- #1

domephilis

- 45

- 3

- Homework Statement
- A mass M hangs vertically at the end of a cable of mass m and length L.

(i) How long will it take for a transverse pulse to travel from bottom to top if you ignore m, the cable mass?

(ii) Now repeat, including m and remembering that the velocity of the signal varies with the distance from the bottom end. Show that the answer reduces to part (i) if you take the limit as m goes to 0.

Source: Problem 9.12 of R. Shankar's Fundamentals of Physics I

- Relevant Equations
- $$v=\sqrt{\frac{T}{\mu}}$$

I had no problem with (i). The tension was ##Mg##. The mass per unit length was ##\frac{m}{L}##. The answer is $$\sqrt{\frac{mL}{Mg}}$$, which was correct.

With (ii), I just tried the same thing but changed the tension to ##(M+m)g##. That did not yield the right answer at all. The solution in the book said that "At any given point along the cable the tension will have to balance both the mass at the end of the cable and any mass from the portion below it (I believe it refers to the travelling pulse)". It then proceeded to set up a differential equation relating the velocity ##\frac{dx}{dt}## to the force at the position of the travelling pulse. This completely negates the premise for the simplified wave equation which assumes that tension is constant across the string for small oscillations. The wave equation my textbook gave is this:$$\frac{\partial^2 \psi}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 \psi}{\partial t^2}$$. Plus, why did it choose to ignore the force exerted by the ceiling which definitely balances out the weight of the entire mass-string system (not just the bottom part)?

With (ii), I just tried the same thing but changed the tension to ##(M+m)g##. That did not yield the right answer at all. The solution in the book said that "At any given point along the cable the tension will have to balance both the mass at the end of the cable and any mass from the portion below it (I believe it refers to the travelling pulse)". It then proceeded to set up a differential equation relating the velocity ##\frac{dx}{dt}## to the force at the position of the travelling pulse. This completely negates the premise for the simplified wave equation which assumes that tension is constant across the string for small oscillations. The wave equation my textbook gave is this:$$\frac{\partial^2 \psi}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 \psi}{\partial t^2}$$. Plus, why did it choose to ignore the force exerted by the ceiling which definitely balances out the weight of the entire mass-string system (not just the bottom part)?