Intuition The one core ideaAnything moving in a circle is turning , and turning means its direction of motion keeps changing — that change is an acceleration that always points to the middle. To make that turn happen, some real force (a string, gravity, friction, a magnet…) must supply exactly that inward pull; "centripetal force" is just the job-title we give to whatever real force is doing the pulling. (We build every symbol below before writing a single formula.)
Before you can read the parent note Centripetal force — what provides it , you must own every symbol it throws at you. We build them one at a time, each from the picture it lives in.
Definition The moving body as apoint
We shrink the whole object (stone, car, planet) down to a single point that traces a path. Everything we say is about where that point is and how fast it moves .
Picture a dot sliding along a curved track. That dot is our whole world for now.
r r center of the circle to the moving point. Its picture: a straight spoke from the hub to the rim of a wheel.
Look at the figure: the orange dot is the moving point, the magenta spoke is r circle and not some wobbly loop. The topic needs r weaker as the circle gets wider (bigger r
Intuition Why an arrow, not just a number?Speed like "5 m/s" is only a size. But which way the point moves matters just as much. An arrow — a vector — stores both the size (its length) and the direction (where it points). We write vectors with a little arrow on top: v
v v glued to the point, pointing along the track — the direction it would fly off in if released this instant.
Look at the figure: at every spot on the circle the violet velocity arrow lies tangent — it just grazes the circle, always at a right angle (9 0 ∘ r
∣ v ∣ v ∣ v ∣ length of the velocity arrow = the plain speed, written simply v v
Δ Δ delta ) in front of anything means "final minus initial" — how much it changed. So Δ t Δ v = v final − v initial
Intuition Subtracting two arrows — the pictureTo do v 2 − v 1 from the tip of v 1 v 2 is the difference Δ v v 1 v 2 same length (same speed), if they point different ways this connecting arrow is not zero.
Look at the figure: the two violet arrows are equal in length (speed unchanged) but rotated apart by a small angle. The orange arrow bridging their tips is Δ v toward the center of the original circle. That is the seed of centripetal acceleration: a non-zero change in velocity even at constant speed.
a a Δ t Δ v Δ v Δ t direction (toward the center) but rescales.
Because Δ v a centripetal ("center-seeking") acceleration.
θ θ theta , plain words: "how far around, or how tilted") measures rotation or tilt. Δ θ small turn of the point around the center during time Δ t
Intuition Why measure angle inradians
A radian is defined so that arc length = × . Picture: bend a piece of string of length r 1 = r θ no extra conversion factor — and the derivation leans on arc ≈
Definition The small-angle rulesin x ≈ x tiny angle x sin x ≈ x ( for small x ) . half-chord is r sin ( Δ θ /2 ) half-arc is r ( Δ θ /2 ) sin ( Δ θ /2 ) ≈ Δ θ /2 chord = 2 r sin ( 2 Δ θ ) ≈ 2 r ⋅ 2 Δ θ = r Δ θ . How good is the swap? The error is of order x 3 Δ θ = 0.1 ≈ 5. 7 ∘ sin ( 0.05 ) = 0.0499979 0.05 0.01% Δ θ → 0 chord / ( r Δ θ ) → 1 why we take the limit of a small turn.
Look at the figure: the magenta arc has length r Δ θ Δ θ ≈ sin x ≈ x ∣Δ v ∣ ≈ v Δ θ
Δ v ∣Δ v ∣ ≈ v Δ θ v 1 , v 2 same length v Δ v isosceles triangle — two equal sides of length v Δ θ Δ v v ∣Δ v ∣ = 2 v sin ( 2 Δ θ ) ≈ v Δ θ . v Δ θ a = v 2 / r
ω ω omega , plain words: "how many radians of turn per second") is Δ t Δ θ ω
See Uniform Circular Motion for more on the v = ω r
m F m F
F c F c shorthand for "the net inward force required" — the total of the inward slices of all the real forces, nothing more. It equals mass times centripetal acceleration:
F c = m a c = m r v 2 = m ω 2 r . Read it carefully: F c requirement to be met , not an extra force you draw. In any problem you hunt for the real providers below and set their combined inward pull equal to F c Newton's Second Law .
You will meet these in the situation-by-situation tour. Each is a real force that might be the one pointing inward — i.e. that might play the role of F c
Definition Quick glossary of providers
T tension , the inward pull of a string (Friction contrasts this with rubbing). Picture: a taut rope yanking the stone toward your hand.g m g weight , the downward pull of Earth.G , M r 2 GM m Gravitation and Orbits .N normal force , the push a surface gives perpendicular to itself. On a tilted road its sideways slice does the turning. See Banking of Roads .f s , μ s static friction and its coefficient; friction can supply at most μ s N Friction .q , B q v B Magnetic Force on Moving Charges ."centrifugal" — the fake outward force felt in a spinning frame; not real in the ground frame. See Pseudo-forces and Non-inertial Frames .
Velocity v tangent to circle
Delta v change of velocity
Small angle rule sin x approx x
Delta v approx v times delta theta
Acceleration a points inward
Centripetal a equals v squared over r
Newton second law F equals m a
Required inward force Fc equals m v squared over r
What real force provides it
A vector stores which two things? Its size (length) and its direction.
What does ∣ v ∣ The length of the velocity arrow — the plain speed v
Which way does the velocity arrow point on a circle? Tangent to the circle, at 9 0 ∘ r
What does Δ The change in it: final minus initial.
Why is Δ v Because the direction of
v changes, so the two arrows differ even with equal length.
State the small-angle rule and why it lets chord ≈ sin x ≈ x x = 2 r sin ( Δ θ /2 ) ≈ r Δ θ
Why is ∣Δ v ∣ ≈ v Δ θ The equal-length velocity arrows form an isosceles triangle;
∣Δ v ∣ = 2 v sin ( Δ θ /2 ) ≈ v Δ θ .
Which way does a Toward the center (centripetal).
Why do we measure the angle in radians? So that arc length = r θ
What is ω v Radians turned per second; v = ω r
State Newton's Second Law as an arrow equation. What does F c The net inward force required , F c = m v 2 / r
