Work Done By A Variable Force

dW = .d (8.5)

The total work done in going from point A to point B as shown in the figure. i.e.

W_{A →B} =

In terms of rectangular components,

= Fx + Fy + Fz

and d = dx + dy + dz

therefore ,

WA → B = (8.6)

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 (8.7)

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 =

orWs = (8.8)

Note

• 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.

Graphically, the work done by the spring force in a displacement from xi to xf is the shaded area (as shown in the figure 8.8) which is the difference in the areas of two triangles.

In general, the work done by a variable force F(x) from an initial point xi to final point xf is given by the area under the force - displacement curve as shown in the figure (8.9).

Area (work) above the x - axis is taken as positive, and vice-versa.

A 5 kg block moves in a straight line on a horizontal frictionless surface under the influence of a force that varies with position as shown in the figure. Find the work done by this force as the block moves from the origin to x = 8m.

Solution

The work from x = 0 to x = 8 m is the area under the curve.