AP EAMCET – Physics
Rotational Motion & Gravitation
Study Notes – Short, clear, and exam‑ready.
Designed to help you understand the why and how and to practise quick calculations.
1. Simple Explanation
| Concept | What it is | Why it matters for AP EAMCET |
|---|---|---|
| Rotational Motion | Motion of a body about an axis. | Many questions involve torque, moment of inertia, angular momentum, and energy. |
| Gravitation | Universal attraction between masses. | Appears in questions on orbits, escape velocity, tidal forces, and gravitational potential energy. |
2. Key Points (Rotational Motion)
| Topic | Formula (Symbol | Units) | Typical Use |
|---|---|---|---|
| Angular displacement | (\theta) | rad | Describes a rotation. |
| Angular velocity | (\omega = \frac{d\theta}{dt}) | rad s⁻¹ | Rate of rotation. |
| Angular acceleration | (\alpha = \frac{d\omega}{dt}) | rad s⁻² | Change in angular speed. |
| Moment of Inertia | (I) | kg m² | Rotational “mass”. |
| Torque | (\tau = I\alpha) | N m | Rotational analogue of force. |
| Angular Momentum | (L = I\omega) | kg m² s⁻¹ | Conserved if no external torque. |
| Rotational Kinetic Energy | (K_{\rm rot} = \frac{1}{2}I\omega^2) | J | Energy stored in rotation. |
| Kinetic Energy (general) | (K = \frac{1}{2}mv^2) | J | Connects linear and angular motion: (v = r\omega). |
| Conservation of Energy | (K_{\rm rot} + U = \text{constant}) | J | Used in problems with rotating systems and potential energy. |
Moment of Inertia for Common Shapes
| Shape | (I) (about symmetry axis) |
|---|---|
| Solid cylinder or disk | ( \frac{1}{2}MR^2 ) |
| Thin hoop or ring | ( MR^2 ) |
| Solid sphere | ( \frac{2}{5}MR^3 ) |
| Thin spherical shell | ( \frac{2}{3}MR^2 ) |
Tip: For a system of particles, (I = \sum m_i r_i^2).
3. Key Points (Gravitation)
| Topic | Formula (Symbol | Units) | Typical Use |
|---|---|---|---|
| Newton’s Universal Law | (F = G \frac{m_1 m_2}{r^2}) | N | Force between two masses. |
| Gravitational Field | (g = G \frac{M}{r^2}) | m s⁻² | Acceleration due to a mass (M). |
| Gravitational Potential Energy | (U = -G \frac{m_1 m_2}{r}) | J | Work needed to separate masses. |
| Escape Velocity | (v_{\rm esc} = \sqrt{2gR}) | m s⁻¹ | Minimum speed to leave a planet. |
| Orbital Velocity (circular) | (v_{\rm orb} = \sqrt{gR}) | m s⁻¹ | Speed for stable orbit. |
| Kepler’s Third Law (for Earth‑satellite) | (T^2 = \frac{4\pi^2 r^3}{GM}) | s | Relation between orbital period and radius. |
| Gravitational Potential | (V = -G \frac{M}{r}) | J kg⁻¹ | Used in energy calculations. |
Escaping Earth: For Earth, (g \approx 9.8) m s⁻² and (R_E = 6.37\times10^6) m → (v_{\rm esc} \approx 11.2) km s⁻¹.
4. Worked Example (Rotational Motion)
Problem – A solid disc of mass (5,\rm kg) and radius (0.2,\rm m) is rotating at (10,\rm rad,s^{-1}).
A constant torque of (2,\rm N,m) is applied.
Find: (a) the angular acceleration, (b) the angular velocity after (3,\rm s), (c) the change in rotational kinetic energy.
Step‑by‑Step Solution
-
Moment of Inertia
[ I = \frac{1}{2}MR^2 = \frac{1}{2}(5)(0.2)^2 = 0.1,\rm kg,m^2 ] -
Angular Acceleration
[ \tau = I\alpha ;;\Rightarrow;; \alpha = \frac{\tau}{I} = \frac{2}{0.1} = 20,\rm rad,s^{-2} ] -
Angular Velocity after 3 s
[ \omega = \omega_0 + \alpha t = 10 + 20\times 3 = 70,\rm rad,s^{-1} ] -
Change in Rotational Kinetic Energy
[ \Delta K = \frac{1}{2}I(\omega^2 - \omega_0^2) = \frac{1}{2}(0.1)(70^2 - 10^2) = 0.05(4900-100)=0.05 \times 4800 = 240,\rm J