Work Done by a Spring

The relationship between the force F _sp exerted by an ideal spring and its exten­sion, or compression, was discovered in the seventeenth century by Robert Hooke and is therefore called Hooke's law:

Hooke's law F _sp = − kx (4.3)

FIGURE 4.3	FIGURE 4.4

where k, measured in N/m, is called the spring constant. The quantity x is the displacement of the free end of the spring from its equilibrium position—the point at which the spring has its natural length (see Fig. 4.3). 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 always tends to restore the system to its equilibrium position.

The work done by the spring in a displacement from x _i to x _f is the shaded area in Fig. 4.4, which is the difference in the areas of two triangles.

The work done by the spring when the displacement of the free end from its equilibrium position changes from x _i to x _f is

(4.4)

This work is positive when the force and the displacement are in the same direc­tion.

Work as an Integral of the F_x versus x Graph

When the force is an arbitrary function of position, we need the techniques of calculus to evaluate the work done by it. Figure 4.5 a shows F_x as some function of the position x. We begin by replacing the actual variation of the force by a series of small steps. The area under each segment of the curve is approximately equal to the area of a rectangle. The height of the rectangle is a constant value of force, and its width is a small displacement ∆ х. Thus, the n th step involves an amount of work . The total work done is given approximately by the sum of the areas of the rectangles:

FIGURE 4.5

As the size of the steps is reduced, the tops of the rectangles more closely trace the actual curve shown in Fig.4.5 b. In the limit ∆x→ 0, which is equivalent to letting the number of steps tend to infinity, the discrete sum is replaced by a continuous integral:

Thus, the work done by a force F_x from an initial point A to final point В is

(4.5)

The sign of this work depends on F_x and the limits of the integral. Interchanging the limits of the integral is equivalent to reversing the direction of the displace­ment. Thus,

W _B→A = − W _A→B (4.6)

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