Part 3. An example of the work done on a body by a variable force.

 

A common example of the work done on a body by a variable force is found in the action of a spring on a movable body to which it is attached. We consider here the common linear spring of stiffness k where the force F in the spring, tension or compression, is proportional to its deformation x, so that F = kx. Figure 3/4 shows  the two cases where the body is moved by a force P so as   to stretch  the spring a distance x or to compress the spring a distance x. The force exerted by the spring on the body in each case is in the sense opposite to the displacement so it does negative work on the body. Thus, for both stretching and compressing the spring, the work done on the body is negative and is given by

U_1-2= - ò ^x2_x1 Fdx = - ò ^x2_x1 kx dx = -½ k(x₂² – x₁²)

 

When a spring under tension is being relaxed or when a spring under compression is being relaxed, then we see from both examples in Fig. 3/4 that the deformation changes from x₂ to a lesser defor­mation x_1. For this condition the force exerted on the body by the spring in both cases is in the samesense as the displacement, and, therefore, the work done onthe body is positive.

The magnitude of the work, positive or negative, is seen to be equal to the shaded trapezoidal area shown for both cases in fig. 3/4. In calculating the work done by a spring force, care must be taken to see that the units for k and x are consistent. Thus, if x is in meters, k must be in N/m.

The expression F ═ kx is actually a static relationship which is true only when elements of the spring have no acceleration. Dynamicbehavior of a spring when its mass is accounted for is a fairly complex problem which will not be treated here. We shall assume that the mass of the spring is small compared with the masses of other accelerating parts of the system, in which case, the linear static relationship will not involve appreciable error.

 

 

We now consider the work done on a particle of mass m, Fig. 3/5, moving along a curved path under the action of the force F, which stands for the resultant Σ F of all forces acting on the particle. The position of m is established by the position vector r, and its displacement along its path during time dt is represented by the change d r in its position vector. The work done by F during a finite movement of the particle from point 1 to point 2 is

U_1-2= ò ²₁ F• d r= ò^S2_S1 F_t ds

where the limits specify the initial and final end points of the inter­val of motion involved. When we substitute Newton's second law F = ma, the expression for the work of all forces becomes

U _1-2 = ò ²₁ F• d r= ò ²₁ ma• d r

But a•d r = a_t ds, where a _t is the tangential component of the ac­celeration of m. In terms of the velocity v of the particle, ds = v dv. Thus, the expression for the work of F becomes

 

U _1-2 = ò ²₁ F• d r= ò ^U2_U1 mv• dv = ½ m (v ₂² – v ₁²)   (2)

where the integration is carried out between points 1 and 2 along the curve, at which points the velocities have the magnitudes v₁ and v₂, respectively.

 

Comprehension  check. (Part 3.)