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Work Done by a Variable Force

When the magnitude and direction of a force vary in three dimensions, it can be expressed as a function of the position vector (r), or in terms of the coordinates (x, y, z). The work done by such a force in an infinitesimal displacement ds is
W = .d

Work done by a Spring

If x be the displacement of the free end of the spring from its equilibrium position then, the magnitude of spring force is given by
Fx = -kx

The negative sign signifies that the force always opposes the extension (x > 0) or the compression (x < 0) of the spring. In other words, the force tends to restore the system to its equilibrium position

The work done by the spring force for a displacement from xi to xf is given by
Ws =

or Ws =

Note that

• The work done by a spring force is negative
• The work done by the spring force only depends on the initial and final points.
• The net work done by the spring force is zero for any path that returns to the initial point.

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