# Relationship between centripetal force and velocity

### The Centripetal Force Requirement

This constantly changing velocity means that the object is accelerating ( centripetal acceleration). For this acceleration to happen there must be a resultant force. of Gravitation Up: Circular Motion and the Previous: Relationship Between Linear and Centripetal acceleration is the rate of change of tangential velocity: . Mass, velocity, and radius are all related when you calculate centripetal force. In fact, when you know this information, you can use physics equations to.

Now, what I wanna do in this video is see if I can connect our centripetal acceleration to angular velocity, our nice variable omega right over here and omega right over here you could use angular speed. It's the magnitude, I could say our magnitude of our angular velocity, so our angular speed here. So, how can we make this connection?

### Deriving formula for centripetal acceleration from angular velocity (video) | Khan Academy

Well, the key realization is to be able to connect your linear speed with your angular speed. So, in previous videos, I think it was the second or third when we introduced ourselves to angular velocity or the magnitude of it which would be angular speed, we saw that our linear speed is going to be equal to our radius, the radius of our uniform circular motion times the magnitude of our angular velocity and I don't like to just memorize formulas. It's always good to have an intuition of why this makes sense.

Remember, angular velocity or the magnitude of angular velocity is measured in radians per second and we typically view radians as an angle but if you think of it as an arc length, a radian you could view it as how many radii in length am I completing per second?

And so, if I multiply that times the actual length of the radii, then you can get a sense of well, how much distance am I covering per second?

## Circular Motion

Hopefully that makes some sense and we actually prove this formula, we get an intuition for this formula in previous videos but from this formula it's easy to make a substitution back into our original one to have en expression for centripetal acceleration, the magnitude of centripetal acceleration in terms of radius and the magnitude of angular velocity and I encourage you, pause this video and see if you can drive that on your own.

All right, let's do this together. So, if we start with this, we have the magnitude of our centripetal acceleration is going to be equal to, instead of putting V squared here, instead of V, I can write R omega, so let me do that, R and then omega. Yet you would have a difficult time identifying such a backwards force on your body.

Indeed there isn't one. The feeling of being thrown backwards is merely the tendency of your body to resist the acceleration and to remain in its state of rest.

The car is accelerating out from under your body, leaving you with the false feeling of being pushed backwards. Now imagine that you are in the same car moving along at a constant speed approaching a stoplight. The driver applies the brakes, the wheels of the car lock, and the car begins to skid to a stop.

There is a backwards force upon the forward moving car and subsequently a backwards acceleration on the car. However, your body, being in motion, tends to continue in motion while the car is skidding to a stop. It certainly might seem to you as though your body were experiencing a forwards force causing it to accelerate forwards. Yet you would once more have a difficult time identifying such a forwards force on your body. 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.

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.

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.