What is the maximum angle that can be reached by a pendulum in an accelerometer?

[radi0headfan]

Hi all, I'm new to the forum and figured someone on here would be able to help me out w/ my question. I'm having trouble understanding something regarding an accelerometer. Here's a brief description of an accelerometer taken from an excerpt I read to get you all up to speed on what I'm dealing with:

"An accelerometer is a device that may be used to measure the acceleration of an object moving horizontally. One type of accelerometer consists of a simple pendulum made up of a small dense body suspended from a "massless" rod. The unweighted end of the rod is fixed, but is allowed to pivot freely. When the system is accelerated to the right, the pendulum swings back toward the left, and makes an angle θ with the vertical. The size of this angle θ gives a measure of the acceleration, so the larger the angle, the greater the acceleration. When there is no displacement from the vertical, the system is traveling with a constant velocity."

So my question has to do with the maximum angle that can be reached by the pendulum in the accelerometer. "Assuming there are no limitations on the acceleration of the object, the maximum angle(theta) that the pendulum of the accelerometer can make with the vertical has to be just under 90". The explanation that I read said that the vertical component of the tension in the rod needs to cancel out the mass of the bob, and in order to do so, the maximum angle would have to be less than 90 in order for there to be a vertical component of the tension. Now I understand the reasoning that was given, but I don't understand the logic behind it. I don't understand why the the tension in the string has to cancel out the mass of the bob and why it can't continue upwards past 90 degrees. Why can't there be a net force in the upward direction that would justify a greater angle? Any ideas? Thanks in advance for any help.

 

[berkeman]

If the acceleration were infinite to the right, the pendulum would be all the way to the left, at 90 degrees. Do you see that? And anything less than infinite acceleration will allow the pendulum to move slightly downward.

Are you familiar with drawing free body diagrams yet?

http://en.wikipedia.org/wiki/Free_body_diagram

 

[radi0headfan]

I don't see that. The way I'm thinking about it is that the object accelerates infinitely, and then the pendulum, initially at 0 degrees and having the capability for 360 degree rotation, feels that force too. I don't understand why the bob cannot exceed 90 degrees though. Is there an equation, maybe dealing with laws of energy conservation, that can justify this or? I just don't see the justification for no net force in the upward direction if there's a large enough force applied to the object.

 

[berkeman]

Oh, I think I see your confusion. The pendulum angle is meant to be in the quiescent position, given some magnitude of acceleration. That's what a free body diagram shows, an equilibrium position of things in response to forces and accelerations. You are correct, that if you diagram the angle theta as you start accelerating the system to the right, that the pendulum could swing past the vertical for finite accelerations...but it won't stay there. Consider the case where the acceleration is slowly ramped up -- the pendulum will always stay below the horizontal then, right?

 

[radi0headfan]

I'm familiar with drawing free-body diagrams. For the free body diagram, I'm guessing there would be an equal and opposite force, assuming no frictional forces, on the accelerometer's pendulum? I'm not sure how that force would be broken down into its components when dealing with the pendulum though.

 

[radi0headfan]

Oh, I think I see your confusion. The pendulum angle is meant to be in the quiescent position, given some magnitude of acceleration. That's what a free body diagram shows, an equilibrium position of things in response to forces and accelerations. You are correct, that if you diagram the angle theta as you start accelerating the system to the right, that the pendulum could swing past the vertical for finite accelerations...but it won't stay there. Consider the case where the acceleration is slowly ramped up -- the pendulum will always stay below the horizontal then, right?

Hmm, my understanding is that if the object is at standstill or already moving at constant velocity that the accelerometer will be at 0 degrees. I'm saying at any point in its motion, if infinite acceleration is applied, the max angle on that pendulum will be slightly less than 90. I'm having trouble correlating the force applied on the object to the resulting tension in the pendulum. If it was true that it cannot go above 90 degrees, then the vertical component of tension always has to cancel out the mass of the bob. I'm trying to figure out why the acceleration of the object(given that its on a flat horizontal surface), cannot result in an overall net force, and therefore net acceleration, in the upward direction of the pendulum. If that happened, then it would go above 90 degrees, but supposedly it can't and I'm wondering where my flaw in logic is. Thanks again for your help.

 

[radi0headfan]

oh i think i just figured it out...ill post what I am thinking and see if it makes sense.

 

[radi0headfan]

I wanted to actually try to figure out an equation that would apply to the different cases, such as acceleration of the object in only the horizontal direction or acceleration in both horizontal and vertical directions. After drawing a free body diagram, I'm seeing that the force on the rod in the pendulum is going to be equal but opposite in direction to the force applied on the object, given that there's no frictional forces. So after the force is applied, there's an equal and opposite horizontal tension and a tension going down due to the mass of the bob. The resultant vector of these forces will always be at an angle less than 90. However, if there was a component of applied force going downwards, the tension due to it would be in the upward direction cancelling out the force due to gravity. Tcos(theta) will now, instead of being equal to mg, will be equal to F - mg and if F is equal to or exceeds mg, the angle will become equal to or larger than 90. This seems correct but please let me know if you think something's wrong. Thanks.