Uniform & non-uniform circular motion, banking, friction, vertical circular motion (JEE Main focus).
1) Basics of Circular Motion
- Uniform circular motion: speed constant, velocity changes.
- Non-uniform circular motion: both speed and velocity change.
- Centripetal force always acts towards the centre.
Centripetal force: F_c = mv² / r
Centripetal acceleration: a_c = v² / r
2) Non-uniform Circular Motion
- Total acceleration has two components:
- Radial (centripetal): v² / r
- Tangential: dv / dt
a = √[(v²/r)² + (dv/dt)²]
tanα = (dv/dt) / (v²/r)
α is the angle between net acceleration and radial direction.
3) Banking of Roads
- Banking helps provide centripetal force using horizontal component of normal reaction.
tanθ = v² / (rg)
This formula assumes no friction (ideal banking).
4) Vehicle on Horizontal Circular Track (With Friction)
- Centripetal force is provided by static friction.
v_max = √(μrg)
Condition for no skidding: v ≤ √(μrg)
5) Cyclist Taking a Turn
- For safe turning, cyclist bends inward.
tanθ = v² / (rg)
Same expression as banking angle.
6) Vertical Circular Motion
- Velocity given at lowest point = u.
- At angle θ from lowest point:
T − mgcosθ = mv² / r
v² = u² − 2gr(1 − cosθ)
T = m[u² − 2gr + 3grcosθ] / r
7) Condition to Just Complete a Vertical Circle
Minimum velocity at bottom: u = √(5gr)
| Position | Velocity | Tension |
|---|---|---|
| Bottom (A) | √(5gr) | 6mg |
| Side (B) | √(3gr) | 3mg |
| Top (C) | √(gr) | mg |
8) Toppling of Vehicle on Circular Track
- Occurs when inner wheels lose contact.
v ≤ √(rg · a / h)
h = height of centre of gravity, a = half width of vehicle.
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Last modified: December 14, 2025
