Cable on a Table (classical mechanics)

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SUMMARY

The discussion centers on the dynamics of a perfectly flexible cable of length l as it slips off a table. The length hanging over the edge after time t is described by the equation x(t) = x0cosh(√(g/l)t). Participants explore the concepts of mass per unit length (D = m/l) and the relationship between force and acceleration during the cable's motion. The use of conservation of energy is suggested as a simpler approach to solving the problem, particularly for determining velocity and acceleration.

PREREQUISITES
  • Understanding of classical mechanics principles, particularly motion and forces.
  • Familiarity with hyperbolic functions, specifically cosh.
  • Knowledge of conservation of energy in physics.
  • Basic calculus, particularly differentiation and second derivatives.
NEXT STEPS
  • Study the application of conservation of energy in dynamic systems.
  • Learn about hyperbolic functions and their properties in physics problems.
  • Explore the derivation of motion equations for flexible bodies.
  • Investigate the relationship between mass distribution and acceleration in non-constant force scenarios.
USEFUL FOR

Students and educators in physics, particularly those focusing on classical mechanics, as well as anyone interested in understanding the dynamics of flexible bodies in motion.

bluedrew
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Homework Statement



A perfectly flexible cable has length l. Initially, the cable is at rest, with a length of it hanging vertically over the edge of a table. Neglecting friction, consider the cable's motion as it slips off the edge of the table. (a) Show that the length hanging over the edge after a time t is given by x(t) = x0cosh (\sqrt{}g/l*t
(b) Find the velocity and the acceleration of the rope as functions of time.
(c) Find the time tend from the start of the rope's slide to the moment at which the cable slides completely off the table, and
(d) the velocity and the acceleration at time tend

Homework Equations



Assume that the sections of cable remain straight during the motion.

The Attempt at a Solution



Clearly, the force is not constant on the whole cable and as it slides off the table, the acceleration is getting closer to g. I've written the mass per unit length as D=m/l (where m is the mass of the portion of the cable hanging over the table and l is the length of the cable), and I know for part (a) I have to solve for Dg=M*(d2x/dt2) (where g is gravity, and M is the mass of the whole cable), but I'm not sure how to do that. What does d2x/dt2 become?

Thanks so much in advance, there's something really simple I'm just not seeing here.
 
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welcome to pf!

hi bluedrew! welcome to pf! :smile:

why not just use conservation of energy? :wink:
 

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