Centripetal

So that's centripetal force causing centripetal acceleration which causes the object to go towards the center

And this force is called the centripetal force

And that centripetal acceleration is the acceleration due to gravity

We'll do more videos on intuition of centripetal acceleration and centripetal forces, inward forces, if this does not make complete intuitive sense for you at this moment.

Another situation like this, once again this involves centripetal acceleration, inward forces, inward acceleration.

What I want to do in this video is a calculus proof of the famous centripetal acceleration formula that tells us the magnitude of centripetal acceleration, the actual direction will change it's always going to be pointing inwards, but the magnitude of centripetal acceleration is equal to the magnitude of the velocity squared divided by the radius

Centripetal force, centri you might recognize as center and then petal is seeking the center.

The magnitude of the centripetal acceleration is equal to your speed, the magnitude of your velocity, squared, divided by your radius.

So this centripetal force, something is pulling on this object towards the center that causes it to go into this circular motion

At least at the bottom half right over here the track itself is actually providing the centripetal force to keep it going in a circle

In fact if you did not have gravity which is causing the centripetal acceleration it would just go in a straight path forever and ever

So calculating the altitude simplifies into calculating the point where the magnitudes of the centripetal acceleration required for orbital motion and the gravitational acceleration provided by Earth's gravity are equal.

So we know that the magnitude of your centripetal acceleration is going to be equal to your speed squared divided by the radius of the circle that you are going around

So we know from our studies of circular motion so far what's keeping it going in circular motion assuming that it has a constant speed is some type of centripetal acceleration

So there's a force, the centripetal force in this yoyo example is the tension in the string that's constantly pulling on the yoyo towards the center and that's why that yoyo goes in a circle

If you speed up a lot beyond that speed, you will slowly spiral away from the Earth because then the centripetal acceleration due to gravity won't be enough to keep you in a perfect circular path

Earth orbit A spacecraft enters orbit when it has enough horizontal velocity for its centripetal acceleration due to gravity to be less than or equal to the centrifugal acceleration due to the horizontal component of its velocity.

So in the example of a satellite or anything in the orbit even the moon in orbit around the Earth the thing that's keeping an orbit as opposed to flying out into space is a centripetal force of Earth's gravity

When you see a car going at a constant speed so on the speedometer, it might say, 60 mph, 40 mph, whatever the constant speed but it's traveling in a circle so what is keeping what is the centripetal force in that example?

So if we square it, this is going to be (v over r) squared, now we saw that in the video on angular velocity, times r and this is all going to be the magnitude of our acceleration, which is really our centripetal acceleration, our inward directed acceleration.

And in the next video we'll figure out how fast does it have to travel in order to stay in orbit, in order for it not to plummet to Earth due to the force of gravity, due to the acceleration that is occurring, this centripetal, center seeking acceleration.

So we know that the magnitude of that centripetal acceleration has to be equal to this speed or the magnitude of the velocity squared divided by r where r is the radius of the circular path the radius of the orbit, which would be the radius of Earth plus the altitude