Class 11 · Chapter 16

Circular Motion

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Circular Motion Short Notes

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Condensed revision points for Circular Motion — for quick recall before exams, not a substitute for the full notes.

ω = dθ/dt = 2πf = 2π/T (rad/s)
α = dω/dt = d²θ/dt² (rad/s²)
Equations of motion: ω = ω₀ + αt | θ = ω₀t + ½αt² | ω² = ω₀² + 2αθ
ω is an axial vector — direction via right-hand rule along the axis of rotation.

Centripetal (radial) acceleration: a_c = v²/r = ω²r — always points toward the centre.
Tangential acceleration: a_t = dv/dt = rα — present only in non-uniform circular motion.
Total acceleration: a = √(a_c² + a_t²); in UNIFORM circular motion, a_t = 0 so a = a_c.

F_c = mv²/r = mω²r — NOT a separate force; it's the name for whatever net real force points toward the centre.
Always identify which real force (tension/friction/gravity/normal reaction) is acting as the centripetal force.

Flat (unbanked) road: max safe speed v_max = √(μₛRg); independent of vehicle mass.
Banked road (no friction): tanθ = v²/(Rg).
Banking lets part of the normal reaction supply centripetal force, reducing reliance on friction.

Pseudo force, magnitude mω²r, directed radially outward.
Appears ONLY in a rotating (non-inertial) frame; ground-frame observers never need it.
No Newton's third law reaction pair — it's fictitious.

Speed at angle θ: v² = v₀² − 2gl(1 − cosθ). Tension: T = mv₀²/l − 2mg + 3mg cosθ.
COMPLETES the loop (string): v₀ ≥ √(5gl); then T_bottom = 6mg, T_top = 0.
OSCILLATES in lower half: v₀ < √(2gl) — speed hits zero before tension does.
LEAVES the track (goes slack, then projectile): √(2gl) ≤ v₀ < √(5gl).
Rigid rod/track (can push, not just pull): only needs v₀ ≥ √(4gl) to complete the loop.

Lift accelerating up: g_eff = g + a. Lift accelerating down: g_eff = g − a.
Horizontally accelerating trolley: g_eff = √(g² + a²), tilted at tan⁻¹(a/g) from vertical.
Once g_eff is found, reuse every vertical-circle formula with g replaced by g_eff.

Don't forget tension/speed both vary continuously around a vertical circle — they are NOT constant like in horizontal circular motion.
v_max on a flat curve does not depend on mass — don't introduce mass into that formula.
Centripetal and centrifugal force are NOT an action-reaction pair — centrifugal force isn't even real in an inertial frame.
Check carefully whether the question gives a string (can go slack) or a rigid rod (cannot go slack) before applying the completing-the-loop condition.

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