Circular Motion – Complete Formula & Concept Revision

Circular motion refers to motion of a particle along a circular path. Although speed may remain constant, velocity always changes due to continuous change in direction.

1. Angular Quantities

• Angular displacement: $\theta$ (radians)
• Angular velocity: $\omega = \dfrac{d\theta}{dt}$
• Angular acceleration: $\alpha = \dfrac{d\omega}{dt}$

Relation with linear quantities:

• $v = r\omega$
• $a_t = r\alpha$
• $a_c = r\omega^2 = \dfrac{v^2}{r}$

2. Time Period & Frequency

• Time period: $T = \dfrac{2\pi}{\omega}$
• Frequency: $f = \dfrac{1}{T}$
• $\omega = 2\pi f$

3. Centripetal Acceleration

$a_c = \dfrac{v^2}{r} = r\omega^2$

• Always directed toward centre
• Changes direction of velocity, not magnitude

Trap: Zero centripetal acceleration means no circular motion.

4. Non-Uniform Circular Motion

Acceleration has two components:

• Centripetal: $a_c = \dfrac{v^2}{r}$
• Tangential: $a_t = \dfrac{dv}{dt}$

$a = \sqrt{a_c^2 + a_t^2}$

5. Angular Kinematics (Constant $\alpha$)

• $\omega = \omega_0 + \alpha t$
• $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$
• $\omega^2 = \omega_0^2 + 2\alpha\theta$

6. Centrifugal Force (Non-Inertial Frame)

$F_{cf} = m\omega^2 r$

• Appears only in rotating frame
• Acts radially outward
• Fictitious force

Rule: Never use centripetal and centrifugal forces together.

7. Vertical Circular Motion

Tension at any point making angle $\theta$ from lowest point:

$T = \dfrac{mv^2}{r} + mg\cos\theta$

Minimum speed at top to complete loop:

$v_{top(min)} = \sqrt{gr}$

Minimum speed at bottom:

$v_{bottom(min)} = \sqrt{5gr}$

8. Energy Conservation in Vertical Circle

$\frac{1}{2}mv_b^2 = \frac{1}{2}mv_t^2 + 2mgr$

$v_b^2 = v_t^2 + 4gr$

9. Conical Pendulum

• $T\cos\theta = mg$
• $T\sin\theta = \dfrac{mv^2}{r}$
• $\tan\theta = \dfrac{v^2}{rg}$

Time period:

$T = 2\pi\sqrt{\dfrac{l\cos\theta}{g}}$

Trap: Radius $r = l\sin\theta$

10. Banking of Roads (No Friction)

$v = \sqrt{rg\tan\theta}$

11. Banking of Roads (With Friction)

$v_{max} = \sqrt{\dfrac{rg(\tan\theta + \mu)}{1 - \mu\tan\theta}}$

$v_{min} = \sqrt{\dfrac{rg(\tan\theta - \mu)}{1 + \mu\tan\theta}}$

12. High-Frequency JEE Traps

• Using diameter instead of radius
• Forgetting tangential acceleration
• Mixing inertial and non-inertial frames
• Assuming speed constant everywhere
• Ignoring direction of forces

✔ Covers JEE Main 2026 + Advanced
✔ Formula-complete + concept-linked
✔ Ideal for last-day revision