posted by Grail .
compute the least acceleration with which a man of mass 75 kg can slide down a rope which can sustain a weight of 580 N
You really mean the minimum deceleration. The man's weight is M g = 735 N. For the man to decelerate at a rate a, the rope must apply a friction force F to the man such that
M g - F = M a
F is also the rope tension, which must not exceed 580 N.
At maximum possible F (580 N), one gets the minimum possible acceleration, a_min
a_min = g - 580/M = 9.8 - (580/75)
= 2.1 m/s^2