Rotational Mechanics
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Chapter 1 of PC5131 is Rotational Mechanics, which is problematic. The topic is just a scam, since it is just all of Mechanics but with Angular (not the painful JS Framework). Surmised below is a list of equations which can probably help one understand Rotational Mechanics in the context of normal Mechanical concepts. Good Luck.
Translational and Rotational Counterparts
It's just everything we have already learnt, but they added angular in front of it. So let's just walk through a list of equations, and their rotational counterparts.
| Translational | Rotational |
|---|---|
| \(v_f = v_i + at\) | \(\omega_f = \omega_i + \alpha t\) |
| \(\Delta x = v_i t + \frac{1}{2} at^2 = \frac{v_f+v_i}{2}t\) | \(\Delta\theta=\omega_it+\frac{1}{2}\alpha t^2 = \frac{\omega_i+\omega_f}{2}t\) |
| \(v_f^2 = v_i^2 + 2a\Delta x\) | \(\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta\) |
| \(F_{net} = ma\) | \(\tau_{net} = I\alpha\) |
| \(p = mv\) | \(L = I\omega\) |
| \(J = F\Delta t\) | \(\Delta L = \tau\Delta t\) |
| \(W = F \cdot s\) | \(W = \tau \cdot \Delta\theta\) |
| \(E_K = \frac{1}{2}mv^2\) | \(E_L = \frac{1}{2}I\omega^2\) |
| \(P = \frac{W}{t} = F \cdot v\) | \(P = \tau \cdot \omega\) |
Moment of Inertia, \(I\)
Do note that \(I\) is slightly more cursed, and the following table demonstrates the definition of \(I\):
| Situation | \(I\) | \(\beta\) |
|---|---|---|
| Moment of Inertia off from \(I_{cm}\) | \(I = I_{cm} + Md^2\) | - |
| Moment of Inertia of Multiple Parts | \(I = \sum I_{part}\) | - |
| Hoop (from the centre) | \(I = MR^2\) | \(1\) |
| Circular Disc (from the centre) | \(I = \frac{1}{2}MR^2\) | \(\frac{1}{2}\) |
| Circular Disc (through diameter) | \(I = \frac{1}{4}MR^2\) | \(\frac{1}{4}\) |
| Long Rod (from the centre) | \(I = \frac{1}{12}ML^2\) | \(\frac{1}{12}\) |
| Long Rod (from end) | \(I = \frac{1}{3}ML^2\) | \(\frac{1}{3}\) |
| Solid Sphere | \(I = \frac{2}{5}MR^2\) | \(\frac{2}5\) |
| Hollow Sphere | \(I = \frac{2}{3} MR^2\) | \(\frac{2}3\) |
Shifting from Translational to Rotational
We also note that we can relate all the quantities to each other via the variable \(r\), as shown below:
| Translational | Rotational |
|---|---|
| \(\Delta x\) | \(r\Delta\theta\) |
| \(v\) | \(r\omega\) |
| \(a\) | \(r\alpha\) |
| \(I\) | \(\beta mr^2\) |
| \(\vec F \times \vec{r}\) | \(\vec \tau\) |
| \(\vec r \times \vec p\) | \(\vec L\) |
Conservation Cases
Cases to be considered for collisions:
| Situation | Translational | Rotational |
|---|---|---|
| Elastic Collision | COE, COM applies | COE, COAM applies |
| Inelastic Collison | COM applies, COE does not | COAM applies, COE does not |
| Completely Inelastic Collision (objects stick together) | COM applies, COE does not | COAM applies, COE does not |
| Some force acting throughout the motion (e.g. Gravity) | COE applies, COM does not | COE applies, COAM does not |
Ramp
| Name | Quantity |
|---|---|
| Rotational Kinetic Energy | \(K_{rot} = \frac{\beta}{1+\beta}mgh\) |
| Translational Kinetic Energy | \(K_{trans} = \frac{1}{1+\beta}mgh\) |
| Velocity of Centre of Mass | \(v_{cm} = \sqrt{\frac{2gh}{1 + \beta}}\) |
| Acceleration of Centre of Mass | \(a_{cm} = \frac{v_{cm}^2}{2\Delta x} = \frac{g\sin\theta}{1+\beta}\) |
| Velocity at Top of Loop-de-Loop (Radius \(R\)) | \(v_{loop} = \sqrt{g(R-r)}\) |
| Minimum height, \(h\) for Loop-de-Loop | \(h = \frac{5+\beta}{2}(R-r)\) |
Loop-de-Loop
Loop-de-Loop for normal circular motion:
For rigid body of radius \(r\):