Question

Consider a rope of mass M and length L rotating on top of a table in a horizontal
plane. One of the ends of the rope is fixed and does not move. The rope moves in a circle with angular frequency w. What is the kinetic energy of the rope?

Answer 1

we can model your problem as below:

now I consider an infinitesimal part of the rope, whose length is \(dx\), it mass is:
\[dm = \lambda dx\]
where \(\lambda=M/L\)
The kinetic energy of this infinitesimal part of rope is:
\[dE = \frac{1}{2}dm{v^2} = \frac{1}{2}dm{\omega ^2}{x^2} = \frac{1}{2}\lambda {\omega ^2}{x^2}dx\]
being \(v=\omega x\)

Answer 2

the requested kinetic energy, is given by the subsequent integral:
\[E = \int_0^L {dE} = \int_0^L {\frac{1}{2}\lambda {\omega ^2}{x^2}dx} \]
please complete