Why does centripetal acceleration have a magnitude?

Assuming that the magnitude of velocity is constant. Why does centripetal acceleration have a magnitude? Since acceleration is the rate of change for velocity and its magnitude remains the same shouldn't we express centripetal acceleration by the angle it changed in the vertical or horizontal over a period of time instead? Also I know the formula for centripetal acceleration but I'm not sure what it actually is. I don't understand the physical meaning behind that formula. Please no calculus explanation because I'm in AP physics 1 and I haven't learned calculus yet.

• You answered your question yourself ,only magnitude of velocity constant ,velocity is not.
– Paul
Feb 15 '17 at 6:44
• Go 5 m/s forward. instantly turn around, and go 5 m/s right. You accelerated. Magnitude of veloctiy remained same but velocity direction changed, so the velocity changed too, hence non-zero acceleration. You question about the angle is correct. The angle is actually corresponding to the direction of velocity. And the angle does have a direction (similar to displacement). Feb 15 '17 at 7:08

The velocity is a vector quantity, i.e: it has a direction as well as a magnitude.

If the velocity has to change, there must be acceleration. There must be an acceleration to change the direction even if it does not change the magnitude of the velocity.

In uniform circular motion, the direction of velocity is continuously changing. If the velocity must change, there must be an acceleration. This acceleration is known as centripetal acceleration.

The centripetal acceleration is always directed towards the center of the circular path and is always perpendicular to the direction of the velocity of the circling body.

Hence, there is no need to explicitly give the direction of the centripetal acceleration.

• Then what does the magnitude of the centripetal acceleration physically represent? Feb 15 '17 at 6:42
• For a given velocity and radius, you need a specific amount of centripetal acceleration to keep the body moving in the circular path. The magnitude of the centripetal acceleration is given by $a = \frac{v^2}{r}$. Feb 15 '17 at 6:43

The purpose of the centripetal acceleration is to change the direction of the velocity of the particle under circular motion.

Although it cannot change the velocity's magnitude (because the work done by the centripetal force on the particle is $0$ as $\cos \theta = 0$), it can change the direction of the velocity, which in turn changes the velocity vector.