t} \\ \end{align} }

For a number of particles, the equation of motion for one particle i is:[7]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}\mathbf{p}_i}{\mathrm{d}t} = \mathbf{F}_{E} + \sum_{i \neq j} \mathbf{F}_{ij} }

where pi = momentum of particle i, Fij = force on particle i by particle j, and FE = resultant external force (due to any agent not part of system). Particle i does not exert a force on itself.

Torque

Torque τ is also called moment of a force, because it is the rotational analogue to force:[8]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\tau} = \frac{{\rm d}\mathbf{L}}{{\rm d}t} = \mathbf{r}\times\mathbf{F} = \frac{{\rm d}(\mathbf{I} \cdot \boldsymbol{\omega})}{{\rm d}t} }

For rigid bodies, Newton's 2nd law for rotation takes the same form as for translation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \boldsymbol{\tau} & = \frac{{\rm d}\mathbf{L}}{{\rm d}t} = \frac{{\rm d}(\mathbf{I}\cdot\boldsymbol{\omega})}{{\rm d}t} \\ & = \frac{{\rm d}\mathbf{I}}{{\rm d}t}\cdot\boldsymbol{\omega} + \mathbf{I}\cdot\boldsymbol{\alpha} \\ \end{align} }

Likewise, for a number of particles, the equation of motion for one particle i is:[9]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}\mathbf{L}_i}{\mathrm{d}t} = \boldsymbol{\tau}_E + \sum_{i \neq j} \boldsymbol{\tau}_{ij} }

Yank Yank is rate of change of force:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{Y} & = \frac{d\mathbf{F}}{dt} = \frac{d^2\mathbf{p}}{dt^2} = \frac{d^2(m\mathbf{v})}{dt^2} \\[1ex] & = m\mathbf{j} + \mathbf{2a}\frac{{\rm d}m}{{\rm d}t} + \mathbf{v}\frac{{\rm d^2}m}{{\rm d}t^2} \end{align} }

For constant mass, it becomes; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{Y} = m\mathbf{j}}

Rotatum

Rotatum Ρ is also called moment of a Yank, because it is the rotational analogue to yank:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\Rho} = \frac{{\rm d}\boldsymbol{\tau}}{{\rm d}t} = \mathbf{r}\times\mathbf{Y} = \frac{{\rm d}(\mathbf{I} \cdot \boldsymbol{\alpha})}{{\rm d}t} }

Impulse Impulse is the change in momentum:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \mathbf{p} = \int \mathbf{F} \, dt }

For constant force F:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \mathbf{p} = \mathbf{F} \Delta t }

Twirl/angular impulse is the change in angular momentum:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \mathbf{L} = \int \boldsymbol{\tau} \, dt }

For constant torque τ:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \mathbf{L} = \boldsymbol{\tau} \Delta t }

Precession

The precession angular speed of a spinning top is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\Omega} = \frac{wr}{I\boldsymbol{\omega}} }

where w is the weight of the spinning flywheel.

Energy

The mechanical work done by an external agent on a system is equal to the change in kinetic energy of the system:

General work-energy theorem (translation and rotation)

The work done W by an external agent which exerts a force F (at r) and torque τ on an object along a curved path C is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = \Delta T = \int_C \left ( \mathbf{F} \cdot \mathrm{d} \mathbf{r} + \boldsymbol{\tau} \cdot \mathbf{n} \, {\mathrm{d} \theta} \right ) }

where θ is the angle of rotation about an axis defined by a unit vector n.

Kinetic energy

The change in kinetic energy for an object initially traveling at speed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_0} and later at speed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} is: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E_k = W = \frac{1}{2} m(v^2 - {v_0}^2) }

Elastic potential energy

For a stretched spring fixed at one end obeying Hooke's law, the elastic potential energy is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E_p = \frac{1}{2} k(r_2-r_1)^2 }

where r2 and r1 are collinear coordinates of the free end of the spring, in the direction of the extension/compression, and k is the spring constant.

Euler's equations for rigid body dynamics

Main article: Euler's equations (rigid body dynamics)

Euler also worked out analogous laws of motion to those of Newton, see Euler's laws of motion. These extend the scope of Newton's laws to rigid bodies, but are essentially the same as above. A new equation Euler formulated is:[10]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{I} \cdot \boldsymbol{\alpha} + \boldsymbol{\omega} \times \left ( \mathbf{I} \cdot \boldsymbol{\omega} \right ) = \boldsymbol{\tau} }

where I is the moment of inertia tensor.

General planar motion

See also: Polar coordinate system (section: vector calculus)

The previous equations for planar motion can be used here: corollaries of momentum, angular momentum etc. can immediately follow by applying the above definitions. For any object moving in any path in a plane,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r} = \mathbf{r}(t) = r\hat\mathbf r }

the following general results apply to the particle.

