Consider a pendulum in it's balance point hanging from ceiling. It can swing in all the directions in the space. The pendulum can only swing in a sphere(the string can't bend). Now, is it possible to release the pendulum in a particular height and with a initial condition that in the first go(the same height as the first place) it doesn't pass the balance point but in the return it does? (don't count air or friction) If it is possible what initial conditions and height(s) are required? If not why? Which laws I need to use to solve this problem?
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2 Answers
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The spherical pendulum conserves angular momentum about a vertical axis running through the fixed point of the pendulum. Consider when the pendulum reaches its lowest elevation, so that its velocity is completely horizontal. Its angular momentum at this point is $L = mvr$, where $m$ is the mass of the pendulum bob, $v$ is its velocity, and $r$ is the distance from the vertical axis. There are two situations (assuming the pendulum is moving):
If $r$ is zero, then the pendulum is passing through the balance point. Its angular momentum is zero, and must always be zero. This means that it will pass through the balance point on every swing because it can only move directly towards and away from that point.
If $r$ is not zero, then the pendulum passes to the side of the balance point and will never cross through it. Angular momentum is conserved, which means the value never changes, so the angular momentum can never be zero. This means that the pendulum cannot cross the balance point because, as we saw in the point above, it would have zero angular momentum.
To think about this another way, if the pendulum passes through the balance point, there are no forces at that point, so the pendulum will continue in a straight line up and back to to the balance point. Now think about what happens if time is reversed. The exact same thing! There are no sideways forces, so the pendulum only goes up and back through the balance point. If it reaches the center point, then at no time in the past could it have missed that point. From this, we can conclude that either the pendulum goes through the center point on every swing, or it never goes through.
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No, you cannot, athe pendulum, given any initial speed and initial height will reach a given balance point. By conservation of energy (and in absense of any loses from friction) it will always return to the same height (or balance point), there no way to it could go higher next time, it would need to acquire energy from some source (but it can reach a lower balance point if it loses energy due to friction).
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$\begingroup$ initial speed can be given. And i think you misunderstood me. I meant you can release the pendulum with initial conditions and then it doesnt pass the balance point. and in return it does. $\endgroup$ Commented Dec 9, 2014 at 3:06
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$\begingroup$ I didnt misundertood you: I wrote "given any initial speed and initial height" $\endgroup$– user65081Commented Dec 9, 2014 at 3:20