7.8: Summary of Linear and Angular Analogs
- Page ID
- 54128
Putting it all together
The chart on below shows all of the linear motion and dynamic variables along with their rotational counterparts. Keep this chart out and handy for ready reference to help you from getting “lost” in all the symbols. You should make sure that you recognize the meaning behind the symbols when you see on of these relationships.
Category | Concept | Translation | Rotation | Relation |
---|---|---|---|---|
Kinematic Variables |
Position Velocity Acceleration |
\(x\) \(v=\dfrac{dx}{dt}\) \(a=\dfrac{dv}{dt}\) |
\(\theta\) \(\omega=\frac{d\theta}{dt}\) \(\alpha=\frac{d\omega}{dt}\) |
\(\theta=\frac{s}{r}\) \(\omega=\frac{v}{r}\) \(\alpha=\frac{a}{r}\) |
Fundamental Dynamic Variables |
Force/Torque Mass/Inertia Momentum Impulse Momentum-Impulse |
\(F\) \(m\) \(p=mv\) \(J=\int Fdt\) \(J_{ext}=\Delta p\) |
\(\tau\) \(I\) \(L=I\omega\) \(\textrm{ang}~J=\int \tau dt\) \(\textrm{ang}~ J_{ext}=\Delta L\) |
\(\tau=rF_{\perp}\) \(I=\sum mr^2\) \(L=rp_{\perp}\)
|
Newton's Laws |
First Law Second Law Third Law |
\(\textrm{if}~ F_{net}=0, \textrm{then}~\Delta p=0\) \(F_{net}=ma ~\textrm{or}~ F_{net}=\frac{dp}{dt}\) \(F_{1~on~2}=-F_{2~on~1}\) |
\(\textrm{if}~ \tau_{net}=0, \textrm{then}~\Delta L=0\) \(\tau_{net}=I\alpha ~\textrm{or}~\tau_{net}=\frac{dL}{dt}\) \(\tau_{1~on~2}=-\tau_{2~on~1}\) |
|
Energy |
Kinetic Energy Work
|
\(KE=\frac{1}{2}mv^2\) \(W=\int \limits_{x_1}^{x_2} \vec F \cdot d\vec s\) |
\(KE=\frac{1}{2}I\omega^2\) \(W=\int \limits_{\theta_1}^{\theta_2} \vec\tau \cdot d\vec\theta\) |