CLS-08

Centripetal acceleration

a = v² / r = ω² r toward the centre.

Reading speed

Circular motionHuygens / Newton

Governing equation

a=\dfrac{v^2}{r}=\omega^2 r

where

v: Speed (m/s)
r: Radius (m)
a: Centripetal acceleration (m/s²)
\omega: Angular speed (rad/s)

Lecture brief

Historical brief

Newtonian mechanics (1687) plus energy, angular momentum, Kepler and the ballistic parabola remain the first language of motion. Every sheet here is a closed-form orbit, throw, spin or oscillator. This sheet (CLS-08 — Centripetal acceleration) is the form associated with Huygens / Newton. Working symbols: $v$ , $r$ $\rightarrow$ $a$ , $\omega$ . Uniform circular motion has a velocity that turns; the time derivative is v²/r directed inward, Huygens' centripetal acceleration.

Purpose

Purpose: compute

a

\omega

from

v

r

in Classical mechanics via

a=\dfrac{v^2}{r}=\omega^2 r

a = v² / r = ω² r toward the centre. Use it when a real classical mechanics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

v = 12.000\,\mathrm{m/s}

r = 4.000\,\mathrm{m}

, the governing relation

a=\dfrac{v^2}{r}=\omega^2 r

yields

a = 36.000\,\mathrm{m/s^{2}}

\omega = 3.000\,\mathrm{rad/s}

. Uniform circular motion, inertial observer. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Speed v12 m/sRadius r4 m

Outputs

Centripetal acceleration a36.000 m/s²
Angular speed \omega3.000 rad/s

Reading speed

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Narration of this film

Uniform circular motion, inertial observer.

Uniform circular motion has a velocity that turns; the time derivative is v²/r directed inward, Huygens' centripetal acceleration.

Reading speed

Watch on YouTube