Question

A 1.0kg object is suspended from a spring with k=16 N/m. The mass is pulled 0.25m downward from its equilibrium position and allowed to oscillate. What is the maximum kinetic energy of the object?

Answer 1

1/2kx^2 = (1/2)(16)(1/4)^2 = .5N

Answer 2

There's a couple ways to do this one. Looking at energy is the easiest I think. When the spring is in equilibrium, it already has some displacement due to gravity. So first you need to find how much displacement it has. Do
\[F = kx_0 = mg\]
And solve for x_0, the initial displacement. That'll tell you how far it's displaced in equilibrium. Now you need to find how much potential energy is stored in the spring when you pull it down the 0.25m in addition to the initial potential energy. Do
\[U = (1/2)kx^2 = (1/2)k(\Delta x + x_0)^2\]
where delta x is the added displacement, 0.25m. The mass has the maximum kinetic energy when all this potential is 0. Therefore, this is the answer. Note that I did not use any minus signs anywhere. The first equation should have a minus sign, depending on choice of coordinate system, but I neglected it because I was looking for energy anyway. Sloppy, but it gives you the answer.