Circular motion (approximation of centripetal acc.)

[V0ODO0CH1LD]

Say I have a body moving in a circle of radius r with a constant velocity v.

The time it takes the body to go around the circumference once is:

T = 2πr/v

Then the time it takes the body to go around a fourth of the circumference is T/4.

Now, imagine a diagram where when the body is at the leftmost portion of the circumference its velocity vector is pointing straight up, and when it is at the upmost portion of the circumference its velocity vector points entirely to the right. That is the body traveling over a fourth of the circumference.

Now, the magnitude of the acceleration vector required to make the body behave that way over a period of T/4 is (v√2)/(T/4). And (v√2)/(T/4) is an approximation of the centripetal acceleration for this case.

My question is: why is that an approximation? Why if I do the same thing over a smaller change in time I get closer to the centripetal acceleration? A circle is symmetric and the velocity is constant, right?

 

[Redbelly98]

Say I have a body moving in a circle of radius r with a constant velocity v.

The time it takes the body to go around the circumference once is:

T = 2πr/v

Then the time it takes the body to go around a fourth of the circumference is T/4.

Now, imagine a diagram where when the body is at the leftmost portion of the circumference its velocity vector is pointing straight up, and when it is at the upmost portion of the circumference its velocity vector points entirely to the right. That is the body traveling over a fourth of the circumference.

Agreed, so far.

Now, the magnitude of the acceleration vector ...

Not quite. This is the magnitude of the average acceleration, where the average acceleration is defined as
\vec{a_{avg}}=\frac{\Delta \vec{v}}{\Delta t}

... required to make the body behave that way over a period of T/4 is (v√2)/(T/4). And (v√2)/(T/4) is an approximation of the centripetal acceleration for this case.

My question is: why is that an approximation?

Because it is an average, and the actual instantaneous acceleration changes substantially (because it changes direction) during the time interval.

Why if I do the same thing over a smaller change in time I get closer to the centripetal acceleration?

Because, for a small time interval, the acceleration does not change direction very much, and is more closely approximated by its average over that time interval.

A circle is symmetric and the velocity is constant, right?

No, velocity is a vector and is not constant if its direction changes. Speed is constant, however.

 

[V0ODO0CH1LD]

That helped a lot! Thanks!

One for additional question though. How does all of that get summarised into (ω^2)*r or (v^2)/r?

 

[Redbelly98]

You're welcome

This explains things pretty well:

http://scienceblogs.com/dotphysics/2009/07/basics-centripetal-acceleration/