### Hooke's Law

Consider an ideal spring suspending a mass from a rigid ceiling, as depicted in Fig.B.1. Assume the mass is at rest, and that its distance from the ceiling is fixed.

If denotes the mass of the earth, and is the distance of mass 's center from the earth's center of mass, then the downward force on the mass is given by Eq.(B.2) as

*acceleration due to gravity*. Changes in due to the motion of the mass are assumed negligible relative to the radius of the earth (about miles), and so is treated as a constant for most practical purposes near the earth's surface. We see that if we double the mass , we double the force pulling on the spring. It is an

*experimental fact*that typical springs exhibit a displacement that is approximately proportional to the applied force for a wide range of applied forces. This is

*Hooke's law*for ideal springs:

where is the

*displacement*of the spring from its natural length. We call the

*spring constant*, or

*stiffness*of the spring. In terms of our previous notation, we have

Note that the force on the spring in Fig.B.1 is
gravitational force. Equal and opposite to the force of gravity is
the *spring force* exerted upward by the spring on the mass
(which is not moving). We know that the spring force is equal and
opposite to the gravitational force because the mass would otherwise
be accelerated by the net force.^{B.4} Therefore, like gravity, a
displaced spring can be regarded as a definition of an applied force.
That is, whenever you have to think of an applied force, you can
always consider it as being delivered by the end of some ideal spring
attached to some external physical system.

Note, by the way, that normal interaction forces when objects touch
arise from the *Coulomb force* (electrostatic force, or repulsion
of like charges) between electron orbitals. This electrostatic force
obeys an ``inverse square law'' like gravity, and therefore also
behaves like an ideal spring for small displacements.^{B.5}

The specific value of depends on the physical units adopted as
well as the ``stiffness'' of the spring. What is most important in
this definition of force is that a doubling of spring displacement
doubles the force. That is, the spring force is a *linear*
function of spring displacement (compression or stretching).

**Next Section:**

Applying Newton's Laws of Motion

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Gravitational Force