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${\displaystyle m=\iint \sigma \,\mathrm {d} S}$

${\displaystyle m=\iiint \rho \,\mathrm {d} V}$

$${\displaystyle \mathbf {m} =\mathbf {r} m}$$

Discrete masses about an axis ${\displaystyle x_{i}}$: $${\displaystyle \mathbf {m} =\sum _{i=1}^{N}\mathbf {r} _{i}m_{i}}$$

Continuum of mass about an axis ${\displaystyle x_{i}}$: $${\displaystyle \mathbf {m} =\int \rho \left(\mathbf {r} \right)x_{i}\mathrm {d} \mathbf {r} }$$

(Symbols vary)

Discrete masses: $${\displaystyle \mathbf {r} _{\mathrm {com} }={\frac {1}{M}}\sum _{i}\mathbf {r} _{i}m_{i}={\frac {1}{M}}\sum _{i}\mathbf {m} _{i}}$$

Mass continuum: $${\displaystyle \mathbf {r} _{\mathrm {com} }={\frac {1}{M}}\int \mathrm {d} \mathbf {m} ={\frac {1}{M}}\int \mathbf {r} \,\mathrm {d} m={\frac {1}{M}}\int \mathbf {r} \rho \,\mathrm {d} V}$$

$${\displaystyle I=\sum _{i}\mathbf {m} _{i}\cdot \mathbf {r} _{i}=\sum _{i}\left|\mathbf {r} _{i}\right|^{2}m}$$

Mass continuum: $${\displaystyle I=\int \left|\mathbf {r} \right|^{2}\mathrm {d} m=\int \mathbf {r} \cdot \mathrm {d} \mathbf {m} =\int \left|\mathbf {r} \right|^{2}\rho \,\mathrm {d} V}$$

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

Torque

where ${\displaystyle \mathbf {q} =\mathbf {q} (t)}$ and p = p(t) are vectors of the generalized coords and momenta, as functions of time

$${\displaystyle \mathbf {v} _{\mathrm {average} }={\Delta \mathbf {r} \over \Delta t}}$$ Instantaneous:

$${\displaystyle \mathbf {v} ={d\mathbf {r} \over dt}}$$

$${\displaystyle \mathbf {a} _{\mathrm {average} }={\frac {\Delta \mathbf {v} }{\Delta t}}}$$

Instantaneous:

$${\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}}$$

$${\displaystyle {\boldsymbol {\alpha }}={\frac {{\rm {d}}{\boldsymbol {\omega }}}{{\rm {d}}t}}=\mathbf {\hat {n}} {\frac {{\rm {d}}^{2}\theta }{{\rm {d}}t^{2}}}}$$

Rotating rigid body:

$${\displaystyle \mathbf {a} ={\boldsymbol {\alpha }}\times \mathbf {r} +{\boldsymbol {\omega }}\times \mathbf {v} }$$

$${\displaystyle \mathbf {j} _{\mathrm {average} }={\frac {\Delta \mathbf {a} }{\Delta t}}}$$

Instantaneous:

$${\displaystyle \mathbf {j} ={\frac {d\mathbf {a} }{dt}}={\frac {d^{2}\mathbf {v} }{dt^{2}}}={\frac {d^{3}\mathbf {r} }{dt^{3}}}}$$

$${\displaystyle {\boldsymbol {\zeta }}={\frac {{\rm {d}}{\boldsymbol {\alpha }}}{{\rm {d}}t}}=\mathbf {\hat {n}} {\frac {{\rm {d}}^{2}\omega }{{\rm {d}}t^{2}}}=\mathbf {\hat {n}} {\frac {{\rm {d}}^{3}\theta }{{\rm {d}}t^{3}}}}$$

Rotating rigid body:

$${\displaystyle \mathbf {j} ={\boldsymbol {\zeta }}\times \mathbf {r} +{\boldsymbol {\alpha }}\times \mathbf {a} }$$

$${\displaystyle \mathbf {p} =m\mathbf {v} }$$

For a rotating rigid body:

$${\displaystyle \mathbf {p} ={\boldsymbol {\omega }}\times \mathbf {m} }$$

Angular momentum is the "amount of rotation":

$${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {I} \cdot {\boldsymbol {\omega }}}$$

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.

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

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

$${\displaystyle {\frac {\mathrm {d} \mathbf {p} _{i}}{\mathrm {d} t}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}}$$

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

$${\displaystyle {\boldsymbol {\tau }}={\frac {{\rm {d}}\mathbf {L} }{{\rm {d}}t}}=\mathbf {r} \times \mathbf {F} ={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\omega }})}{{\rm {d}}t}}}$$

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

$${\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&={\frac {{\rm {d}}\mathbf {L} }{{\rm {d}}t}}={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\omega }})}{{\rm {d}}t}}\\&={\frac {{\rm {d}}\mathbf {I} }{{\rm {d}}t}}\cdot {\boldsymbol {\omega }}+\mathbf {I} \cdot {\boldsymbol {\alpha }}\\\end{aligned}}}$$

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

$${\displaystyle {\frac {\mathrm {d} \mathbf {L} _{i}}{\mathrm {d} t}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}}$$

