Classical Mechanics - Energy+Circular Motion problem

In summary, the conversation discussed a point particle of mass m moving on a frictionless surface described by polar coordinates r and phi. The particle is connected to the origin by a spring with spring constant k, providing a force of magnitude k r directed towards the origin. The potential energy due to the spring was found to be kr^2/2. The kinetic energy was written as mv^2/2, with v being r_dot^2 + (r*phi_dot)^2. The angular momentum L was derived as mr^2*d(phi)/dtheta, and used to eliminate phi and d(phi)/dtheta in the total energy expression E. The motion r(t) was solved for in terms of t, E, and L
  • #1
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Homework Statement


A point particle of mass m moves on a frictionless surface. Its position can be described with polar coordinates r and phi. The particle is connected to the origin of the coordinate system by a spring with spring constant k and unstretched length zero, which provides a force of magnitude k r directed toward the origin of the coordinate system.
(a) Find the potential energy due to the spring as a function of r.
(b) Write the kinetic energy as a function of r, dr/dt, and d(phi)/dt.
(c) Write the angular momentum L about the origin as a function of r and d(phi)/dt.
(d) Combine your results to write an expression for the total energy E which depends on r and dr/dt. Your expression should not depend on phi or d(phi)/dt -- you eliminate that in favor of the angular momentum.
(e) Solve for the motion r(t) in the form t - t0 equals an integral of a function of r, where that integral involves the constants E and L.
(f) For a fixed non-zero value of L, find the smallest value of E that is possible. Show that this corresponds to a circular orbit, and that the circular orbit obeys the expected F = m v^2 / R rule that you learned in first-year physics.
(g) For a fixed non-zero value of L, assume E = (13/12)*Sqrt(k/M)*L . Find the minimum and maximum values of r.

Homework Equations


See "attempt..." below

The Attempt at a Solution


part a. PE in this case is kr^2/2 (due to spring only)
part b. KE = mv^2/2, where v is r_dot^2+(r*phi_dot)^2, see attachment for how I derived this--I apologize for not being familiar with the formatting here!

L=Iw=mr^2w, w=d(phi)/dtheta

what I did: part c->solve for d(phi)/dtheta
part d->use d(phi)/dtheta from part c to replace the same thing in KE equation. Add KE and PE=Total E
part e-> solve for r_dot. Then make it into separable DE, solve the left side (integral dt=t-t_0) and it equals to a crazy right hand side below
part f->set 2E-k^r2-L^2/(m*r^2) to 0 because it can't be 0, or otherwise there is a problem. checked it and it llokd OK
(-----part g->(not sure part) set r_dot=0, solve for r, except I couldn't. I know that I should be getting 2 r values, but wolfram spits out imaginary answer. And I need to find r_dot_dot, plug in the r values. If positive=minima, negative=maxima (my plan, anyway)------)

I have work for part a-f, currently stuck in part g. And now I question my whole work.
30hlp1s.jpg

(here is the link, just in case the picture doesn't show: http://i58.tinypic.com/30hlp1s.jpg)

Thank you very much in advance. Any enlightenment or confirmation if I am in the right direction or suggestion or advice would be very much appreciated.
 
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Picture doesn't work, but at least the link works.
 

Related to Classical Mechanics - Energy+Circular Motion problem

1. What is classical mechanics?

Classical mechanics is a branch of physics that deals with the motion of macroscopic objects under the influence of forces. It is based on the laws of motion and gravitation developed by Sir Isaac Newton in the 17th century.

2. What is energy in classical mechanics?

In classical mechanics, energy is a property of a physical system that allows it to do work. It can exist in various forms such as kinetic energy (energy of motion), potential energy (energy of position), and thermal energy (energy due to temperature).

3. How is energy conserved in circular motion?

In circular motion, the total mechanical energy (kinetic + potential) of the object remains constant as long as there is no external force acting on it. This is known as the conservation of energy principle.

4. How does energy affect circular motion?

The energy of an object in circular motion determines its speed and radius of the circular path. The higher the energy, the faster the object will move and the larger the radius of the circular path will be.

5. What is the relationship between energy and centripetal force in circular motion?

In circular motion, the centripetal force is responsible for keeping an object in its circular path. The magnitude of this force is directly proportional to the amount of kinetic energy the object possesses. This means that the higher the kinetic energy, the greater the centripetal force required to maintain the circular motion.

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