Rotational Motion & Gravitation
Prepared for SRMJEEE – Physics
1. Simple Explanation
| Concept | What it means | Everyday analogy |
|---|---|---|
| Rotational motion | Motion of a rigid body around an axis | A spinning top |
| Torque | Rotational “force” that changes angular speed | Turning a door knob |
| Moment of inertia | Resistance of a body to change its rotation | Spinning a bowling ball vs a tennis ball |
| Angular momentum | Rotational “quantity of motion” | A spinning ice‑skater extending arms slows down |
| Gravitation | Attractive force between masses | Earth pulling us to the ground |
| Orbital motion | Path of a body under gravity keeping it from falling | A satellite circling Earth |
2. Key Points
2.1 Rotational Dynamics
| Symbol | Definition | Unit |
|---|---|---|
| (I) | Moment of inertia | kg·m² |
| (\tau) | Torque | N·m |
| (\alpha) | Angular acceleration | rad/s² |
| (\omega) | Angular velocity | rad/s |
| (L) | Angular momentum | kg·m²/s |
| (K_{\text{rot}}) | Rotational kinetic energy | J |
2.1.1 Fundamental Relations
- Newton’s 2nd Law (rotational)
[ \tau = I,\alpha ] - Angular Momentum
[ L = I,\omega ] - Work–Energy Theorem (rotational)
[ W = \Delta K_{\text{rot}} = \frac{1}{2}I(\omega_f^2-\omega_i^2) ] - Conservation of Angular Momentum
If (\tau_{\text{ext}}=0), then (L=\text{constant}).
⇒ (I_i\omega_i = I_f\omega_f)
2.1.2 Common Moments of Inertia
| Body | (I) |
|---|---|
| Solid sphere about centre | (\displaystyle \frac{2}{5}MR^2) |
| Thin spherical shell | (\displaystyle \frac{2}{3}MR^2) |
| Solid cylinder about central axis | (\displaystyle \frac{1}{2}MR^2) |
| Thin circular hoop | (MR^2) |
| Rod about centre (axis ⟂ to rod) | (\displaystyle \frac{1}{12}ML^2) |
2.2 Gravitation
| Symbol | Definition | Unit |
|---|---|---|
| (G) | Universal gravitational constant | (6.67\times10^{-11} \text{N m}^2\text{/kg}^2) |
| (M, m) | Masses of two bodies | kg |
| (r) | Separation distance | m |
| (g) | Gravitational field at a point | m/s² |
| (V_g) | Gravitational potential | J/kg |
| (E_g) | Gravitational potential energy | J |
2.2.1 Newton’s Law of Gravitation
[ F_g = G\frac{Mm}{r^2} ]
2.2.2 Gravitational Field
[ g = \frac{F_g}{m} = G\frac{M}{r^2} ]
2.2.3 Gravitational Potential Energy
[ E_g = -G\frac{Mm}{r} ]
2.2.4 Escape Velocity
[ v_{\text{esc}} = \sqrt{2gR} = \sqrt{\frac{2GM}{R}} ]
2.2.5 Circular Orbital Velocity
[ v_{\text{orb}} = \sqrt{\frac{GM}{r}} ]
2.2.6 Kepler’s Laws (for planetary motion)
- Elliptical Orbits – planets move in ellipses with the Sun at one focus.
- Equal Areas in Equal Times – area swept out per time is constant.
- Period–Radius Relation – (T^2 \propto r^3).
3. Worked Example
Problem
A solid disk of mass (2.0;\text{kg}) and radius (0.2;\text{m}) is rotating at (10;\text{rad/s}). A small mass (0.05;\text{kg}) is slid from the rim to the centre.
(a) Find the new angular speed.
(b) Find the change in rotational kinetic energy.
Solution
-
Initial moment of inertia (disk only)
[ I_i = \frac{1}{2}MR^2 = \frac{1}{2}(2.0)(0.2)^2 = 0.04;\text{kg·m}^2 ] -
Moment of inertia of added mass at radius 0.2 m
[ I_m = mR^2 = 0.05(0.2)^2 = 0.002;\text{kg·m}^2 ] -
Initial total (I)
[ I_{\text{initial}} = I_i + I_m = 0.042;\text{kg·m}^2 ] -
After sliding to centre
The small mass contributes almost nothing: (I_m^{\text{final}}\approx 0).
[ I_{\text{final}} \approx I_i = 0.04;\text{kg·m}^2 ] -
Angular momentum conservation
[ I_{\text{initial}}\omega_i = I_{\text{final}}\omega