Boundary Conditions for Waves in Joined Strings with a Knot of Mass

[mathlete]

I'm given the fact that two strings under tension T are joined by a knot of mass m... I'm supposed to find the appropriate boundary conditions. I know that the tensions are the same in both ropes and that the boundary will be continuous. I know the "trick" in this problem is knowing the derivatives don't equal because of the knot of mass m, but I don't really know how to formulate it. Can anyone provide a nudge in the right direction?

 

[Physics Monkey]

You can make progress by making of use of Newton's laws for the massive knot while remembering that the string is continuous. In other words, the boundary conditions have to insure that the massive knot, which is a point but with non-infinitesimal mass, stays with the string.

 

[mathlete]

I understand that part, but I'm not sure how to formulate it... I get df/dz (from the left) - df/dz (from the right) = m/T but I don't think that's right.

 

[Meir Achuz]

I understand that part, but I'm not sure how to formulate it... I get df/dz (from the left) - df/dz (from the right) = m/T but I don't think that's right.

Close.
1. Should =-mg/T.
2. You need sin\theta. (Unless you can make a small angle approx.)df/dz is tan\theta.