Centripetal Acceleration

Centripetal Acceleration

According to Newton's first law, a body is either at rest, or moves in a straight line at fixed speed, unless a force acts on it. So how can a body move in a circular path?

If Newton's law is true, then there is only one answer: a force is acting on the body.

If you are walking slowly in a circle, you would not feel any extra force pushing at you. However, if you are running or cycling in a circle, you will need to tilt yourself towards the centre of the circle. The effect is to push yourself in the direction. Of course, you push yourself by pushing against the ground, so it is really the ground that pushes you. That's the force we are looking for.

Without this force, you would just go straight. With this force pushing you sideways, you would change direction. If a force of the same magnitude keeps pushing you sideways, you would move sideways by the same amount every second. The resulting path would be a circle.

A force that makes a body go in a circle is called a centripetal force. This sideways force is effectively pointing to the centre of the circle. It is also perpendicular to the direction of the motion.

According to Newton's second law, a force produces acceleration. If a body goes round a circle at uniform speed, where is the the acceleration?

The abstract answer is, the direction changes, so the velocity changes, therefore there is acceleration, since acceleration is the change of velocity per unit time. We can also calculate this acceleration by doing a vector subtraction of the initial and final velocities, over a short time.

We can also think of a more intuitive answer. Suppose I am sliding ahead on a frozen pond. If there is no force, I would just go straight. There is no sideways movement. If a force pulls me to the left for a short time, I would still move ahead, but at the same time a little bit to the left. So I have gained some velocity to the left, where previously there is none. So I have accelerated to the left, even if my forward velocity remains the same.

We see that it is the speed perpendicular to the direction of motion that changes. For circular motion, this is the direction to the centre of the circle. So the acceleration is directed to the centre, and is called the centripetal acceleration.

The Physics Notes