Hooke's law is the relationship between the force exerted on the mass and its position x. Consider a object with mass m, that is on a frictionless surface and is attached to a spring with spring constant k. The force the spring exerts on the mass depends on how much the spring is stretched or compressed, and so this force is a function of the mass's position.

The idea behind Hooke's Law

Any object that is initially displaced slightly from a stable equilibrium point will oscillate about its equilibrium position. It will, in general, experience a restoring force that depends on the displacement x from equilibrium.

Hooke's Law is written:

The left particle serves as reference point. Its mass is set to 1, the minimum value possible for the algorithm. The friction is set to a maximum value to prevent oscillations. The top end of the spring is connected to fixed objects.

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In the following simulation the quantities displacement and Force are changed either by a constant factor or in a linear manner.



The spring constant is increased by a factor of 2 from left to right.

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Mass (weight) and spring constant are increased by a factor of 2 from left to right.

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Mass (weight) and spring constant are linearly increased.

Energy of a Spring

The total mechanical energy of a mass m, on a spring, when no other forces do work, is given by the sum of the kinetic and potential energies.

It is easy to evaluate E in terms of the amplitude A of the motion, because v = 0 when x is at its maximum value of A.

^{To find the velocity of an object the following equation can easily be used:}

Amplitude ( A ): The maximum distance that an object moves from its equilibrium position. A simple harmonic oscillator moves back and forth between the two positions of maximum displacement, at x = A and x = - A .

Period ( T ): The time that it takes for an oscillator to execute one complete cycle of its motion. If it starts at t = 0 at x = A , then it gets back to x = A after one full period at t = T .

Frequency ( f ): The number of cycles (or oscillations) the object completes per unit time.

The unit of frequency is usually taken to be 1 Hz = 1 cycle per second.

The negative sign in Hooke's law ensures that the force is always opposite to the direction of the displacement and therefore back towards the equilibrium position (restoring force).

The constant k in Hooke's law is traditionally called the "spring constant" for the system, even when the restoring force is not provided by a simple spring.

http://encarta.msn.com/index/concise/0vol16/029b0000.asp

http://www-lj.eb.com:82/index.htcl/aDB/articles_alpha/thisRow/30601/

http://www.ae.msstate.edu/~masoud/Teaching/SA2/def.proportional_limit.html

http://theory.uwinnipeg.ca/physics/shm/node2.html

http://loner.ccsr.uiuc.edu/cyberprof/physics/101/Lecture/L10P2.html

http://www.ph.utexas.edu/~phy-demo/demo-txt/1r10-10.html

http://guernsey.uoregon.edu/~phdemo/demo/Mechanics/Mech-Elasticity.html

The pictures demonstrating Hooke's Law were found at http://www.ipn.uni-kiel.de/work/a7.1/xyzet/Experiments/E_hook.htm