Kinematics Dynamics
Position

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r} =\mathbf{r}\left ( r,\theta, t \right ) = r \hat\mathbf r }

Velocity

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} = \hat\mathbf r \frac{\mathrm{d} r}{\mathrm{d}t} + r \omega \hat\mathbf\theta }

Momentum

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p} = m \left(\hat\mathbf r \frac{\mathrm{d} r}{\mathrm{d}t} + r \omega \hat\mathbf\theta \right) }

Angular momenta Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L} = m \mathbf{r}\times \left(\hat\mathbf{r} \frac{\mathrm{d} r}{\mathrm{d}t} + r\omega\hat\mathbf\theta\right) }

Acceleration

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a} =\left ( \frac{\mathrm{d}^2 r}{\mathrm{d}t^2} - r\omega^2\right )\hat\mathbf r + \left ( r \alpha + 2 \omega \frac{\mathrm{d}r}{{\rm d}t} \right )\hat\mathbf\theta }

The centripetal force is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}_\bot = - m \omega^2 R \hat\mathbf r= - \omega^2 \mathbf{m} }

where again m is the mass moment, and the Coriolis force is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}_c = 2\omega m \frac{{\rm d}r}{{\rm d}t} \hat\mathbf\theta = 2\omega m v \hat\mathbf\theta }

The Coriolis acceleration and force can also be written:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}_c = m\mathbf{a}_c = -2 m \boldsymbol{ \omega \times v}}

Central force motion

For a massive body moving in a central potential due to another object, which depends only on the radial separation between the centers of masses of the two objects, the equation of motion is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2}{d\theta^2}\left(\frac{1}{\mathbf{r}}\right) + \frac{1}{\mathbf{r}} = -\frac{\mu\mathbf{r}^2}{\mathbf{l}^2}\mathbf{F}(\mathbf{r})}

Equations of motion (constant acceleration)

These equations can be used only when acceleration is constant. If acceleration is not constant then the general calculus equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).

Linear motion Angular motion
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v-v_0}=\mathbf at } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\omega - \omega_0} = \boldsymbol\alpha t }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x - x_0} = \tfrac{1}{2}(\mathbf{v_0+v})t } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\theta - \theta_0} = \tfrac{1}{2}(\boldsymbol{\omega_0 + \omega})t}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x - x_0} = \mathbf v_0t + \tfrac{1}{2}\mathbf at^2 } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\theta - \theta_0} = \boldsymbol\omega _0 t + \tfrac{1}{2} \boldsymbol\alpha t^2}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x - x_0} = \mathbf vt - \tfrac{1}{2}\mathbf at^2 } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\theta - \theta_0} = \boldsymbol\omega t - \tfrac{1}{2} \boldsymbol\alpha t^2}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x_{n^{th}} = \mathbf v_0 + \mathbf a(n-\tfrac{1}{2})} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol\theta_{n^{th}} =\boldsymbol\omega_0+\boldsymbol\alpha(n-\tfrac{1}{2})}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v^2 - v_0^2 = 2\mathbf{a(x-x_0)} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega^2 - \omega_0^2 = 2\boldsymbol{\alpha(\theta-\theta_0)}}
See also: Linear motion § Analogy between linear and rotational motion

Galilean frame transforms

For classical (Galileo-Newtonian) mechanics, the transformation law from one inertial or accelerating (including rotation) frame (reference frame traveling at constant velocity - including zero) to another is the Galilean transform.

Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative accelerations.

Motion of entities Inertial frames Accelerating frames
Translation

V = Constant relative velocity between two inertial frames F and F'.
A = (Variable) relative acceleration between two accelerating frames F and F'.

 Relative position

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}' = \mathbf{r} + \mathbf{V}t }

Relative velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}' = \mathbf{v} + \mathbf{V} }

Equivalent accelerations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}' = \mathbf{a} }

Relative accelerations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}' = \mathbf{a} + \mathbf{A} }

Apparent/fictitious forces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}' = \mathbf{F} - \mathbf{F}_\mathrm{app} }

Rotation

Ω = Constant relative angular velocity between two frames F and F'.
Λ = (Variable) relative angular acceleration between two accelerating frames F and F'.

