I haven’t done a “basics” topic in quite some time. It’s odd, I have used centripetal acceleration quite often, but I never derived the expression that I use. To get to the point, the magnitude of the acceleration of an object moving in a circle is:
Also, the direction of this acceleration vector is always towards the center of the circle the object is moving in. This is really not too difficult to derive (but it does use at least one “trick”). Let me start with an object moving in a circle at a constant speed. I am going to show to instances of the object one and then again after a very short time.
To find the acceleration, I will just use the same normal definition of average acceleration:
From the diagram above, I can determine the change in velocity. Let me write the vectors for v1 and v2:
So, the second velocity has components in both the x- and y-directions (which I didn’t label, but you get the idea). Ok, now for the first trick. This angle ? is really small. It has to be if I want to get an expression for the acceleration of an object moving in a circle. Just think, what if I took ? = 2?? In this case, I would be finding the average acceleration for the object going all the way around the circle. Clearly this would be zero since the two velocity vectors would be identical.
Since ? is small, I can say that cos(?) is about equal to 1 and sin(?) is about equal to ? (where ? must be in radians). This means I can write v2 as:
Hopefully, you won’t feel cheated with those tricks. The first one should be clear. Look at the diagram with the vectors. The second velocity vector is moving in the x-direction essentially just as much as the first vector. The smaller ? is, the truer this is.
For the sin trick, look at the sin function. Near ? = 0, the function is really really really straightish looking. Right? If you plot y = sin(?) and y = ?, near the origin, these two functions are right on top of each other. Here is a plot showing that:
Now, I can start putting this stuff together. Let me calculate the change in velocity:
Boom. I just got the direction of the acceleration for an object moving in a circle. In this particular case, it is in the negative y-direction. If you look at the vector diagram you can see that this is towards the center of the circle. Now, for the magnitude of the acceleration, I need to first find the time that this change in velocity takes place. I know the speed (v) and I know the distance of the arc-length. This gives a time of:
Putting the change in velocity together with the change in time (in non-vector form):
Done. This is the value I have been using all along. Of course, if you know the angular speed instead of the linear velocity this can be written differently. First, the relationship between linear velocity and angular velocity for something moving in a circle:
Substituting in for v, it is easy to see that the acceleration (magnitude) can be written as:
That is it. The acceleration of an object moving in a circle.