List of Equations in Classical Mechanics - Dynamics

Dynamics

	Translation	Rotation
Momentum	Momentum is the "amount of translation" For a rotating rigid body:	Angular momentum Angular momentum is the "amount of rotation": 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: $\begin{align} \mathbf{F} & = \frac{d\mathbf{p}}{dt} = \frac{d(m\mathbf{v})}{dt} \\ & = m\mathbf{a} + \mathbf{v}\frac{{\rm d}m}{{\rm d}t} \\ \end{align} \,\!$ For a number of particles, the equation of motion for one particle i is: where p_i = momentum of particle i, F_ij = force on particle i by particle j, and F_E = 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, becuause it is the rotational analogue to force: For rigid bodies, Newton's 2nd law for rotation takes the same form as for translation: $\begin{align} \boldsymbol{\tau} & = \frac{{\rm d}\bold{L}}{{\rm d}t} = \frac{{\rm d}(\bold{I}\cdot\boldsymbol{\omega})}{{\rm d}t} \\ & = \frac{{\rm d}\bold{I}}{{\rm d}t}\cdot\boldsymbol{\omega} + \bold{I}\cdot\boldsymbol{\alpha} \\ \end{align} \,\!$ Likewise, for a number of particles, the equation of motion for one particle i is:
Yank	Yank is rate of change of force: $\begin{align} \mathbf{Y} & = \frac{d\mathbf{F}}{dt} = \frac{d^2\mathbf{p}}{dt^2} = \frac{d^2(m\mathbf{v})}{dt^2} \\ & = m\mathbf{j} + \mathbf{2a}\frac{{\rm d}m}{{\rm d}t} + \mathbf{v}\frac{{\rm d^2}m}{{\rm d}t^2} \\ \end{align} \,\!$ For constant mass, it becomes;	Rotatum Rotatum Ρ is also called moment of a Yank, becuause it is the rotational analogue to yank:
Impulse	Impulse is the change in momentum: For constant force F:	Angular impulse is the change in angular momentum: For constant torque τ:

Translation

Rotation

Momentum

Momentum is the "amount of translation"

For a rotating rigid body:

Angular momentum

Angular momentum is the "amount of rotation":

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:

$\begin{align} \mathbf{F} & = \frac{d\mathbf{p}}{dt} = \frac{d(m\mathbf{v})}{dt} \\ & = m\mathbf{a} + \mathbf{v}\frac{{\rm d}m}{{\rm d}t} \\ \end{align} \,\!$

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

where p_i = momentum of particle i, F_ij = force on particle i by particle j, and F_E = 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, becuause it is the rotational analogue to force:

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

$\begin{align} \boldsymbol{\tau} & = \frac{{\rm d}\bold{L}}{{\rm d}t} = \frac{{\rm d}(\bold{I}\cdot\boldsymbol{\omega})}{{\rm d}t} \\ & = \frac{{\rm d}\bold{I}}{{\rm d}t}\cdot\boldsymbol{\omega} + \bold{I}\cdot\boldsymbol{\alpha} \\ \end{align} \,\!$

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

Yank

Yank is rate of change of force:

$\begin{align} \mathbf{Y} & = \frac{d\mathbf{F}}{dt} = \frac{d^2\mathbf{p}}{dt^2} = \frac{d^2(m\mathbf{v})}{dt^2} \\ & = m\mathbf{j} + \mathbf{2a}\frac{{\rm d}m}{{\rm d}t} + \mathbf{v}\frac{{\rm d^2}m}{{\rm d}t^2} \\ \end{align} \,\!$

For constant mass, it becomes;

Rotatum

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

Impulse

Impulse is the change in momentum:

For constant force F:

Angular impulse is the change in angular momentum:

For constant torque τ:

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Famous quotes containing the word dynamics:

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—Cathy Rindner Tempelsman (20th century)

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