If you hang a weight from a scale ten feet up and the weight descends 2 feet, is the loss in gravitational potential energy equal to the elastic potential energy gained?
Not quite. When you first let go of the weight, it falls freely because the spring isn’t stretched and doesn’t exert any upward force on the weight. The spring won’t support the weight fully until the weight has fallen 2 feet. By then, the weight has acquired a lot of kinetic energy and it overshoots the 2-foot level. The weight begins to bounce up and down around that 2 foot point and takes a while to settle down. The weight is vibrating up and down because it has too much energy at the 2-foot point. Eventually, it converts its extra energy into thermal energy and becomes motionless at the 2-foot point. At that point, it has turned exactly 1/2 of the missing gravitational potential energy into elastic potential energy and the other 1/2 into thermal energy. This 50/50 conversion is a remarkable result related to the exact proportionality between the spring’s distortion and the force it exerts.