Hi, all! I just registered here, and hope to enjoy enlightening discussions about the subject of physics in the future.(adsbygoogle = window.adsbygoogle || []).push({});

The spesific reason for the post is a project I am currently concerning myself with. I hit a snag recently, and hope to resolve it with help from others here. Here goes:

Background

I'll ignore friction and air resistance in the following.

I'm making an apparatus consisting of a beam (B) rotating about a pivot (A), loaded with two masses M and m; one attached to each end. It looks like a sort of see-saw.

Now, there are three forces in play, all due to gravity:

-The force on the largest mass M, G_{M}

-The force on the smaller mass m, G_{m}

-The force on the beam itself, G_{B}.

Please note that A may be different from the beam's centre of gravity.

The system is designed so that of the three resulting torques (only two if the beam's weight doesn't set up a torque about A), T_{M}will always be bigger than the other(s).

The result is that the system will be set in rotational motion if held in an initial position, and then released. The mass M will move in an arc downwards, and then hit some kind of blockage (e.g. the floor).

I have calculated the system's moment of inertia (I), and the magnitude of the three torques are easily found. The angle between the beam and the horizontal axis will be denoted as θ.

Problem:

Once the system is released, the total torque will accelerate the system from rest because of ΣT = Iα.

In so doing, the mass M will move through a small angle dθ, and the lever arm of T_{M}will be shortened since the arm is equal to R*cos(θ). Thus the torque decreases, and also the acceleration. The same is true for the mass m on the other side of the pivot, only that its torque in the other (negative) direction is decreased.

The net result I'm guessing will be a steadily decreasing acceleration and an angular speed that keeps growing throughout the motion, but less and less with time.

How do I model this change in acceleration and speed mathematically?

My goalis to calculate the final speed of the masses M and m, but I can't quite relate the varying quantities to time. My guess is that this will lead to some differential equations, and I'm not steady enough with my math to crack this one.

Any help will be greatly appreciated!

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# Variable torque due to rotation

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