Accelerometer Version-2

So, I built a new accelerometer. Why? The jelly-jar one was just not doing it for me. Plus, the cork was starting to make the water all yellowy. It was a good start, but I can do better. What was wrong with the jelly-jar one? First, it didn't let the cork move very far before hitting the wall. Second, it was kind of hard to see exactly where the cork was. Lastly, there was no way to get a reading of the acceleration from the jelly-jar. Now, I am going to fix that.

My new design uses a sphericalish glass flask. The floating bob is anchored in (near) the center of this sphere. Here is a picture:

Maybe it is not clear from the picture, here is a diagram.

(shown at some acceleration so the bob is not vertical) Now, how do I make markings on this so I can get some measurements. Clearly, the angle the bob makes with the vertical is going to be related to the acceleration of the accelerometer. The best way to look at this is first to consider the acceleration as a fake force. Usually, this fake force idea is discouraged in introductory classes. However, in this case I am dealing with a non-inertial frame - if you know what you are doing it is ok. Suppose I modify my device so that instead of floating, it is a ball hanging (this way I don't need to worry about the buoyant force). Here is a free body diagram:

So, this fake force is going to be in the opposite direction as the acceleration and have a value of ma. Note that this hanging mass is not as good of an accelerometer for two reasons. First, the mass is going to swing (there is no dampening force). Second, the mass swings in a direction opposite to the acceleration. But anyway, there is a relationship between the angle the mass hangs at and the acceleration (assuming I could stop it from swinging). In this frame, the horizontal (x) forces must add up to zero and also for the vertical (y) forces. This gives:

I hope you realize that I am calling the tension F_T and the angle the string makes with the vertical is ?. Using the y-equation, I can solve for F_T. This can be plugged into the x-equation to get:

The whole point is to get an expression relating the acceleration to the angle the thing is hanging at. Let me put in ma instead of the fake force and solve for a:

This obviously has the correct units for acceleration. Also, when ? = 0, a = 0 m/s². What about for really high accelerations? When ? approaches ?/2, the acceleration approaches infinity.

This leads to the other way of thinking about this accelerometer. I can just say as Einstein says that the bob can not tell the difference between gravity and acceleration. I can re-define the local "gravity" as the vector sum of g(vector) and a(vector). This is the same as what I did above. However, using this second method it can be seen that the floating cork would have the same relationship between angle and acceleration as the hanging mass.

To finish my second-gen accelerometer, I need to add some lines for acceleration. How much would the bob be deflected if the acceleration (perpendicular to gravity) is 1/2 g?

What about other accelerations? 1 g would be 45°, 2 g's would be 63.4°, 3 g's would be 71.6° and so on. What about 10 g's (don't do that or you will likely hurt yourself), but that would be a deflection of 84.3°. You get the idea.

So that is a much better accelerometer? But does it really work? It is really difficult to get reliable accelerations just holding it. Any reasonable acceleration makes you go too fast, people just can't do that very easily. It should work fairly well in a car and I am going to make a cup-holder adapter for it. However, the easiest acceleration to reproduce AND measure is centripetal acceleration. When I get time, I will make a video of this on an rotating platform so that we can check how well it measures.