Tangential velocity and centripetal acceleration

The speed at which the body travels around the circumference, the tangential velocity, is obtained from the angular velocity by multiplying it by the radius. And since the direction of the velocity is constantly changing, an acceleration exists even at constant speed: the centripetal acceleration.

Key formulae

$v = \omega\,r$ $a_c = \frac{v^2}{r} = \omega^2 r$

Centripetal acceleration is always directed towards the centre of the trajectory: it is what “bends” the motion, preventing the body from continuing in a straight line. Velocity, on the other hand, is always tangent to the circumference. Velocity and centripetal acceleration are therefore perpendicular at every instant.

Uniform circular motion: the velocity $\vec{v}$ is tangent to the trajectory, the centripetal acceleration $\vec{a}_c$ always points towards the centre O.

Example — The seconds hand

Place a blue dot halfway along the seconds hand of a clock and a red dot at the tip; the hand is $3\;\text{m}$ long. Both dots complete one revolution every $60\;\text{s}$, so they have the same period $T = 60\;\text{s}$ and the same angular velocity: $\omega = \frac{2\pi}{60} = 0{,}105\;\text{rad/s}$ But the red dot ($r = 3\;\text{m}$) has a greater speed than the blue one ($r = 1{,}5\;\text{m}$): $v_{\text{rosso}} = \omega \cdot 3 = 0{,}314\;\text{m/s}, \qquad v_{\text{blu}} = \omega \cdot 1{,}5 = 0{,}157\;\text{m/s}$

Same $\omega$ does not mean same speed

Same period and same angular velocity do not imply the same tangential speed! Speed depends on the radius: the further you are from the centre, the faster you go.

Uniform circular motion will underpin the understanding of centripetal force, to be tackled after introducing Newton’s laws.

Collegamenti

Argomenti: Cinematica Concetti: Moto circolare uniforme · Accelerazione centripeta · Velocità angolare

Esercizi collegati: Dimostrazione dell’accelerazione centripeta · Accelerazione centripeta della rotazione terrestre · Ranking di punti su un disco rotante