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

For constant mass, it becomes; $${\displaystyle \mathbf {Y} =m\mathbf {j} }$$

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

$${\displaystyle {\boldsymbol {\mathrm {P} }}={\frac {{\rm {d}}{\boldsymbol {\tau }}}{{\rm {d}}t}}=\mathbf {r} \times \mathbf {Y} ={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\alpha }})}{{\rm {d}}t}}}$$

$${\displaystyle \Delta \mathbf {p} =\int \mathbf {F} \,dt}$$

For constant force F:

$${\displaystyle \Delta \mathbf {p} =\mathbf {F} \Delta t}$$

$${\displaystyle \Delta \mathbf {L} =\int {\boldsymbol {\tau }}\,dt}$$

For constant torque τ:

$${\displaystyle \Delta \mathbf {L} ={\boldsymbol {\tau }}\Delta t}$$

$${\displaystyle \mathbf {r} =\mathbf {r} \left(r,\theta ,t\right)=r{\hat {\mathbf {r} }}}$$

$${\displaystyle \mathbf {v} ={\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}}$$

$${\displaystyle \mathbf {p} =m\left({\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}\right)}$$

Angular momenta $${\displaystyle \mathbf {L} =m\mathbf {r} \times \left({\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}\right)}$$

$${\displaystyle \mathbf {a} =\left({\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\omega ^{2}\right){\hat {\mathbf {r} }}+\left(r\alpha +2\omega {\frac {\mathrm {d} r}{{\rm {d}}t}}\right){\hat {\mathbf {\theta } }}}$$

$${\displaystyle \mathbf {F} _{\bot }=-m\omega ^{2}R{\hat {\mathbf {r} }}=-\omega ^{2}\mathbf {m} }$$

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

$${\displaystyle \mathbf {F} _{c}=2\omega m{\frac {{\rm {d}}r}{{\rm {d}}t}}{\hat {\mathbf {\theta } }}=2\omega mv{\hat {\mathbf {\theta } }}}$$

The Coriolis acceleration and force can also be written:

$${\displaystyle \mathbf {F} _{c}=m\mathbf {a} _{c}=-2m{\boldsymbol {\omega \times v}}}$$

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

$${\displaystyle \mathbf {r} '=\mathbf {r} +\mathbf {V} t}$$

Relative velocity $${\displaystyle \mathbf {v} '=\mathbf {v} +\mathbf {V} }$$

Equivalent accelerations $${\displaystyle \mathbf {a} '=\mathbf {a} }$$

$${\displaystyle \mathbf {a} '=\mathbf {a} +\mathbf {A} }$$

Apparent/fictitious forces $${\displaystyle \mathbf {F} '=\mathbf {F} -\mathbf {F} _{\mathrm {app} }}$$

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

$${\displaystyle \theta '=\theta +\Omega t}$$ Relative velocity $${\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}+{\boldsymbol {\Omega }}}$$

Equivalent accelerations $${\displaystyle {\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}}$$

$${\displaystyle {\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}+{\boldsymbol {\Lambda }}}$$

Apparent/fictitious torques $${\displaystyle {\boldsymbol {\tau }}'={\boldsymbol {\tau }}-{\boldsymbol {\tau }}_{\mathrm {app} }}$$

$${\displaystyle {\frac {{\rm {d}}\mathbf {T} '}{{\rm {d}}t}}={\frac {{\rm {d}}\mathbf {T} }{{\rm {d}}t}}-{\boldsymbol {\Omega }}\times \mathbf {T} }$$

• x = Transverse displacement
• θ = Angular displacement
• A = Transverse amplitude
• Θ = Angular amplitude

Solution: $${\displaystyle x=A\sin \left(\omega t+\phi \right)}$$

Solution: $${\displaystyle \theta =\Theta \sin \left(\omega t+\phi \right)}$$

• b = damping constant
• κ = torsion constant

Solution (see below for ω'): $${\displaystyle x=Ae^{-bt/2m}\cos \left(\omega '\right)}$$

Resonant frequency: $${\displaystyle \omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {b}{4m}}\right)^{2}}}}$$

Damping rate: $${\displaystyle \gamma =b/m}$$

Expected lifetime of excitation: $${\displaystyle \tau =1/\gamma }$$

Solution: $${\displaystyle \theta =\Theta e^{-\kappa t/2m}\cos \left(\omega \right)}$$

Resonant frequency: $${\displaystyle \omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {\kappa }{4m}}\right)^{2}}}}$$

Damping rate: $${\displaystyle \gamma =\kappa /m}$$

Expected lifetime of excitation: $${\displaystyle \tau =1/\gamma }$$

• k = spring constant
• m = mass of oscillating bob

• k = spring constant
• b = Damping coefficient

• I = Moment of inertia about oscillating axis
• κ = torsion constant

• L = Length of pendulum
• g = Gravitational acceleration
• Θ = Angular amplitude

$${\displaystyle \omega ={\sqrt {\frac {g}{L}}}}$$

Exact value can be shown to be: $${\displaystyle \omega ={\sqrt {\frac {g}{L}}}\left}$$

• T = kinetic energy
• U = potential energy
• E = total energy

$${\displaystyle U={\frac {m}{2}}\left(x\right)^{2}={\frac {m\left(\omega A\right)^{2}}{2}}\cos ^{2}(\omega t+\phi )}$$ Maximum value at x = A: $${\displaystyle U_{\mathrm {max} }{\frac {m}{2}}\left(\omega A\right)^{2}}$$

Kinetic energy $${\displaystyle T={\frac {\omega ^{2}m}{2}}\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}={\frac {m\left(\omega A\right)^{2}}{2}}\sin ^{2}\left(\omega t+\phi \right)}$$

Total energy $${\displaystyle E=T+U}$$