Concepts from Zero
Start here. Build the intuition before memorizing formulas.
What is Circular Motion?
When an object moves along a circular path, it's in circular motion. The key insight — even if speed is constant, direction keeps changing.
Core Insight: Changing direction = changing velocity. Changing velocity = acceleration. Acceleration needs a force. That force is Centripetal Force — always pointing inward toward the centre.
⚡ Uniform Circular Motion
Speed = constant
Direction = changing
Velocity = changing
|a| = constant
e.g. Earth around Sun
Direction = changing
Velocity = changing
|a| = constant
e.g. Earth around Sun
🎢 Non-Uniform Circular Motion
Speed = changing
Direction = changing
Both v and a change
|a| varies
e.g. Roller coaster loop
Direction = changing
Both v and a change
|a| varies
e.g. Roller coaster loop
Build Intuition — Step by Step
Core Concepts Deep Dive
🔥 High Weight
1
Angular Velocity (ω)
ω = Δθ / Δt | Unit: rad/s
How fast the angle is changing. Think of it as "rotational speed."
2
Linear ↔ Angular Speed
v = rω
Outer edge of a spinning disc moves faster than the inner edge — same ω, bigger r, bigger v!
3
Centripetal Acceleration Exam Fav
a = v²/r = rω²
Always points towards the centre. It's what keeps bending the path into a circle. Without it, the object flies off tangentially.
4
Centripetal Force
F = mv²/r = mrω²
NOT a new force! It's whatever existing force points inward:
🪨 Stone on string → Tension | 🚗 Car on road → Friction | 🌍 Planet → Gravity
🪨 Stone on string → Tension | 🚗 Car on road → Friction | 🌍 Planet → Gravity
5
Time Period & Frequency
T = 2π/ω = 2πr/v f = 1/T = ω/2π
T = time for one complete revolution. f = revolutions per second (Hz).
Live Visualizer
Interact with the simulation to see velocity and acceleration in real time.
v (m/s)
—
a=v²/r
—
ω rad/s
—
Speed
1.5
Radius
120
Show / Hide
Observe: Blue (v) is always tangent. Red (a) always points inward. They are always 90° apart.
Increase radius → same ω, more linear speed (v = rω)
Increase speed → more centripetal acceleration (a = v²/r)
Increase speed → more centripetal acceleration (a = v²/r)
Direction Rules — Board Style
Formula Sheet
Every formula you need — with units and when to use each.
Core Formulas 🔥 Memorise
v = rω
Linear ↔ Angular speed
a = v²/r
Centripetal acceleration
a = rω²
Centripetal acc (angular)
F = mv²/r
Centripetal force
F = mrω²
Centripetal force (angular)
T = 2π/ω
Time period from ω
T = 2πr/v
Time period from v
f = 1/T
Frequency (Hz)
ω = 2πf
Angular ↔ Frequency
ω = 2π/T
Angular ↔ Period
Units & When to Use
| Quantity | Symbol | Unit | When to use |
|---|---|---|---|
| Angular velocity | ω | rad/s | RPM or revolutions given |
| Linear velocity | v | m/s | Speed in m/s given directly |
| Centripetal acc. | a | m/s² | Find force or check circular condition |
| Centripetal force | F | N (Newton) | Tension, friction, gravity problems |
| Time period | T | seconds | Time for one full revolution given |
| Frequency | f | Hz (hertz) | Revolutions per second given |
Problem Solving — 5 Step Strategy NCERT
1
Identify — Is this circular motion? Uniform or non-uniform? What is the radius?
2
Find Radius — Length of string, radius of road curve, orbit radius, or half-diameter.
3
Choose Formula — v given → use a = v²/r. ω given → use a = rω² and v = rω first.
4
Identify Inward Force — Tension? Friction? Normal force component? Gravitational pull?
5
Equate and Solve — Set that force = mv²/r. Substitute values. Solve for unknown.
💡 NEET Shortcut for RPM problems: If n RPM given → f = n/60 Hz → ω = 2πn/60 rad/s → use F = mrω²
Special Cases
High-scoring topics in CBSE and NEET — learn these deeply.
Vertical Circular Motion 🔥 NEET Favourite
A stone on a string completing vertical circles. Speed is NOT constant — gravity speeds it at bottom, slows at top.
