This month's Challenge Question: Specs & Techs from IHS Engineering360:
You have a rope of length L and a frictionless table with height 1.5L. Put the rope on the top of the table and pull one end to the border of the table so that a small piece hangs down. Release the rope, so that the rope starts to fall down. What is the speed of the rope at the very instant when it loses contact with the top of the table?
And the answer is:
Let’s apply the conservation of energy principle. Assume that the linear mass density of the rope is . The total mass of the rope is
For the purpose of this problem, the total mass is concentrated in the center of mass, or
The following figure shows the stages of the problem. Figure (a) shows the rope on top of the table at the moment when it starts to fall down. Figure (b) shows the rope at the instance when the rope loses contact with the top of the table. Figure (c) depicts the position of the center of mass corresponding to figure (b).
According to the conservation of energy principle, the kinetic energy of the rope in figure (b) is equal to the loss in potential energy. The loss in potential energy is the distance the center of mass moves in figure (b) multiplied by the mass and the gravitational constant (g). So, in equation form we have:
Solving for the speed, we get:
Notice that this is the same result obtained by an object of any mass that falls a distance L/2.
