Rotatum

In physics, rotatum is the derivative of torque with respect to time. Expressed as an equation, rotatum Ρ is:

${\displaystyle {\vec {P}}={\frac {d{\vec {\tau }}}{dt}}}$

where τ is torque and ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}}$ is the derivative with respect to time ${\displaystyle t}$.

The term rotatum is not universally recognized but is commonly used. This word is derived from the Latin word rotātus meaning to rotate.[citation needed] The units of rotatum are force times distance per time, or equivalently, mass times length squared per time cubed; in the SI unit system this is kilogram metre squared per second cubed (kg·m2/s3), or Newtons times meter per second (N·m/s).

Relation to other physical quantities

Newton's second law for angular motion says that:

${\displaystyle \mathbf {\tau } ={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}}$

where L is angular momentum, so if we combine the above two equations:

${\displaystyle \mathbf {\mathrm {P} } ={\frac {\mathrm {d} \mathbf {\tau } }{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}\right)={\frac {\mathrm {d} ^{2}\mathbf {L} }{\mathrm {d} t^{2}}}={\frac {\mathrm {d} ^{2}(I\cdot \mathbf {\omega } )}{\mathrm {d} t^{2}}}}$

where ${\displaystyle I}$ is moment of Inertia and ${\displaystyle \omega }$ is angular velocity. If the moment of inertia is not changing over time (i.e. it is constant), then:

${\displaystyle \mathbf {\mathrm {P} } =I{\frac {\mathrm {d} ^{2}\omega }{\mathrm {d} t^{2}}}}$

which can also be written as:

${\displaystyle \mathbf {\mathrm {P} } =I\zeta }$

where ς is Angular jerk.