Notes on the application of Newton's law to small transverse displacements of a rope under tension newtonlawrope_wiki.ppt

Show that the non-dispersive wave equation results when Newton's law is applied to small transverse displacements of a rope under tension.

The derivation should not be labored, but it should be driven home that THIS is the physics part of the endeavor! This is where the system is defined and the laws of physics are applied. A beautiful mathematical form then follows, and we solve a differential equation, and interpret its consequences.

Set up the problem as an application of Newton's Law to transverse motion. Students will invariably say “$F=ma$”. Ask them to identify “$a$”. It may take prodding to have them come up with $\frac{\partial^{2}\psi \left( t \right)}{\partial t^{2}}$. They must be confident in identifying what is accelerating. The concept of mass density is important to identify “$m$”.

Now work on “$F$”. What is the transverse force that produces the transverse acceleration? Under what conditions can the parallel force be neglected? Write down the condition $\cos \theta \approx 1$ explicitly. ($\theta$ is the angle with the horizontal.) It is this approximation that allows us to say $\sin \theta \approx \tan \theta$, and make the connection to $\frac{\partial\psi }{\partial tx}$.

Finally obtain $T\frac{\partial^{2}\psi \left( x,t \right)}{\partial x^{2}}=\mu \frac{\partial^{2}\psi \left( x,t \right)}{\partial t^{2}}$. Identify the velocity of propagation as $v=\sqrt{\frac{T}{\mu }}$. Stress that $v$ was part of the “model”; $T$ and $\mu$ were physical parameters of the system. We've now modeled the system, and coupled with the previous discussion of the solutions to the differential equation, we are now in a position to predict the motion.