Relative angular position

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta' = \theta + \Omega t } Relative velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\omega}' = \boldsymbol{\omega} + \boldsymbol{\Omega} }

Equivalent accelerations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\alpha}' = \boldsymbol{\alpha} }

Relative accelerations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\alpha}' = \boldsymbol{\alpha} + \boldsymbol{\Lambda} }

Apparent/fictitious torques Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\tau}' = \boldsymbol{\tau} - \boldsymbol{\tau}_\mathrm{app} }

Transformation of any vector T to a rotating frame

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{{\rm d}\mathbf{T}'}{{\rm d}t} = \frac{{\rm d}\mathbf{T}}{{\rm d}t} - \boldsymbol{\Omega} \times \mathbf{T} }

Mechanical oscillators

SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively.

Equations of motion
Physical situation Nomenclature Translational equations Angular equations
SHM
  • x = Transverse displacement
  • θ = Angular displacement
  • A = Transverse amplitude
  • Θ = Angular amplitude
 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = - \omega^2 x }

Solution: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = A \sin\left ( \omega t + \phi \right ) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}^2 \theta}{\mathrm{d}t^2} = - \omega^2 \theta }

Solution: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = \Theta \sin\left ( \omega t + \phi \right ) }

Unforced DHM
  • b = damping constant
  • κ = torsion constant
 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + b \frac{\mathrm{d}x}{\mathrm{d}t} + \omega^2 x = 0 }

Solution (see below for ω'): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=Ae^{-bt/2m}\cos\left ( \omega' \right )}

Resonant frequency: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_\mathrm{res} = \sqrt{\omega^2 - \left ( \frac{b}{4m} \right )^2 } }

Damping rate: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = b/m }

Expected lifetime of excitation: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau = 1/\gamma}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}^2 \theta}{\mathrm{d}t^2} + b \frac{\mathrm{d}\theta}{\mathrm{d}t} + \omega^2 \theta = 0 }

Solution: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta=\Theta e^{-\kappa t/2m}\cos\left ( \omega \right )}

Resonant frequency: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_\mathrm{res} = \sqrt{\omega^2 - \left ( \frac{\kappa}{4m} \right )^2 } }

Damping rate: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \kappa/m }

Expected lifetime of excitation: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau = 1/\gamma}

Angular frequencies
Physical situation Nomenclature Equations
Linear undamped unforced SHO
  • k = spring constant
  • m = mass of oscillating bob
 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \sqrt{\frac{k}{m}} }
Linear unforced DHO
  • k = spring constant
  • b = Damping coefficient
 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega' = \sqrt{\frac{k}{m}-\left ( \frac{b}{2m} \right )^2 } }
Low amplitude angular SHO
  • I = Moment of inertia about oscillating axis
  • κ = torsion constant
 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \sqrt{\frac{\kappa}{I}}}
Low amplitude simple pendulum
  • L = Length of pendulum
  • g = Gravitational acceleration
  • Θ = Angular amplitude
 Approximate value

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \sqrt{\frac{g}{L}}}

Exact value can be shown to be: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \sqrt{\frac{g}{L}} \left [ 1 + \sum_{k=1}^\infty \frac{\prod_{n=1}^k \left ( 2n-1 \right )}{\prod_{n=1}^m \left ( 2n \right )} \sin^{2n} \Theta \right ]}

Energy in mechanical oscillations
Physical situation Nomenclature Equations
SHM energy
  • T = kinetic energy
  • U = potential energy
  • E = total energy
 Potential energy

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = \frac{m}{2} \left ( x \right )^2 = \frac{m \left( \omega A \right )^2}{2} \cos^2(\omega t + \phi)} Maximum value at x = A: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_\mathrm{max} = \frac{m}{2} \left ( \omega A \right )^2 }

Kinetic energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T = \frac{\omega^2 m}{2} \left ( \frac{\mathrm{d} x}{\mathrm{d} t} \right )^2 = \frac{m \left ( \omega A \right )^2}{2}\sin^2\left ( \omega t + \phi \right )}

Total energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = T + U }

DHM energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \frac{m \left ( \omega A \right )^2}{2}e^{-bt/m} }

See also

Notes

  1. ↑ Mayer, Sussman & Wisdom 2001, p. xiii
  2. ↑ Berkshire & Kibble 2004, p. 1
  3. ↑ Berkshire & Kibble 2004, p. 2
  4. ↑ Arnold 1989, p. v
  5. ↑ "Section: Moments and center of mass".
  6. ↑ R.P. Feynman; R.B. Leighton; M. Sands (1964). Feynman's Lectures on Physics (volume 2). Addison-Wesley. pp. 31–7. ISBN 978-0-201-02117-2.
  7. ↑ "Relativity, J.R. Forshaw 2009"
  8. ↑ "Mechanics, D. Kleppner 2010"
  9. ↑ "Relativity, J.R. Forshaw 2009"
  10. ↑ "Relativity, J.R. Forshaw 2009"

References

Template:Classical mechanics derived SI units

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