a mass M$=3\mathrm{.}0$ kg is hanging from a long string of length L$=26\mathrm{.}0$ m$.$ The string by itself has a mass of m$=19$ kg$.$ A student gives the mass a quick horizontal shake to set up a wave which travels upwards along the string.

Part $($a$)$ what is the speed of the wave at the bottom of the string, in meters per second?

Part $($b$)$ express the tension of the chain at a height y above the bottom in terms of M$,$ m$,$ and g$.$

Part $($c$)$ what is the speed of the wave, in meters per second, at the top of the string?

Part $($d$)$ how long does it take, in seconds, for the wave to travel up the string?