Rotational motion deals with objects rotating about an axis. It’s analogous to linear motion with different physical quantities.
Overview
graph TD
A[Rotational Motion] --> B[Kinematics]
A --> C[Dynamics]
A --> D[Rolling Motion]
C --> C1[Torque]
C --> C2[Moment of Inertia]
C --> C3[Angular Momentum]Linear vs Rotational Quantities
| Linear | Rotational | Relation |
|---|---|---|
| Displacement $s$ | Angular displacement $\theta$ | $s = r\theta$ |
| Velocity $v$ | Angular velocity $\omega$ | $v = r\omega$ |
| Acceleration $a$ | Angular acceleration $\alpha$ | $a_t = r\alpha$ |
| Mass $m$ | Moment of inertia $I$ | - |
| Force $F$ | Torque $\tau$ | $\tau = r \times F$ |
| Momentum $p$ | Angular momentum $L$ | $L = r \times p$ |
| $F = ma$ | $\tau = I\alpha$ | - |
| $KE = \frac{1}{2}mv^2$ | $KE = \frac{1}{2}I\omega^2$ | - |
Rotational Kinematics
For constant angular acceleration:
$$\omega = \omega_0 + \alpha t$$ $$\theta = \omega_0 t + \frac{1}{2}\alpha t^2$$ $$\omega^2 = \omega_0^2 + 2\alpha\theta$$Centre of Mass
$$\boxed{\vec{r}_{cm} = \frac{\sum m_i\vec{r}_i}{\sum m_i}}$$For continuous body:
$$\vec{r}_{cm} = \frac{\int \vec{r}\,dm}{\int dm}$$Properties
- Total external force causes acceleration of CM
- If no external force, CM moves with constant velocity
- Internal forces don’t affect CM motion
Moment of Inertia
$$\boxed{I = \sum m_i r_i^2 = \int r^2\,dm}$$Standard Moments of Inertia
| Body | Axis | Moment of Inertia |
|---|---|---|
| Point mass | Distance r | $mr^2$ |
| Rod (length L) | Through center, perpendicular | $\frac{mL^2}{12}$ |
| Rod | Through end, perpendicular | $\frac{mL^2}{3}$ |
| Ring | Through center, perpendicular | $mR^2$ |
| Disc | Through center, perpendicular | $\frac{mR^2}{2}$ |
| Solid sphere | Through center | $\frac{2mR^2}{5}$ |
| Hollow sphere | Through center | $\frac{2mR^2}{3}$ |
| Cylinder | Through axis | $\frac{mR^2}{2}$ |
Radius of Gyration
$$I = mk^2 \Rightarrow k = \sqrt{\frac{I}{m}}$$Theorems
Parallel Axis Theorem:
$$\boxed{I = I_{cm} + md^2}$$Perpendicular Axis Theorem (for planar bodies):
$$\boxed{I_z = I_x + I_y}$$Torque
$$\boxed{\vec{\tau} = \vec{r} \times \vec{F}}$$Magnitude: $\tau = rF\sin\theta = r_\perp F = rF_\perp$
Newton’s Second Law (Rotation)
$$\boxed{\tau = I\alpha}$$Angular Momentum
$$\boxed{\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}}$$Conservation of Angular Momentum
If $\tau_{ext} = 0$:
$$\boxed{L = I\omega = \text{constant}}$$Applications:
- Ice skater spinning
- Diver tucking
- Angular velocity of planets
Relation with Torque
$$\tau = \frac{dL}{dt}$$Rotational Kinetic Energy
$$\boxed{KE_{rot} = \frac{1}{2}I\omega^2}$$Rolling Motion
Rolling = Translation + Rotation
Pure Rolling Condition
$$\boxed{v_{cm} = R\omega}$$(No slipping at contact point)
Kinetic Energy in Rolling
$$KE_{total} = KE_{trans} + KE_{rot} = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I\omega^2$$For rolling: $KE = \frac{1}{2}mv^2\left(1 + \frac{k^2}{R^2}\right)$
Acceleration on Incline (Pure Rolling)
$$\boxed{a = \frac{g\sin\theta}{1 + \frac{I}{mR^2}} = \frac{g\sin\theta}{1 + \frac{k^2}{R^2}}}$$Time to reach bottom of incline (length L):
$$t = \sqrt{\frac{2L(1 + k^2/R^2)}{g\sin\theta}}$$Comparison of Rolling Objects
| Object | $I/mR^2$ | Acceleration | Speed at bottom |
|---|---|---|---|
| Ring/Hoop | 1 | $\frac{g\sin\theta}{2}$ | Slowest |
| Disc/Cylinder | 1/2 | $\frac{2g\sin\theta}{3}$ | Medium |
| Solid Sphere | 2/5 | $\frac{5g\sin\theta}{7}$ | Fastest |
Practice Problems
Find the moment of inertia of a disc about a tangent in its plane.
A uniform rod of mass 2 kg and length 1 m rotates about one end. Find angular momentum when ω = 10 rad/s.
A disc and ring of same mass and radius roll down an incline. Which reaches bottom first?
A spinning top has angular momentum 10 kg⋅m²/s. If moment of inertia is 0.5 kg⋅m², find angular velocity.