Conservation of Energy + circular motion

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To determine the minimum initial velocity for a mass to complete two circular motions, one must consider the conservation of energy and the forces acting on the mass at the top of the circle. At the top, the centripetal force requirement, mv^2/r = mg, ensures the mass maintains circular motion. While theoretically, if mechanical energy were perfectly conserved, the mass could continue indefinitely, real-world factors like friction and air resistance lead to energy loss, preventing perpetual motion. Thus, additional initial velocity is necessary to account for these losses and ensure the mass completes multiple revolutions. Understanding these dynamics is crucial for accurately applying conservation of energy principles in circular motion scenarios.
anotherghost
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Quick question:

I'm having trouble understanding a concept: say you've got a mass hanging from a peg by a string. Using conservation of energy, you can figure out what the minimum velocity is that the mass has to have initially so that it goes around the circle: at the top, mv^2/r = mg, so it just makes it around once. However, I'm having trouble with this - how do you calculate the minimum initial velocity so that it goes around the circle twice?

Confusion.
 
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Say it goes around once. If mechanical energy is truly conserved, then it should go around the circle twice, thrice, four times and so on. In other words, if it has enough kinetic energy at the bottom to make it through the top and energy is not lost anywhere, then it will keep on going forever. However, in real life, energy is lost to friction and air resistance so this does not happen
 

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