Answer Happy

Suppose a spring of length l is suspended from a ceiling, and after attaching a weight of mass m the spring stretches to have a length of l + L. Notice there are two forces acting on the weight: the force of gravity pulling the weight down (i.e., the object's "weight") and the tension in the spring pulling the object back up. If we denote these quantities W and T, respectively, we then have W+T = 0. Notice the weight of the object equals W mg (and we will choose units so this quantity is positive), and by Hooke's law the tension in the spring is T = --kl for some constant k that depends on the spring. We then have the equation mg - kl = 0. Suppose that when a weight of 32 pounds is attached to a five foot long spring, the spring stretches by two feet. The weight is then pulled down four feet below its initial position and is released. Find the position u(t) of the weight from its equilibrium position t seconds after being released. (The acceleration due to gravity in Imperial units is 32 feet per second, and so the mass of our weight is one slug.)