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
To compute the least acceleration with which a man can slide down a rope, we need to consider the forces involved.
The force acting on the man is his weight, which can be calculated using the equation:
Weight = mass × acceleration due to gravity
Given that the mass of the man is 75 kg, and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the weight of the man as follows:
Weight = 75 kg × 9.8 m/s² = 735 N
To slide down the rope, the force exerted on the rope by the man must be less than or equal to the maximum weight the rope can sustain, which is given as 580 N.
Therefore, we can write the equation of forces as:
Force on rope ≤ Maximum weight the rope can sustain
or
Force on rope ≤ 580 N
Since the force on the rope is equal to the weight of the man, we have:
Weight of the man ≤ 580 N
So the maximum weight the man can have to slide down the rope is 580 N.
Now we need to find the maximum acceleration at which the man can slide down the rope. This can be calculated using Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration:
Force = mass × acceleration
In this case, the force is the weight of the man, and the mass is given as 75 kg. Therefore:
Weight of the man = mass × acceleration
Since we want to find the least acceleration, we can rearrange the equation as:
Acceleration = Weight of the man / mass
Substituting the given values, we have:
Acceleration = 580 N / 75 kg ≈ 7.73 m/s²
Therefore, the least acceleration with which a man of mass 75 kg can slide down a rope which can sustain a weight of 580 N is approximately 7.73 m/s².