Final answer:
The least time required to raise a load 'W' through a height 'h' with the maximum allowable rope tension of 'nW' is derived using the kinematic equation and Newton's second law, and is given by sqrt(2h/g(n-1)), where 'g' is the acceleration due to gravity.
Step-by-step explanation:
The student is asking about the least time required to raise a load, using a massless inextensible rope, from rest to rest through a height 'h' with a maximum allowable tension of nW, where 'W' is the weight of the load. To determine the least time, one would apply the principles of classical mechanics, specifically utilizing Newton's second law of motion, which relates the net force on an object to its acceleration (Fnet = ma).
When the load is lifted with maximum tension nW, this implies that the acceleration of the load is maximum just before the rope reaches its maximum tension. Therefore, the net force is Fnet = nW - W = (n-1)W, and using Newton's second law, we have (n-1)W = ma, where 'm' is the mass of the load and 'a' is its acceleration. Since W = mg, we can simplify to a = g(n-1).
Using the kinematic equation h = 1/2 a t^2 (assuming constant acceleration), and solving for 't', the time, we can find the least time 't' by substituting the acceleration 'a' with g(n-1). Therefore, the formula to find the least time 't' will be t = sqrt(2h/g(n-1)).