Lab 8: Centripetal Acceleration

The commonly believed centrifugal force has long been rendered fictitious. Although it is true that an object spun in circular motion tends to fly outwards, it must be made clear that this is not a force externally acted upon an object, but comes from the object's moment of inertia. Instead, an object confined into circular motion must be acted upon by an external force pulling it towards the center. As such, centripetal force is not some independent physical phenomenon, but the force named for causing circular motion; this force could be tension (when the object is spun by a string), or normal (for example, in banked curves). It might be more clear, then, to call it centripetal acceleration (and also centrifugal acceleration that an object causes about a system) caused by other forces. Since centripetal does not refer to an independent force, we conclude that there must be some relationship such that we could find centripetal acceleration without introducing any additional variables.

If we consider an object swung uniformly about with a string, its tendency to move in a straight line (in one direction) due to inertia is perpendicular to the string. In order to determine this relationship, we sample some initial point in the flight path, and call the velocity direction and magnitude v_i. Then, we sample another point and call this the final velocity v_f, such that the angle formed in between them is θ:

Note that both v_i and v_f have the same magnitude, but different directions. And since they are both exactly perpendicular to the points that form the angle θ, the angle between them must also be θ. We can place them together to calculate a vector difference, or Δv.

The change of this velocity over the change of time is the definition of acceleration:

 a = Δv / Δt

To find Δv, we consider that the vector diagram forms an isosceles triangle because both v_i and v_f have the same magnitude. Therefore, we can divide θ down the middle, forming 2 identical right angles, with v_i and v_f being the hypotenuses. Then, we can say that half of Δv is v*sin(θ/2), as illustrated by the following:

Therefore:

Δv = 2v sin(θ / 2)

And since time is the arc length divided by speed, we have (where r is radius):

Δt = r θ / v

We now have all we need to find the relationship. Acceleration restated becomes:

a = [2v sin(θ / 2)] / [r θ / v]

a = (v² / r) [sin(θ / 2) / (θ / 2)]

And since we're interested in instantaneous acceleration:

a = v² / r

This means that if we devise an experiment testing the acceleration and velocity squared of an object, we should note a linear relationship, and that relationship (slope) should be the radius. And we're going to do just that!

Pictured above is a heavy iron plate used for esoteric training by Shaolin monks. We attached an accelerometer to it, so that when we spin it, the accelerometer would pick up the acceleration and record it into Logger Pro. Also required are 5 pieces of masking tape and a messy desk. In order to get the velocity, we use rudimentary (compared to the accelerometer) handheld timers, timing 2-4 revolutions of the plate. If we divide the number of revolutions by the time, we should get the angular velocity (rev/s). On that note, we know that angular velocity ω is:

ω = v / r
a = r ω²

Here's the data from the timers, the calculations, and the data recorded by the accelerometer (wow, it's that much easier to copy your own spreadsheet directly into a blog instead of messing with copying and pasting from those Macbooks):

Time (s)	Rev	Velocity (ω)	Acceleration (m/s²)
4.4	2	0.455	1.40
2.5	2	0.800	5.00
3.7	2	0.541	2.20
1.4	2	1.429	12.90
1.6	4	2.500	37.41

 If we plot angular velocity against acceleration, this is what we get:

The linear fit has an pretty good correlation, meaning the experiment is pretty consistent. The slope is 0.1493, so we should expect the radius to be about 14.9 cm, if our calculations were correct.

Measuring the plate with a ruler, we find the radius to be about 15 cm. That confirms our results.