$(25$ points$)$ A $5\mathrm{.}00$ kg bucket of is suspended over a well shaft by a rope and cylindrical winch. The winch has a radius of $10\mathrm{.}0$ cm and mass of $25\mathrm{.}0$ kg $.$ When the bucket is at a height $30\mathrm{.}0$ m above the well water, the winch breaks and the bucket falls down the well into the water, unwinding the rope around the winch without slipping as it falls.