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Aug 30th 2012, 09:03 PM

#1  Member
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 angular momentum prob
Two point masses, with masses of m and 2m, are connected through a string that has a length of L. The two bodies are put on an horizontal frictionless table as in the figure, so that m is at the origin. The body 2m is above it (on the y axis) at a distance of L/2 m. At a certain moment they give 2m a velocity of v0 in the positive x axis direction.
What's the kinetic energy of the system after the string is streched and m starts to move?
I realize that conservation of both linear and angular momentum applies here  but I don't understand something: Am I suppose to understand that the system starts to rotate about an axis which passes through the center of mass just as the string is streched? And if so  was there any "mathmatical way" for me to know the system starts rotating? Because it seems from the solution that when they calculated the angular momentum of the system as the string was streched  they used Icm*w to describe its angular momentum, while when they calculated the angular momentum of the system at the beginning (The picture phase) they used L=mvrsin(q) to describe the angular momentum as the sum of the angular momentum of 2m and m with respect to the center of mass...

 
Aug 31st 2012, 07:43 AM

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Join Date: Jun 2010 Location: Morristown, NJ USA
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You asked about the kinetic energy of the system, so I don't understand why you're trying to find angular momentum. Conservation of energy says that the initial KE is 1/2 mv_0^2, and so that should be the system's KE just as m starts to move (at that point the string hasn't stretched, so there's no PE in it yet).
Last edited by ChipB; Aug 31st 2012 at 08:17 AM.

 
Aug 31st 2012, 08:01 AM

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Originally Posted by ChipB You asked about the kinetic energy of the system, so I don't understand why you're trying to find angular momentum. Conservation of energy says that the initial KE is 1/2 mv_0^2, and so that should be the system's KE just as m starts to move (at that point the strung hasn't stretched, so there's no PE in it yet). 
According to the solution there's no conservation of mechanical energy. I don't fully understand the reason because as you said the string isn't streched yet but according to their solution, that follows KE=0.5MVcm^2+0.5Icm*w^2, you get in the final stage that KE=0.75mv0^2, while the inital is mvo^2 (notice that 2m has the speed of vo and not m like you calculated)
edit: the question asks us to give the KE after the string is streched and m starts to move from the origin. What happens is that just when the string becomes streched and is not loose anymore  a kind of plastic collision occures between the two bodies that now move "together" and thats why theres no conservation of energy
Last edited by assaftolko; Aug 31st 2012 at 11:27 PM.

 
Sep 1st 2012, 08:05 AM

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Chip don't give up on me 
 
Sep 2nd 2012, 05:35 AM

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It seems as if the solution you posted assumes that the string somehow becomes rigid once it reaches full extension, and hence the system starts to rotate about its center of mass as it moves. This would indeed be a form of inelastic collison, and hence KE is not constant. But as the problem is written, I don't see how this could happen. A string is assumed to always be flexible. In this problem the impulse received by the mass m is equal and opposite to the impulse provided by the 2m mass, and consequently the m mass starts to move faster than the 2m mass. The result is the string goes slack as the smaller mass starts to catch up to the larger. During this phase both objects are moving along straight paths and there is no angular momentum. So again, I don't get it. Sorry.
Last edited by ChipB; Sep 2nd 2012 at 07:50 AM.

 
Sep 2nd 2012, 05:50 AM

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Originally Posted by ChipB It seems as if the solution you posted assumes that the string somehow becomes rigid once it eaches full extension, and hence the system starts to rotate about its center of mass as it moves. This would indeed be a form of inelastic collison, and hence KE is not constant. But as the problem is written, I don't see how this could happen. A string is assumed to always be flexible. In this problem the impulse received by the mass m is equal and opposite to the impulse provided by the 2m mass, and consequently the m mass starts to move faster than the 2m mass. The result is the string goes slack as the smaller mass starts to catch up to the larger. During his phase both objects are moving along straight paths and there is no angular momentum. So again, I don't get it. Sorry. 
Angular momentum is a quantity that exist even where there's no rotational movement  like in the start phase: 2m has the velocity of v0 and so it has angular momentum, so I don't see why you are saying there's no angular momentum...
You are correct that as the string streches the body that we get from 2m, m and the string is now considered as rigid  I think since this question appears in the dynamic of rigid body chapter, they assumed I'd understand it by myself. About what you said about the string  I always thought we assumed the string is not flexible so it doesn't get shorter or longer if you apply force on it...
By the way how do you know the system starts to rotate about it's center of mass? isn't it possible that the rotation axis will pass through another point than the cm?
Also chip I've posted another "logic" question that deals with conservation of angular momentum with respect to the cm while the cm position changes in space  I'd be very grateful if you could find the time to look at it :)

 
Sep 2nd 2012, 08:15 AM

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Originally Posted by assaftolko Angular momentum is a quantity that exist even where there's no rotational movement  like in the start phase: 2m has the velocity of v0 and so it has angular momentum, so I don't see why you are saying there's no angular momentum... 
Angular momentum exists only if the object is moving in a curved path. If it is moving in a straight line it has linear momentum, but not angular momentum. An essential difference is that momentum is a vector, which for linear momentum is in the direction of motion (p= mv) and for angular momentun is in the direction w, which is perpendicular to the arc of the object (p=Iw).
Originally Posted by assaftolko About what you said about the string  I always thought we assumed the string is not flexible so it doesn't get shorter or longer if you apply force on it... 
I meant that the string can bend, and hence cannot carry any torque or sideways force to either object. I did not mean that the string can stretch.
Originally Posted by assaftolko By the way how do you know the system starts to rotate about it's center of mass? isn't it possible that the rotation axis will pass through another point than the cm? 
An unconstrained object has motion components consisting of translation of its cm plus rotation about the cm. If there are constraints involved (such as rotation about a pivot point) then the center of rotation is controlled by the constraint and would not necessarily be the center of mass.
Last edited by ChipB; Sep 4th 2012 at 11:36 AM.

 
Sep 2nd 2012, 09:47 PM

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Originally Posted by ChipB Angular momentum exists only if the object is moving in a curved path. If it is moving in a straight line it has linear momentum, but not angular momentum. An essential difference is that momentumis a vector, which for linear momentum is in the direction of motion (p= mv) and for angular momentun is in the direction w, which is perpendicular to the arc of the object (p=Iw).
I meant that the string can bend, and hence cannot carry any torque or siways force to either object. I did not mean that the string can stretch.
An unconstrained object has motion components consisting of translation of its cm plus rotation about the cm. If there a constraints involved (such as rotation about a pivot point) then the center of rotation is controlled by the constraint and would not necessarily be the center of mass. 
Chip thanks a lot! I did read that angular momentum has to do with rotational movement  that's an understatement, but so why does the system have angular momentum in the start phase where 2m moves in a straight line along the x direction?

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