Rope Sliding Over Friction less Peg

[bmb2009]

Homework Statement

A uniform rope of mass M and length L is hung off a small peg. The rope moves without friction. Obtain an equation for the length of the rope hanging to the right of the peg, which is denoted x(t).

Homework Equations

The Attempt at a Solution

I set up a net force equation: F_net= Axg-A(L-x)g where A is defined as M/L
with manipulation that equation is equivalent to:
M(d^2x/dt^2)= 2Axg-AgL... now how do I solve that equation for x(t)... Please help.. I know I have to separate the variables but I do not know how to solve a second order differential...Thanks

 

[ehild]

Homework Statement

A uniform rope of mass M and length L is hung off a small peg. The rope moves without friction. Obtain an equation for the length of the rope hanging to the right of the peg, which is denoted x(t).

Homework Equations

The Attempt at a Solution

I set up a net force equation: F_net= Axg-A(L-x)g where A is defined as M/L
with manipulation that equation is equivalent to:
M(d^2x/dt^2)= 2Axg-AgL... now how do I solve that equation for x(t)... Please help.. I know I have to separate the variables but I do not know how to solve a second order differential...Thanks

You can eliminate M and A by substituting A= M/L.
Rearranging, it becomes

d² x/dt²-(2g/L)x=-g.

That is a second order, linear differential equation.
The left hand side is the homogeneous part. You get the general solution of such an inhomogeneous equation by solving the homogeneous equation

d² x/dt²-2(g/L)x=0

and adding a particular solution to the general solution of the homogeneous equation.

For the homogeneous equation, try the solution in the form

x=e^kt. , with k a constant. Find the possible values of k.

A particular solution of the equation is when the rope is in equilibrium and the acceleration is zero.

See, for example, http://en.wikipedia.org/wiki/Linear_differential_equation

ehild