Cable on a Table (classical mechanics)

In summary, the conversation discusses the motion of a perfectly flexible cable as it slips off the edge of a table. The length of the cable hanging over the edge after a certain time is given by the equation x(t) = x0cosh (\sqrt{}g/l*t). The velocity and acceleration of the rope can also be determined as functions of time. Using conservation of energy, the time t[SUB]end and the velocity and acceleration at time t[SUB]end can be calculated.
  • #1
bluedrew
1
0

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 ([tex]\sqrt{}g/l[/tex]*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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  • #2
welcome to pf!

hi bluedrew! welcome to pf! :smile:

why not just use conservation of energy? :wink:
 

1. What is Cable on a Table in classical mechanics?

Cable on a Table is a classic problem in the field of classical mechanics that involves studying the motion of a cable or string that is attached to a table and has a weight hanging from it. It is used to understand the principles of tension, gravity, and friction in a 2D system.

2. How is the motion of the cable determined in this problem?

The motion of the cable is determined by balancing the forces acting on it. This includes the tension force from the weight of the object, the normal force from the table, and the friction force between the cable and the table. By applying Newton's second law, the equations of motion for the cable can be derived.

3. What is the significance of studying Cable on a Table in classical mechanics?

Studying Cable on a Table allows scientists to understand and apply the principles of classical mechanics to real-world systems. It also helps in understanding the concept of equilibrium, where all forces acting on the cable are balanced, resulting in no net acceleration.

4. What factors can affect the motion of the cable in this problem?

The motion of the cable can be affected by various factors such as the length and weight of the cable, the weight of the object, the angle at which the cable is hung, and the coefficient of friction between the cable and the table. These factors can alter the equations of motion and change the behavior of the cable.

5. Can the Cable on a Table problem be applied to real-life situations?

Yes, the principles and equations used in the Cable on a Table problem can be applied to real-life situations, such as understanding the behavior of a cable supporting a hanging object, or the motion of a cable car on a slope. It is a fundamental concept in classical mechanics and has practical applications in engineering and physics.

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