in this illustration we are, going to calculate
the speed of a falling rope supported by a nail. here the figure shows a uniform smooth
and flexible rope of mass m and length 2 l. which is symmetrically supported on a nail
ay, here if due to a slight jerk rope starts falling. we are required to find its speed
when it leaves off the nail. here in solution, we can see. if jerk is provided on the right
part and say it starts falling. and if this is nail ay and 1 rope will be, moving off
the nail. the situation would be this its total length 2 l, is on 1 side of the nail.
and here if we just have look on it center of mass this is, the center of mass, of this
rope. which is at a distance, l bellow the nail. earlier if we have a look the center
of mass of rope would be at the midpoint as rope is uniform. and the center of mass is
at a depth l by 2, from the nail ay, so here we can see, in, the process of motion. of
rope. the displacement of center of mass. of rope is. this h, by which the center of
mass is fallen down can be written as l by 2. so we can directly write using work energy
theorem. as g is uniform we can consider, the work done by gravity on the rope. in the
process of motion we’ll be m g h. so initial kinetic energy of the rope was zero. plus
work done on it is, m g l by 2. must be equal to the final kinetic energy which is half
m v square. where v is the speed attain by the rope as., from the initial moment it was
starting from rest. so, m gets cancelled out this 2 gets cancelled out and the value of
sped we are getting is root g l. that will be the final answer for this problem.