# Relationship of centripetal force and acceleration

### What is centripetal acceleration? (article) | Khan Academy

This constantly changing velocity means that the object is accelerating ( centripetal acceleration). For this acceleration to happen there must be a resultant force. Centripetal Acceleration and Centripetal Force. Ready For More Physics Fun? Circular Motion. When an object moves in a circle at constant speed, we describe . The direction of the net force is in the same direction as the acceleration. So for an object moving in a circle, there must be an inward force acting upon it in order .

Click to break the string and observe the ball move off at a constant velocity. When the string holding the ball to the centre is broken, the ball travels in a straight line along what would have been a tangent to the circle.

Having the string in place is essential to keep the ball moving with circular motion. In this unit we will look at how the force in the string keeps the ball moving in a circle.

Acceleration As the ball in the above diagram moves around the circle, it is travelling at a constant speed. However, the direction in which the ball moves is changing, so its velocity is changing since velocity An object's velocity states both the speed and direction of motion relative to a fixed reference point. Since the velocity is changing, the ball is accelerating. Click through the stages in the analysis of Fig. Determining the acceleration of a body moving in a circle.

The stages in determining the acceleration of the ball in Fig. Consider an initial instant when the velocity of the ball is u.

## Centripetal force

A fraction of a second later, the ball has moved and its velocity has changed to v. Note that the magnitude of the velocity is the same, but its direction has changed. Acceleration is defined as the rate of change of velocity. Indeed there is no physical object accelerating you forwards.

The feeling of being thrown forwards is merely the tendency of your body to resist the deceleration and to remain in its state of forward motion.

This is the second aspect of Newton's law of inertia - "an object in motion tends to stay in motion with the same speed and in the same direction You are once more left with the false feeling of being pushed in a direction which is opposite your acceleration.

These two driving scenarios are summarized by the following graphic. In each case - the car starting from rest and the moving car braking to a stop - the direction which the passengers lean is opposite the direction of the acceleration. This is merely the result of the passenger's inertia - the tendency to resist acceleration.

The passenger's lean is not an acceleration in itself but rather the tendency to maintain the state of motion while the car does the acceleration. The tendency of a passenger's body to maintain its state of rest or motion while the surroundings the car accelerate is often misconstrued as an acceleration.

This becomes particularly problematic when we consider the third possible inertia experience of a passenger in a moving automobile - the left hand turn. Suppose that on the next part of your travels the driver of the car makes a sharp turn to the left at constant speed. During the turn, the car travels in a circular-type path. That is, the car sweeps out one-quarter of a circle. The friction force acting upon the turned wheels of the car causes an unbalanced force upon the car and a subsequent acceleration.

The unbalanced force and the acceleration are both directed towards the center of the circle about which the car is turning.

Your body however is in motion and tends to stay in motion. It is the inertia of your body - the tendency to resist acceleration - that causes it to continue in its forward motion. While the car is accelerating inward, you continue in a straight line. If you are sitting on the passenger side of the car, then eventually the outside door of the car will hit you as the car turns inward. This phenomenon might cause you to think that you are being accelerated outwards away from the center of the circle.

In reality, you are continuing in your straight-line inertial path tangent to the circle while the car is accelerating out from under you. The sensation of an outward force and an outward acceleration is a false sensation. There is no physical object capable of pushing you outwards.

You are merely experiencing the tendency of your body to continue in its path tangent to the circular path along which the car is turning.

## What is centripetal acceleration?

You are once more left with the false feeling of being pushed in a direction that is opposite your acceleration. The Centripetal Force and Direction Change Any object moving in a circle or along a circular path experiences a centripetal force. That is, there is some physical force pushing or pulling the object towards the center of the circle. This is the centripetal force requirement. The word centripetal is merely an adjective used to describe the direction of the force. We are not introducing a new type of force but rather describing the direction of the net force acting upon the object that moves in the circle.

Whatever the object, if it moves in a circle, there is some force acting upon it to cause it to deviate from its straight-line path, accelerate inwards and move along a circular path.

Three such examples of centripetal force are shown below. As a car makes a turn, the force of friction acting upon the turned wheels of the car provides centripetal force required for circular motion.

### Centripetal Forces and Accelerations

As a bucket of water is tied to a string and spun in a circle, the tension force acting upon the bucket provides the centripetal force required for circular motion. As the moon orbits the Earth, the force of gravity acting upon the moon provides the centripetal force required for circular motion. The centripetal force for uniform circular motion alters the direction of the object without altering its speed.

• Circular Motion
• The Centripetal Force Requirement

The idea that an unbalanced force can change the direction of the velocity vector but not its magnitude may seem a bit strange. How could that be?

There are a number of ways to approach this question.

### The Centripetal Force Requirement

One approach involves to analyze the motion from a work-energy standpoint. Recall from Unit 5 of The Physics Classroom that work is a force acting upon an object to cause a displacement. As the centripetal force acts upon an object moving in a circle at constant speed, the force always acts inward as the velocity of the object is directed tangent to the circle.

This would mean that the force is always directed perpendicular to the direction that the object is being displaced. The angle Theta in the above equation is 90 degrees and the cosine of 90 degrees is 0. Thus, the work done by the centripetal force in the case of uniform circular motion is 0 Joules.

Recall also from Unit 5 of The Physics Classroom that when no work is done upon an object by external forces, the total mechanical energy potential energy plus kinetic energy of the object remains constant.