Question

At the spring meeting of the American Potential Energy Society, a debate erupts over the choice of a reference point for elastic potential energy. One faction (group #1) feels it is only natural to choose the reference point at the location at which the spring is neither stretched nor compressed and to measure the elastic potential energy relative to that unstrained configuration. Another faction (group #2), whose members always like to think "outside of the box", wants to choose the reference point by defining the elastic potential energy to be zero at the point of maximum spring stretch for a given situation. Consider a "standard" horizontal spring/block system on a frictionless surface as shown in Fig. 8.5.1; the spring constant is k, and the point of maximum spring stretch is a distance Xm to the right of the point of zero spring strain. 2xm3 *m3 A B m point of zero strain x = 0 point of maximum stretch Interactive Figure 8.5.1: A block/spring system on a horizontal, frictionless surface. The change in elastic potential energy for a change in the block's coordinate from x1 to xz is given by: AU, = U2 - U1 = = (x3 – «?) To define a reference point for elastic potential energy, let's choose a value for X1 and define the elastic potential energy to equal zero at that location. Such a choice simplifies the expression for AU, as follows: U2 - 0) = (x3 - x) (use the notation X1 + Xref) U2 = (x2 – xer) (change the "2" subscript to "s") U = (2-) (A) where Xref refers to the reference point at which the potential energy is zero. Equation (A) is the key equation for this analysis, because the two factions mentioned in the problem have chosen different locations for the reference point. Question 1 What does Group #1 compute for the elastic potential energy of the spring/block system when the block is located at point A in Fig. 8.5.1? This will be a symbolic result in terms of k and Xm.

Answer 1

a) The potential energy at point A as calculated by group 1 is

U 1 k(Im/3 2 kon 18

2) Point A is 2x_m/3 distance away from point of maximum stretch. Therefore the potential energy at A according to group 2 is

2kim U = 5k(22m/3)2 9

3) The potential energy at point B as calculated by group 1 is

2kim U = 5k(22m/3)2 9