Normal (Centripetal) Acceleration

a_{n} = \frac{v ^{2}}{ρ} = ω^{2} ρ

On a curved path the normal component of acceleration points toward the center of curvature. Even at constant speed, a particle on a circle of radius ρ needs an_n = v²/ρ. Use ω = v/ρ to switch forms.

$a_{n}$ (m/s²): Normal acceleration
$v$ (m/s): Speed
$ρ$ (m): Radius of curvature
$ω$ (rad/s): Angular speed

Step-by-step solved example

A car travels at 20 m/s on a 50 m radius curve. Find an and ω.

Normal acceleration v²/ρ toward the center of curvature.

1. Normal acceleration
an = v²/ρ = 400/50 = 8.00 m/s².
$a_{n} = 8.00 m/s^{2}$
2. Angular speed
ω = v/ρ = 20/50 = 0.40 rad/s. Check: ω²ρ = 0.16×50 = 8.00 m/s².
$ω = 0.40 rad/s$

Answer: an = 8.00 m/s², ω = 0.40 rad/s

10 practice questions

0/10 correct

1.v = 10 m/s, ρ = 5 m. an =

2.ω = 4 rad/s, ρ = 2 m. an =

3.Uniform circular motion still requires

4.an is directed

5.v = 30 m/s, ρ = 90 m. an =

6.If ρ doubles at constant v, an

7.The relation between v, ω, and ρ is

8.At constant speed the tangential acceleration at is

9.Car at 15 m/s on ρ = 25 m. an =

10.ω = 3 rad/s, v = 6 m/s. ρ =