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Springs and Hooke's Law

Everyone deals with springs all the time. Go find one and play with it, a shoelace works enough. Pull on one end and you'll feel the same amount force on your other hand and see the spring stretch outward. Using Newton's 1st law, the force the spring produces is opposite what your hand produces, and the other end has the same situation as shown below.


It might be a bit confusing, but there are two ways to draw the free body diagram (FBD) for this situation. In a statics problem, the forces are balanced on every object in the system and therefore are balanced at every point (often called a node) in the system. The FBD on the objects is simple, and it tells us that all the forces we can see have the same magnitude, but nothing about the spring force, it only sees normal forces where it contacts the hands. The other diagram looks at the atoms at the ends of the springs, one side is pushing on the hand atoms and the other side is pulling on other spring atoms. The point of this was to show that the applied force on a spring is going to be equal and opposite to the internal forces, the middleman of normal forces can be ignored.

Looking at the point on the right, the applied force would be considered positive because it points in the positive x direction and the spring would be under tension. The material would stretch to the right as shown below.


As shown above, the restoring force points in the opposite direction of the applied force and displacement, so a positive displacement has a negative spring force (restoring force is the same thing). This gives the relationship discovered by Robert Hooke, referred to as Hooke's law. The variable "k" is called a spring constant, and is a function of material properties, shape, mechanical advantages, temperature, and more. Determining spring constant for basic shapes like solid bars is covered on the solid mechanics page (not written yet), but shapes like coil springs are far more complicated and not covered in basic physics courses, so empirical (meaning experimental) data will be used in your classes.

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