Angular Acceleration: Definition, Relations & Practice
Explore angular acceleration, its relation to tangential acceleration, circular motion, formulas, solved problems, and FAQs.
1. Angular Acceleration
Angular acceleration α is the rate of change of angular velocity ω with respect to time t.
2. Relation with Tangential Acceleration
Tangential acceleration at relates to α by:
where r is the radius from axis to the point.
3. Average & Instantaneous
Average Angular Acceleration
Over interval Δt, from ω1 to ω2:
Instantaneous Angular Acceleration
4. Total Acceleration in Circular Motion
Total acceleration atotal is vector sum of tangential and centripetal accelerations:
ac = ω²r, at = rα
5. Uniform vs Non-Uniform Circular Motion
| Parameter | UCM | NUCM |
|---|---|---|
| ω | Constant | Variable |
| α | Zero | Non-zero |
| at | Zero | Non-zero |
| ac | Non-zero | Non-zero |
6. Analogy: Linear vs Angular
| Linear | Angular |
|---|---|
| Displacement: s | Angle: θ |
| Velocity: v = ds/dt | ω = dθ/dt |
| Acceleration: a = dv/dt | α = dω/dt |
7. Equations of Motion (Constant α)
- ω = ω₀ + αt
- θ = ω₀t + ½αt²
- ω² = ω₀² + 2αθ
8. Practice Problems
Q1.
A wheel’s speed increases from 2 rad/s to 10 rad/s in 4 s. Find α.
Ans: α = (10 – 2) / 4 = 2 rad/s².
Q2.
A pulley of radius 0.1 m has α = 3 rad/s². What is the tangential acceleration?
Ans: at = rα = 0.1 × 3 = 0.3 m/s².
Q3.
A disk starting from rest with α = –5 rad/s² stops after turning 20 rad. Find final ω.
Ans: ω² = 0 + 2(–5)(20) = –200 (unphysical negative → it stops before 20 rad).
9. FAQs
Q1. What are dimensions of α?
α = dω/dt, ω in rad/s, t in s → dimensions: T⁻².
Q2. If ω is constant, α = ?
Zero, since dω/dt = 0.
Q3. What causes constant α?
No torque change → constant net torque yields constant α.
Q4. Does α depend on radius?
No, α = dω/dt; ω is angular displacement rate, independent of r.
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