Centripedal Acceleration

Centripedal Acceleration

Centripedal acceleration is a topic that has come up numerous times and I think it deserves a little attention. It is something that can be observed all the time in real life, whether it be driving your car around a corner, reaching altitude in an airplane or maybe going on an amusement park ride like the pirate ship. What isn't so obvious is how the acceleration is calculated or which direction the acceleration is acting. To determine how centripedal acceleration is calculated we must first look to the definition of acceleration.

a = dv / dt

Hopefully this equation looks familiar. It states that acceleration is determined by how a velocity changes with time. Remember that acceleration is a vector quantity; it has both a magnitude and a direction (ie. for every second the velocity changes by three meters per second, to the right --> a = 3m/s^2 to the right). Since we know that time is not a vector because it has no direction we can deduce that in order for acceleration to be a vector the velocity must also be a vector. Okay now we see that the magnitude and the direction of acceleration depends on how the magnitude and the direction of the velocity change with time.

If the block in the diagram above is travelling around the circle at a constant speed then the velocity magnitude is not changing... but the direciton is! This is how we define centripedal acceleration.

You can see from the image above that two points in time are noted with a velocity tangent to the circle in each. Never mind all the numbers and try to see just how the acceleration is determined. The two velocities are of equal magnitude but have slightly different directions. To determine how the velocity changed in this point in time we use the tale to tale idea and find that the change in the velocities is the small vector pointing to the middle of the circle. This is easiest to see in the diagram in the top right corner. The theory to find the equation for centripedal acceleration relies on the principle of taking an instantanious change in time so that the two vectors are at nearly the same point. This may allow you to imagine the diagram a bit better.

I recommend that you skim through the derivation from the first equaiton above to the second equation to understand how the equation for centripedal acceleration is determined.