Rotational Motion – Study Notes
(Prepared for TS EAMCET – Physics – Beginner Level)
Tip: Think of everything that spins – a bicycle wheel, a merry‑go‑round, a spinning fan.
The same rules that govern straight‑line motion also apply when objects turn.
1. What is Rotational Motion?
- Rotation = turning around an axis.
- Example: A bicycle wheel keeps turning when you pedal.
- We talk about how far it turns, how fast it turns, and how quickly its turning speed changes.
2. Basic Quantities
| Quantity | Everyday Meaning | Symbol | Unit |
|---|---|---|---|
| Angular displacement | How many degrees / radians the wheel has turned | θ | rad |
| Angular velocity | Speed of turning (how many rad/s) | ω | rad s⁻¹ |
| Angular acceleration | How fast the turning speed changes (rad s⁻²) | α | rad s⁻² |
| Moment of inertia | How hard it is to spin an object (depends on shape & mass) | I | kg m² |
| Torque | “Turning force” that makes it spin | τ | N m |
| Rotational kinetic energy | Energy stored in a spinning object | Krot | J |
| Angular momentum | “Rotational momentum” (mass × speed × radius) | L | kg m² s⁻¹ |
Note: In the notebooks we always use radians for angles.
1 rad = 57.3°, so 360° = 2π rad.
3. Simple Analogies
| Straight‑line | Rotational |
|---|---|
| Distance | Angular displacement (θ) |
| Speed (v) | Angular velocity (ω) |
| Acceleration (a) | Angular acceleration (α) |
| Force (F) | Torque (τ) |
| Mass (m) | Moment of inertia (I) |
| Linear momentum (p = m v) | Angular momentum (L = I ω) |
| Linear kinetic energy (½ m v²) | Rotational kinetic energy (½ I ω²) |
4. Key Relationships
| Relationship | Formula | Short Explanation |
|---|---|---|
| Angular velocity | ω = Δθ / Δt | How fast the angle changes. |
| Angular acceleration | α = Δω / Δt | Rate of change of ω. |
| Torque–Angular acceleration | τ = I α | Like F = m a, but for turning. |
| Angular momentum | L = I ω | “Mass × speed × radius” for rotation. |
| Rotational kinetic energy | Krot = ½ I ω² | Energy of a spinning object. |
| Energy in a rotating system | Krot + U = constant (if no external torque) | Conservation of energy. |
| Conservation of angular momentum | L₁ = L₂ (no external τ) | If no outside torque, L stays the same. |
5. Tiny Example – A Spinning Wheel
- Wheel radius: 0.3 m
- Mass: 2 kg (uniform disc)
- Moment of inertia for a disc:
(I = \frac{1}{2} m r^2 = \frac{1}{2} \times 2 \times 0.3^2 = 0.18) kg m² - It starts from rest and is given a torque of 0.6 N m.
- Angular acceleration:
(α = τ / I = 0.6 / 0.18 ≈ 3.33) rad s⁻² - After 2 s, angular velocity:
(ω = α t = 3.33 × 2 ≈ 6.66) rad s⁻¹ - Rotational kinetic energy:
(Krot = ½ I ω² = 0.5 × 0.18 × 6.66² ≈ 4.0) J
So a small push (torque) makes the wheel spin faster, gaining rotational energy.
6. Common Pitfalls to Avoid
| Pitfall | Fix |
|---|---|
| Mixing degrees and radians | Always convert to radians when using formulas. |
| Forgetting that torque = force × radius, not force alone | Use (τ = r × F) (cross product). |
| Confusing linear speed with angular speed | Remember (v = r ω). |
| Using linear mass instead of moment of inertia | Use (I), not (m), in rotational equations. |
7. Quick Practice Problems
-
A spinning wheel of radius 0.2 m has a moment of inertia 0.02 kg m².
If a torque of 0.1 N m is applied, what is the angular acceleration?
Answer: (α = τ/I = 0.1/0.02 = 5) rad s⁻² -
A figure skater pulls in her arms, reducing her moment of inertia from 3 kg m² to 1.5 kg m².
If her initial angular speed was 4 rad s⁻¹, what is her new angular speed?
Answer: (L) conserved → (I₁ ω₁ = I₂ ω₂) → (ω₂ = (I₁/I₂) ω₁ = (3/1.5)×4 = 8) rad s⁻¹ -
A wheel turns through 180° (π rad) in 2 s, starting from rest.
Find its angular