# Use Hooke's Law to determine the variable force in the spring problem. An overhead garage door has two springs, one on each side of the door. A force of 15 pounds is required to stretch each spring 1 foot. Because of the pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the work done by the pair of springs.

As the problem stated, we first need to figure out the spring constant.  Hooke's Law says that the spring constant can be calculated by dividing the force by the distance of movement.  It takes 15 pounds of force to move the spring 1 foot, so the spring constant is 15.

A stretched spring has elastic potential energy.  The potential energy that is stored in the spring is equal to the work done to stretch the spring in the first place.  That means the potential energy in the spring is equal to the work that the spring can do; therefore, calculating the spring's potential energy will give you the work done by the spring when the door opens.

The formula for calculating a spring's potential energy (work) is PE = (kx^2)/2. K is the spring constant.  X is the total movement of the spring from its rest position. The door moves a total of 8 feet, and the problem stated that the springs move half that distance.

(15 x 4^2)/2 = work

(15 x 16)/2 = work

240/2 = work

120 foot pounds = work

Remember, there are two springs both doing the work, so the answer needs to be doubled.  The total work being done by the springs is 240 foot pounds.

One last detail.  The work is technically a negative number (-240 foot pounds) in this case.  That's because the motion of the springs is opposite to that of the garage door.