"See saw" balance/instability 3

[pizza]

Can someone help me with an explanation to a semmingly simple balancing problem. I would like to have some show me why a balance (or say a see-saw) returns to its equilibrium point mathematically speaking.

If you draw a simple diagram of a see-saw with the fulcrum at the center and two equal weights at equal distance from the fulcrum BUT the see-saw is not horizontal in the starting position, what torque imbalance is causing the see-saw to return to its balanced position? If you do the sum of the torques about the center fulcrum, the force vectors times the distance to the fulcrum, pointing downward, cancel eachother, don't they? So why wouldn't the balance just stay in the "out-of-horizontal" position. My reference x and y are each pointing horizontally to the right and vertically upward respectively.

How do you prove MATHEMATICALLY that the see-saw is NOT in equilibrium when initially positioned as described above?

Thanks for any help.

 

[ivymike]

Hint: Who put your fulcrum on the line between the masses? Move it a hair in either direction and your problem will probably become a bit easier to figure out.

 

[ivymike]

What I was trying to say was that perhaps you should make your balance a bit T-shaped, with the fulcrum offset vertically from the horizontal line that connects the masses.

 

[pizza]

Thanks for the suggestions ivymike. But it still doesn't answer my question. Given the conditions I've described (without adjusting them)I still need the answer.

 

[Hush]

Are you sure the see-saw would return to equilibrium?

 

[trihydrate]

Let me offer an illustration of what I think ivymike is trying to say. If you take a tee with a short branch and hang it from the end of the branch, it will hang level if the center of gravity of the tee exists at the intersection of the runs and the branch. If the tee is pushed down on one run, the center of gravity is shifted away from the vertical line under the end of the branch and when released will oscillate back and forth until the tee stops in the level position again.

The balance would stay in the "out-of-horizontal" position only if the point it is hanging from and the center of gravity are the same.

 

[dvd]

My rub on this is that when you perform this experiment with real objects there is no such thing as a force vector. You set real weights on the see-saw, with centroids above the beam. The distance from the fulcrum to the centroid is not the same from one side to the other. The angle the beam makes is equal above and below horizontal, but the angle from fulcrum to centroid is not the same. Draw a simple experiment on your cad program and you can prove it.

 

[SAP]

Here's something else to consider, albeit totally impractical and highly unlikely to be a consideration in the real world....The gravitational force acting on the see-saw is a function of the square of the distance from the centre of the earth to the centre of the mass of the see-saw. Therefore, if one side of the see saw is pushed down, the attraction to the earth is greater than the other side, and therefore it should not return to the "horizontal equilibrium" at all!!
SAP

 

[MikeMech]

I think DVD's got it.

Picture an empty seesaw with constant cross section and weight per foot. If the fulcrum is positioned AT the centroid of the cross section, midway along the seesaw, the seesaw will theoretically be in equillibrium in any position. If it is slightly below or above the centroid, the moment arms change slightly with rotation.

Then again, most of the seesaws I ever played on weren't that well balanced.

 

[Speedy]

A diagram might help.

Always return to equilibrium (level);

0
IIIIIIIIIIIIIIIIIII



Will stabilise anywhere;

IIIIIIIII0IIIIIIIII


Where 'O' is the fulcrum.

Correct?

Speedy

 

[Ralph2]

Look at it from this perspective... Support a long pipe with a crane, choked with a sling so that it is "balanced" (level). Now a force down on one end will cause the pipe to move up or down. but remove the force and the pipe will level out again... Same as your see-saw.. Why? When the pipe is level the sling is directly above the center of gravity but as the pipe goes off level this point is no longer directly over the C.G. Now the sum of the forces are no longer equal and the resulting force acts in favour of returning the pipe level. The thicker the pipe the higher self leveling force because the distance from the C.G. and the fulcrum of the sling is greater. Moving towards what ivymike was talking about.
Hope this helps......
Ralph

 

[pizza]

Hey, I just wanted to thank everyone for helping me out on this. Greatly appreciated!! It really helped.

 

[vtl]

you can perform a small dynamic mathematic derivation to find the largest angle you can swing the seasaw before it becomes 'unstable'.

Reguards, consider the inertial point at the fulcrum.