Mechanics- Conservation of energy

[AlonZ]

Homework Statement

Need to find the angle from which a pendulum with the length L that reaches its lowest point will complete a full circle around a pivot which distance is from the lowest point is l

Relevant Equations

assume Tension at the highest point of the loop is 0, because that's the minimun for a full loop.

My, supposedly rational thought is that if the pendulum will drop from a height higher than the top of the loop's height, by the law of conservation of energy, it'll have enough velocity to complete the loop.
The teacher's final result shows a different approach.
Am I right? Wrong? Thanks

 

[PeroK]

Homework Statement:: Need to find the angle from which a pendulum with the length L that reaches its lowest point will complete a full circle around a pivot which distance is from the lowest point is l
Relevant Equations:: assume Tension at the highest point of the loop is 0, because that's the minimun for a full loop.

My, supposedly rational thought is that if the pendulum will drop from a height higher than the top of the loop's height, by the law of conservation of energy, it'll have enough velocity to complete the loop.
The teacher's final result shows a different approach.
Am I right? Wrong? Thanks

What about the kinetic energy needed by the pendulum to complete the loop? If the pendulum has no KE at the top of the loop, it will fall straight down.

 

[AlonZ]

that's why I say mgL(1-cos(a) ) > mg2l, once it's larger there is still some energy left as kinetic for the pendulum not to fall

 

[PeroK]

that's why I say mgL(1-cos(a) ) > mg2l, once it's larger there is still some energy left as kinetic for the pendulum not to fall

What's $\cos(a)$?

 

[AlonZ]

the expression for the height of the pendulum before the drop, you can see at the SS I sent

 

[PeroK]

the expression for the height of the pendulum before the drop, you can see at the SS I sent

Can you explain the question? How could a pendulum ever complete a loop?

 

[AlonZ]

yes, as you can see, the red circle is a new pivot for the pendulum. once the rope hits the red pivot, it spins around it and with enough velocity, completes a loop. the question is what should be the minimum angle from which I'd have to drop the pendulum for it to complete that loop.

 

[PeroK]

yes, as you can see, the red circle is a new pivot for the pendulum. once the rope hits the red pivot, it spins around it and with enough velocity, completing a loop. the question is what should be the minimum angle from which I'd have to drop the pendulum for it to complete that loop.

I'll repeat my first answer: you need kinetic energy at the top of the smaller loop. What you have calculated is the energy to reach the top of the loop, stop and fall straight down. You need the energy to complete a loop.

 

[AlonZ]

I see your point, my thought was that if the pendulum reaches the small loop's highest point and still has the tiniest amount of energy it'll not fall.
Still a bit hard for me to understand why that is wrong.

 

[PeroK]

I see your point, my thought was that if the pendulum reaches the small loop's highest point and still has the tiniest amount of energy it'll not fall.
Still a bit hard for me to understand why that is wrong.

If it has a tiny amount of energy, it will fall in a very steep parabola, which is nearly a straight vertical line.

In order to complete the loop, the natural parabola of the motion must be outside the circle of the loop at the top. Which means the pendulum bob must have at least the KE for circular motion with $g$ as the centripetal force.

If it has less KE than this it will naturally fall inside the circle. If it has more KE it try to escape the circle, but be held in the circular trajectory by the tension in the string.

 

[haruspex]

assume Tension at the highest point of the loop is 0, because that's the minimun for a full loop.

If the pendulum consists of a bob on a rigid rod, it could even be under compression. But from the given answer it appears it is on a string. For that case, your assumption is correct, but it seems you did not actually write that equation. Pity.

 

[AlonZ]

alright think I'm getting the idea.. Thx for the help