Problem

In pulley problems, we have assumed that the ropes are inextensible and massless. In some...

In pulley problems, we have assumed that the ropes are inextensible and massless. In some engineering problem, such as in the design of fast elevators, this may not be a reasonable assumption. Here we will confront one aspect of any design process concerning the quality of the information provided by models of different complexity. In this problem, we will consider the effects of the rope’s deformation. However, to avoid excessive complexity, we will model the rope’s elasticity as being “lumped” at one end.

Let the system be released from rest and the velocity of A be controlled so that it accelerates uniformly downward to a speed of 3 m/s in 1.2 s. Over this time interval, determine and plot, for different values of k, the position of B as a function of time as well as the tension in the rope as a function of time. Keep in mind that the inextensible rope model can be viewed as a special case of the deformable rope model with an infinite stiffness k. Consider various values of k, starting with extremely large values (to simulate infinity) and gradually decreasing the value of k to understand at what point the two models provide significantly different answers (you will need to decide what significant means in this context).