Chapter 6 | Applications of Integration
689
W = lim n →∞ ∑ i =1 n
F ( x i * )Δ x = ∫ a b
F ( x ) dx .
Thus, we can define work as follows.
Definition If a variable force F ( x ) moves an object in a positive direction along the x -axis from point a topoint b , then the work done on the object is (6.12) W = ∫ a b F ( x ) dx . Note that if F is constant, the integral evaluates to F · ( b − a ) = F · d , which is the formula we stated at the beginning of this section. Now let’s look at the specific example of the work done to compress or elongate a spring. Consider a block attached to a horizontal spring. The block moves back and forth as the spring stretches and compresses. Although in the real world we would have to account for the force of friction between the block and the surface on which it is resting, we ignore friction here and assume the block is resting on a frictionless surface. When the spring is at its natural length (at rest), the system is said to be at equilibrium. In this state, the spring is neither elongated nor compressed, and in this equilibrium position the block does not move until some force is introduced. We orient the system such that x =0 corresponds to the equilibrium position (see the following figure).
Figure 6.51 A block attached to a horizontal spring at equilibrium, compressed, and elongated.
According to Hooke’s law , the force required to compress or stretch a spring from an equilibrium position is given by F ( x ) = kx , for some constant k . The value of k depends on the physical characteristics of the spring. The constant k is called the spring constant and is always positive. We can use this information to calculate the work done to compress or elongate a spring, as shown in the following example. Example 6.25
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