Mechanics: Equation of Motion Don't understand teacher's solution

[xxbigelxx]

Homework Statement

A flexible rope of length 1.0 m slides from a frictionless table top. The rope is initially released from rest with 30 cm hanging over the edge of the table. Find the time at which the left end of the rope reaches the edge of the table.

Homework Equations

The Attempt at a Solution

I have attached the solutions that the teacher gave us, but I don't understand the first step (that is all I attached). I understand all of the work after it, just not how they got the initial equation.

 

[fzero]

What's the weight of the part of the rope that's hanging off of the table at t=0? If the length of rope hanging at time t is x(t), what's the weight of the hanging part at time t?

 

[xxbigelxx]

Is the g/L term just the percentage of the rope that's hanging over?

 

[fzero]

Is the g/L term just the percentage of the rope that's hanging over?

Not exactly. x/L is the percentage of the rope that's hanging over. The complete term is the weight of that portion of the rope.

 

[xxbigelxx]

Ohh right that's what I meant. Ok I got it now. Also for the second part of my attachment, how did they know to look for a solution in that form?

 

[fzero]

Ohh right that's what I meant. Ok I got it now. Also for the second part of my attachment, how did they know to look for a solution in that form?

The equation is telling you that the 2nd derivative of x is equal to a constant times x. The exponential function is the only function whose derivatives have that property.

 

[xxbigelxx]

Ohh ok I think I got it now. If anything else for this question pops up I will ask you. Thanks.