🔻 At Bottom
Tension: T_b = mg + mv²/r
T opposes weight → T is MAXIMUM
Both T and gravity, T acts up, gravity down
T opposes weight → T is MAXIMUM
Both T and gravity, T acts up, gravity down
🔺 At Top
Tension: T_t = mv²/r − mg
T and mg both inward → T is MINIMUM
Net inward: T + mg = mv²/r
T and mg both inward → T is MINIMUM
Net inward: T + mg = mv²/r
Min speed at top (string stays taut, T = 0):
v_top_min = √(gr)
v_top_min = √(gr)
Min speed at bottom (energy conservation):
v_bot_min = √(5gr)
v_bot_min = √(5gr)
Banking of Roads 🔥 CBSE Exam Must
Roads are tilted (banked) on curves so vehicles can turn safely even without friction.
tan θ = v²/rg
θ = banking angle | v = optimal speed | r = radius | g = 9.8 m/s²
With friction (NEET level):
v_max = √(rg × (μ + tanθ)/(1 − μtanθ))
v_min = √(rg × (tanθ − μ)/(1 + μtanθ))
v_max = √(rg × (μ + tanθ)/(1 − μtanθ))
v_min = √(rg × (tanθ − μ)/(1 + μtanθ))
Conical Pendulum ⚠ Tricky
A ball on a string moving in horizontal circles. String makes angle θ with vertical.
Vertical equilibrium
T cosθ = mg
Horizontal (centripetal)
T sinθ = mv²/r
T = 2π√(L cosθ / g)
L = string length
Well of Death (Cylinder / Circus)
Cyclist rides inside a vertical cylinder. Normal force from wall acts as centripetal force; friction holds them up.
Condition (no sliding down):
μN ≥ mg → μ(mv²/r) ≥ mg
∴ v_min = √(rg/μ)
μN ≥ mg → μ(mv²/r) ≥ mg
∴ v_min = √(rg/μ)
Common Mistakes
These 6 mistakes cost students the most marks. Learn the corrections.
❌ Myth: "Centripetal force is a special new force"
✅ Centripetal force is NOT a new type of force. It's the label for whatever net inward force exists in the situation — tension, friction, gravity, or normal force. Never draw it separately on a free body diagram!
❌ Myth: "Velocity points towards the centre"
✅ Velocity is always TANGENTIAL — along the tangent to the circle. Acceleration points towards the centre. Remember: velocity "wants to escape," acceleration pulls it back in.
❌ Confusing ω (rad/s) with v (m/s)
✅ They relate by v = rω. For same angular velocity ω, outer points have MORE linear speed. Always convert RPM → ω = 2πn/60 before substituting.
❌ Writing T_top = mg + mv²/r (wrong sign at top of loop)
✅ At the TOP, both tension AND gravity point inward (towards centre), so T + mg = mv²/r. This gives T = mv²/r − mg. At bottom, only T points inward while mg points outward → T − mg = mv²/r.
❌ Not converting RPM to rad/s before using formulas
✅ ω = 2πf, where f = RPM/60. So 300 RPM → f = 5 Hz → ω = 10π rad/s. Never plug RPM directly into the formula!
❌ Not knowing which force acts as centripetal force
✅ Always ask: "What force is pointing towards the centre?" Car on flat road = friction. Car on banked road = N sinθ. Planet around star = gravity. Roller coaster top = weight + Normal. Identify first, equate to mv²/r.
Practice MCQs
Attempt questions, check your score, and read explanations.
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Quick Revision Sheet
Everything you need — one page. Study this the night before.
Core Formulas
v = rω
a = v²/r = rω²
F = mv²/r = mrω²
T = 2π/ω = 2πr/v
f = 1/T, ω = 2πf
a = v²/r = rω²
F = mv²/r = mrω²
T = 2π/ω = 2πr/v
f = 1/T, ω = 2πf
Direction Rules
Velocity → tangent to circle
Acceleration → towards centre
v ⊥ a — always 90°
Speed = constant (UCM only)
ω = constant (UCM only)
Acceleration → towards centre
v ⊥ a — always 90°
Speed = constant (UCM only)
ω = constant (UCM only)
Force Selection
String/rope → Tension
Road curve → Friction
Banked road → N sinθ
Planet/satellite → Gravity
Well of death → Normal force
Road curve → Friction
Banked road → N sinθ
Planet/satellite → Gravity
Well of death → Normal force
Vertical Circle
T_bottom = mg + mv²/r
T_top = mv²/r − mg
v_min(top) = √(gr)
v_min(bottom) = √(5gr)
tan θ = v²/rg (banking)
T_top = mv²/r − mg
v_min(top) = √(gr)
v_min(bottom) = √(5gr)
tan θ = v²/rg (banking)
The VATO Rule
V
elocity
A
lways tangent
T
owards centre = a
O
nly net inward force
Velocity → tangent | Acceleration → centre | Always 90° between them | Only existing inward force = centripetal
Exam Strategy — CBSE: Always draw FBD. Label all forces. Write formula. Substitute. Solve. Show every step for full marks.
Exam Strategy — NEET: Use direction logic to eliminate options instantly. If a force is shown pointing outward — eliminate. Check units match. Banking → tan θ = v²/rg. "String goes slack" → v_top = √(gr).
Exam Strategy — NEET: Use direction logic to eliminate options instantly. If a force is shown pointing outward — eliminate. Check units match. Banking → tan θ = v²/rg. "String goes slack" → v_top = √(gr).