Object on a string with a string wrapping around

[WraithM]

I saw a teacher today with a piece of chalk tied to the end of a string (for drawing circles and such on a chalk board) swing the string around, and the string wrapped around his finger and the string became shorter until there was no more string left. His finger was pointed parallel to the floor so that the plane that the string spins in is perpendicular to the floor. Is there a name for this sort of thing? Is this used in physics problems or anything like that? I'm just interested in more information about this specific situation.

 

[Count Iblis]

Classical mechanics students are sometimes asked questions about this on exams. The idea being that almost all students will use conservation of angular momentum to solve the problem, which is wrong.

So, this allows the Prof. to give the "very smart" students an edge over the students who merely are "ordinary smart".

 

[vin300]

Classical mechanics students are sometimes asked questions about this on exams. The idea being that almost all students will use conservation of angular momentum to solve the problem, which is wrong.

So, this allows the Prof. to give the "very smart" students an edge over the students who merely are "ordinary smart".

The law of conservation of momentum doesn't work, it is a rotational phenomenon.
Due to gravitational deceleration, for a given radius, after some circulations becomes insufficient to complete the circle, but since the radius goes on decreasing, the reqirement of velocity to complete the circle goes on decreasing as sqrrt of it.
If the mass, velocity, radius initally at a point given the total energy can be calculated.
The total energy remains conserved.

 

[WraithM]

The law of conservation of momentum doesn't work, it is a rotational phenomenon.
Due to gravitational deceleration, for a given radius, after some circulations becomes insufficient to complete the circle, but since the radius goes on decreasing, the reqirement of velocity to complete the circle goes on decreasing as sqrrt of it.
If the mass, velocity, radius initally at a point given the total energy can be calculated.
The total energy remains conserved.

Could you show me what you mean? (with math. I know calc and a good deal of classical mechanics, so I'll understand.)

 

[vin300]

Could you show me what you mean? (with math. I know calc and a good deal of classical mechanics, so I'll understand.)

Let me tell you something, I am sure about everything except the last line.
The object is rotating in a vertical plane.
The minimium velocity at the topmost point to complete the circle is v=sqrrt.(g*r).
Normally, the energy required is exhausted after a few rotations, so it has to be supplied energy continuously.
The radius decreases, the maximum velocity required decreases proportionallyas sqrrrt of redius.
The man does not need to continually supply energy.
The radius, velocity and mass given at a point, you can calculate the total energy as
E=K.E.+P.E.
=1/2 mv^2 +mgx
x is the vertical distance of the point from the bottom of circle drawn with radius at that point.
I think this energy must be conserved throughout the process.

 

[rcgldr]

The name of the path is involute of circle. Angular momentum of the system is conserved, but you need to include the angular momentum of whatever the post is attached to, because the string tension results in a torque on the post which transfers the torque to whatever it's attached to (usually the earth).

This was covered in a previous thread:

Comparason of a puck sliding on a frictionless surface attached to string wrapping around a post versus being pulled or released via a hole is covered in post #17:
thread_185178_post_17.htm

link to post with links to animated pictures:
thread_185178_post_21.htm

link to post with the math of involute of circle:
thread_185178_post_32.htm

For the string pulled through hole case, #34 covers this case. The tension versus path while being drawn inwards is slightly forwards and while being released outwards is slightly backwards, so the puck speeds up as it's pulled in and slows down as it's released. Since there's no torque applied to the (assumed frictionless) hole, all of the angular momentum is in the puck (ignoring the string).

The math in post #34 is correct, but the paragraph at the end of post #34 about the involute of circle case (post) is wrong. The tension from the post to the puck is perpendicular to the puck (explained in post #32), not "slightly backwards" as mentioned at the end of post #34. The speed of the puck with string wrapped around a post case remains constant while it's is spiraling inwards or outwards. The angular momentum issue is resolve if you include the angular momentum of whatever the post is attached to as part of the system.

thread_185178_post_34.htm