Class 11 · Chapter 16

Circular Motion

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Circular Motion Formula Sheet

16 formulas across 3 topics in Circular Motion.

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Updated 2026-07-08 · v1.0.0

Angular Kinematics

Angular kinematics mirrors linear kinematics exactly — every SUVAT-style equation has a direct rotational twin, with θ, ω, α replacing x, v, a.

ω = dθ/dt (instantaneous); ω(avg) = Δθ/Δt

Angular velocity

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
ω	angular velocity	rad/s	[T⁻¹]
θ	angular position	rad	[M⁰L⁰T⁰]

Valid when

A vector quantity; direction along the rotation axis via the right-hand rule

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α = dω/dt

Angular acceleration

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
α	angular acceleration	rad/s²	[T⁻²]

Valid when

Zero for uniform circular motion (ω constant) — non-zero only when speed along the circle changes

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ω = ω₀ + αt ; θ = ω₀t + ½αt² ; ω² = ω₀² + 2αθ

Equations of motion for constant angular acceleration

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Direct rotational analogues of the linear SUVAT equations, ω↔v, θ↔x, α↔a

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
ω₀	initial angular velocity	rad/s	[T⁻¹]

Valid when

Valid only for CONSTANT angular acceleration — same restriction as the linear SUVAT set

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v = rω ; a(tangential) = rα

Relation between linear and angular quantities

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Arc length s = rθ, differentiated once for v = rω and again for aₜ = rα

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
r	radius of the circular path	m	[L]
v	linear (tangential) speed	m/s	[LT⁻¹]

Valid when

r is measured from the axis of rotation, not from some arbitrary point

Common mistakes

For points at different radii on the same rigid rotating body, ω is the SAME for all of them, but v = rω differs — a very common source of error in rigid-body problems

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T = 2π/ω = 1/f ; ω = 2πf

Time period and frequency of circular motion

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
T	time for one complete revolution	s	[T]
f	frequency (revolutions per second)	Hz	[T⁻¹]

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Centripetal Acceleration & Force

Moving in a circle at constant speed still means accelerating — the velocity's DIRECTION is constantly changing, and that requires a net inward force at every instant.

a(c) = v²/r = ω²r

Centripetal (radial) acceleration

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Rate of change of the velocity vector's DIRECTION (not magnitude) for motion along a circle

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
a(c)	centripetal acceleration, directed toward the centre	m/s²	[LT⁻²]

Valid when

Present even in UNIFORM circular motion (constant speed) — direction alone changing is enough to require this acceleration

Common mistakes

Centripetal acceleration is NOT a separate force by itself — it's the acceleration that some real force (tension, gravity, friction, normal force) must supply

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a(net) = √(a(c)² + a(t)²)

Net acceleration in non-uniform circular motion

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Radial (centripetal) and tangential acceleration components are always perpendicular, so they combine via Pythagoras

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
a(t)	tangential acceleration, from the changing SPEED	m/s²	[LT⁻²]

Valid when

a(t) = 0 for uniform circular motion, leaving only a(c) — total acceleration then points purely radially inward

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F(c) = mv²/r = mω²r

Centripetal force

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Newton's second law applied along the radial direction: F = ma(c)

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
m	mass of the object moving in the circle	kg	[M]

Valid when

Not a NEW type of force — it's whichever real force (or component of it) happens to point toward the centre

Common mistakes

Never add 'centripetal force' as an extra arrow on a free-body diagram alongside gravity/tension/normal — it IS the net inward component of those real forces

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F(cf) = −mω²r (pseudo force, rotating frame only)

Centrifugal force (pseudo force)

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Valid when

Only appears when analysing motion from within the ROTATING (non-inertial) frame — has no place in an inertial-frame free-body diagram

Common mistakes

Mixing centrifugal force into an inertial-frame analysis alongside real forces is a direct contradiction — pick ONE frame and stay consistent

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Worth remembering

Every circular-motion problem starts the same way: identify which real force(s) — or components of them — supply the required centripetal force, then apply F(net, radial) = mv²/r

Banking of Roads, Conical Pendulum & Vertical Circle

These three classic setups are the same centripetal-force idea applied to increasingly clever geometries — each one is a guaranteed numerical question in JEE/NEET.

v(max) = √(μrg)

Maximum speed on a flat (unbanked) circular road

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Friction alone supplies the centripetal force: μmg = mv²/r

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
μ	coefficient of friction between tyres and road	dimensionless	[M⁰L⁰T⁰]
g	acceleration due to gravity	m/s²	[LT⁻²]

Valid when

Beyond this speed, the car skids outward — friction alone cannot provide enough centripetal force

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tanθ = v²/rg

Banking angle for a frictionless banked road

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Only normal force provides centripetal force: N sinθ = mv²/r, with N cosθ = mg vertically

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
θ	angle of banking of the road	rad	[M⁰L⁰T⁰]

Valid when

This is THE ONE ideal safe speed for a given banking angle when friction is completely absent

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v(max) = √[rg(tanθ + μ) / (1 − μtanθ)]

Maximum safe speed on a banked road WITH friction

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Both normal force and friction (acting down the incline, at the verge of outward skidding) contribute to the centripetal force

Valid when

Minimum safe speed (verge of sliding DOWN and inward) replaces + with − in both places: v(min) = √[rg(tanθ − μ)/(1+μtanθ)]

Common mistakes

This formula reduces correctly to both simpler cases: set μ = 0 to recover tanθ = v²/rg, or θ = 0 to recover the unbanked-road result

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tanθ = v²/rg ; T(period) = 2π√(l cosθ / g)

Conical pendulum — angle and time period

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Tension's horizontal component supplies centripetal force; vertical component balances gravity — identical structure to frictionless banking

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
l	length of the string	m	[L]
θ	angle the string makes with the vertical	rad	[M⁰L⁰T⁰]

Valid when

r = l sinθ relates the circle's radius to the string length and angle

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v(top, min) = √(gr)

Minimum speed at the top of a vertical circle

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At the critical (minimum) speed, gravity alone supplies the entire centripetal force — tension/normal force just touches zero

Valid when

Applies to a string/track providing tension/normal force only inward — below this speed, the object cannot maintain contact at the top and falls out of the circular path

Common mistakes

Below v(top,min), the object does NOT slow down and reverse — it leaves the circular path entirely and becomes a projectile

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v(bottom, min) = √(5gr)

Minimum speed at the bottom to complete a full vertical circle

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Energy conservation from bottom to top (height gained = 2r) combined with v(top,min) = √(gr)

Valid when

This is THE single most tested numerical result in vertical circular motion — memorise both the value and its energy-conservation derivation

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T = mv²/r − mg cosφ (φ measured from the TOP of the circle)

Tension at a general point on a vertical circle

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Newton's second law along the radial direction, with gravity's component toward/away from the centre depending on position

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
φ	angle of the current position measured from the top of the circle	rad	[M⁰L⁰T⁰]

Valid when

At the top (φ=0): T = mv²/r − mg; at the bottom (φ=180°): T = mv²/r + mg — sign of the gravity term flips

Common mistakes

The gravity term's sign depends on position — always re-derive it from a free-body diagram at that specific point rather than memorising one fixed sign

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JEE Advanced frequently combines vertical-circle problems with energy conservation across multiple points, not just top/bottom — always set up energy conservation between the two points actually asked about, not just the standard top-bottom pair.

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