Chapter 5 – Rotational Motion (JEE Physics)
1. Introduction to Rotational Motion
Rotational motion is the motion of a rigid body about a fixed axis.
Every point of the body moves in a circle whose center lies on the axis of rotation.
Examples: rotating wheel, ceiling fan, Earth’s rotation.
2. Rigid Body
A rigid body is a system of particles in which the distance between any two particles remains constant during motion.
3. Angular Displacement
Angular displacement:
$$\theta = \frac{s}{r}$$
- SI unit: radian
- Dimensionless quantity
4. Angular Velocity
$$\omega = \frac{d\theta}{dt}$$
Relation with linear velocity:
$$v = r\omega$$
5. Angular Acceleration
$$\alpha = \frac{d\omega}{dt}$$
Relation with linear acceleration:
$$a_t = r\alpha,\quad a_c = r\omega^2$$
6. Equations of Rotational Motion
$$\omega = \omega_0 + \alpha t$$
$$\theta = \omega_0 t + \frac{1}{2}\alpha t^2$$
$$\omega^2 = \omega_0^2 + 2\alpha\theta$$
7. Torque
Torque is the turning effect of a force.
$$\vec{\tau} = \vec{r} \times \vec{F}$$
Magnitude:
$$\tau = rF\sin\theta$$
8. Moment of Inertia (MOI)
Moment of inertia is the rotational analogue of mass.
$$I = \sum m_ir_i^2$$
9. Radius of Gyration
$$I = Mk^2$$
$k$ represents how mass is distributed relative to axis.
10. Standard Moments of Inertia
| Body | MOI |
|---|---|
| Rod (center) | $\frac{ML^2}{12}$ |
| Rod (end) | $\frac{ML^2}{3}$ |
| Solid sphere | $\frac{2}{5}MR^2$ |
| Ring | $MR^2$ |
11. Parallel Axis Theorem
$$I = I_{cm} + Md^2$$
12. Perpendicular Axis Theorem
$$I_z = I_x + I_y$$
Applicable only for planar bodies.
13. Angular Momentum
$$\vec{L} = \vec{r} \times \vec{p}$$
For rigid body:
$$L = I\omega$$
14. Relation Between Torque and Angular Momentum
$$\vec{\tau} = \frac{d\vec{L}}{dt}$$
15. Conservation of Angular Momentum
If net external torque on a system is zero, angular momentum remains constant.
$$I_1\omega_1 = I_2\omega_2$$
Classic example: skater pulling arms inward.
16. Rotational Kinetic Energy
$$K = \frac{1}{2}I\omega^2$$
17. Rolling Motion
Rolling motion is a combination of translational and rotational motion.
$$v_{cm} = R\omega$$
18. Kinetic Energy in Rolling
$$K = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2$$
19. Rolling Without Slipping
Condition:
$$v = R\omega$$
20. Common JEE Traps
- Confusing torque with force
- Using wrong MOI
- Forgetting parallel axis theorem
- Ignoring translational KE in rolling
21. Final Revision Checklist
You have mastered this chapter if you can:
- Apply rotational equations confidently
- Calculate torque and MOI correctly
- Use angular momentum conservation
- Solve rolling motion problems