Question

Why can gravitational potential energy be ignored in a verticle spring with a hanging mass? My TA mentioned something about no work being done by GPE, but that doesn't really make sense. Also, the assignment is asking for it to be explained conceptually, not with equations. Thanks!

Answer 1

Thee (massless) spring hangs vertically with a block of mass m attached at the bottom. We could calculate how much the spring is stretched by equating the gravitational and spring forces (kx1=mgkx1=mg).

Now, during the pulling process ,positive work is being done on the system, which means that the energy in the system increases.

Let's use the work-energy theorem,

Wnet,external (Positive) =?Etot =?Ugrav (Negative) + ?Uelastic (Positive)

The gravitational potential energy decreases. Where does it go? Well, the only other term that could (mathematically) compensate for this decrease in gravitational potential energy is the increase in elastic potential energy. The spring is not storing gravitational potential energy; rather, gravitational potential energy was converted to elastic potential energy.

The left-hand side of the equation above is positive, the absolute value of ?Uelastic is greater than that of ?Ugrav. So, not only was the gravitational potential energy converted to elastic potential energy, the positive work done on the system also adds to the increase in elastic potential energy.