Classical Mechanics Problem: Central force

[roffelos]

If anyone could help me with this problem I would be very grateful, its been annoying me for a day or two now! Its one of those exam questions that lead you through a derivation, giving you blanks to fill in. Although these are designed to make the derivation easier, sometimes it makes a problem even more frustrating considering you know that you’ve missed something obvious, or found your own tangent and gotten lost, when you realize you can get the result they ask for!

I’m trying to solve a past exam question about 2 particles joined by a string, one of which is resting on a frictionless table, the other hanging underneath the table (the string being threaded through a small hole a distance r from the first particle). The particle on the table is in circular motion with angular velocity theta dot. I’ve seen this situation before, usually asking for the use of Lagrangian's and energy approaches, I’m stumped here and can not get the equation they ask for. So..

Using polar co-ordinates, with r(hat) pointing away from the hole the equation of motion for m1 is the sum of the centrifugal force in the radial direction Fr=m*r*theta dot and the Tension T in the string in the negative r direction so:

m1*a1=m1*r*thetadot^2 - T 1

Angular momentum conservation requires angular momentum Ptheta=r^2*m*thetadot satisfy

d/dt(ptheta)=0 3

so:

m*r^2*thetadot=Po

where po is some initial angular momentum, the problem states that at time t=0 r=a and velocity in the thetahat direction is vo therefore: Po=a*m*vo and

thetadot = a*vo/(r^2) 3

The question then defines the length of the string as l and the length underneath the table as y so that

y + r = l 4

The only forces on m2 are gravity m2*g and the Tension T acting in opposite directions, taking gravity to act in the negative z direction (z perpendicular to the table i.e. rhat) so:

m2*a2=T - m2*g 5


Now the question asks to use equations 1,3,4 and 5 to eliminate y, theta, and T. This is where I think I've missed something because I don’t have a y to eliminate, and so far the only blank spaces I’ve filled in are equations 1, 3 and 5. Eliminating T by rearranging 5 and substituting into 1 and realising that a1 = a2 = ar (and also substituting for thetadot^2 from 3) I get:

(m1 + m2)*ar = m1*alpha^2/r^3 - m2*g where alpha=vo*a

Which just looks like the equation of motion for a particle of mass m1 + m2 acted on by a net force given by the RHS of the above equation; which seems reasonable. Up until this point I think I'm ok, but the next blank is as follows:

d/dt( )=0

preceded by a comment that implies transforming the last equation. By rewriting ar as rddot and rearranging I come to:

rddot - m1*alpha^2/[(m1 + m2)r^3] + m2*g/(m1 + m2) = 0

pulling out a d/dt (where here I may have made a mistake) I get:

d/dt( rdot + [1/(2*rdot)]*m1*alpha^2/[(m1 + m2)*r^2] + m2*g*t/(m1 + m2) + b) 6

where b is some constant

Here is where the trouble comes, as the next question is to integrate equation 6 with the initial conditions at t=0, r=a, and rdot=0. The form of the answer they give is:

rdot^2 = {- 2*m2*g/[(m2-m1)*r^2]}*(r-a)*(r-b)*(r+c) 7

with the last two blank boxes to be filled in being b= ? and c=?

Now there seems to be something obvious I’ve missed, as this question seems overly difficult compared to the others in the exam, but I can't for the life of me find it. When I integrate my equation 6 with those initial conditions I come out with a rdot, that is totally irreconcilable with their equation 7. So If anyone can shed some light on what they seem to be driving at in this problem I would be grateful! Even forgetting my equation 6, their equation 7 seems to be indicating three stable points in the system, namely r= a, b and -c, and presumably they want b and c in terms of l, vo and a. Even this I don’t see how to do unless with comparison to my equation 6 and their equation 7?

 

[Andrew Mason]

It is not apparent to me what the question is here.

The principal points are:

1. angular momentum is conserved as the hanging weight falls, $L = mv(t)r(t)$ and dL/dt = 0
2. The increase in energy of the system as the horizontal radius shortens is provided by the gravitational work done by the falling weight.

From that you should be able to solve the problem.

AM

 

[roffelos]

Thanks for the reply Andrew, the question is a bit long winded but the main problem is I can't get an equation for rddot^2 which allows me to find the constants b and c? I have tried from an energy point of view as well, where I get in the end a cubic equation for the stable points where rdot = 0. This equation also I cannot manipulate into the form they require, which facilitates finding b and c by comparison. My feeling is that i have made a simple error somewhere and that the intergration from equation 6 should be a simple one leading naturaly to the rdot^2 equation they give. So the main problem is simply stated as find b and c in equation 7? I will continue try and hoping maybe someone can shed some light on this annoying exam question.