Subsection 7.3.1 Rotational Overview
The table below summarizes some of the physical quantities as rotational anologue to a translation motion.
| Linear Motion | Unit | Rotational Motion | Unit | Relation |
|---|---|---|---|---|
| \(\vec{F}=m\vec{a}\) | \(N\) | \(\vec{\Gamma}=I\vec{\alpha}\) | \(Nm\) | \(\vec{\Gamma} =\vec{r}\times\vec{F}\) |
| \(v_{f}=v_{i}+a t\) | \(m/s\) | \(\omega_{f}=\omega_{i} +\alpha t\) | \(rad/s \) | \(\vec{v}=\vec{r}\times \vec{\omega}\) |
| \(v^{2}_{f}=v^{2}_{i}+2a s\) | \(\omega^{2}_{f}=\omega^{2}_{i}+2\alpha\theta\) | \(a_{t}=r\alpha\) | ||
| \(s=v_{i}t+\frac{1}{2}at^{2}\) | \(m\) | \(\theta=\omega_{i}t+\frac{1}{2}\alpha t^{2}\) | \(rad\) | \(s=r\theta\) |
| \(P =\frac{W}{t} = \frac{\vec{F}\cdot \vec{s}}{t} = \vec{F}\cdot\vec{v}\) | \(Watt \) | \(P =\frac{W}{t}=\frac{\vec{\Gamma}\cdot \vec{\theta}}{t} = \vec{\Gamma}\cdot\vec{\omega}\) | W | |
| Mass (M) | kg | Inertia (I) | \(kgm^{2}\) | \(I=Mr^{2}\) |
| p=mv | kgm/s | \(L=I\omega\) | \(kgm^{2}/s \) | \vec{L}=\vec{r}\times\vec{p} |
| \(KE=\frac{1}{2}mv^{2} \) | Joule | \(KE=\frac{1}{2}I\omega^{2}\) | Joule | \(a_{c} =a_{r}=\frac{v^{2}}{r}\) |
| \(\Delta p=F\Delta t=\mathscr{I}\) | \(Ns \) | \(\Delta L=\Gamma \Delta t\) | Nms | |
| \(W=F s\) | \(J\) | \(W=\Gamma\theta\) | J | |
| \(F=\frac{\Delta p}{\Delta t}\) | \(\Gamma=\frac{\Delta L}{\Delta t}\) |
Although the mathematical forms of the equations are quite similar, rotational dynamics differs from linear dynamics in the following respects. It applies to systems consisting of a single rotating object rather than a point particle; The motion of a body is due to rotational velocity rather than the translational velocity; and the agent which cause the motion is the net torque not the net force. In rotational motion axis of rotation must be specified to characterize the rotational motion.
