Question Video: Identifying Whether the Equation for Work Done is Correct Mathematics

True or False: If an object moves along a straight line by a varying force 𝑓(𝑥), then the work done 𝑊 to move the object from 𝑥 = 𝑎 to 𝑥 = 𝑏 is given by 𝑊 = ∫^(𝑎)_(𝑏) 𝑓(𝑥) d𝑥.

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Video Transcript

True or False: If an object moves along a straight line by a varying force 𝑓 of 𝑥, then the work done 𝑊 to move the object from 𝑥 equals 𝑎 to 𝑥 equals 𝑏 is given by 𝑊 is the integral of 𝑓 of 𝑥 with respect to 𝑥 from 𝑏 to 𝑎.

We begin by recalling that if the force acting on an object is described by a continuous function, such as in the graph shown, we have to use integration to find the area under the curve and hence the work done. In the graph drawn, 𝐹 is the magnitude of the force, 𝑆 is the magnitude of the displacement, and 𝜃 is the angle between the force acting on the object and its displacement.

In this question, we are told that the object moves along a straight line by a varying force. And we know that if the force and displacement are in the same direction, then 𝜃 is equal to zero degrees. The cos of zero degrees is one. We can then relabel our axes as the displacement 𝑥 and the varying force 𝑓 of 𝑥. To calculate the work done or the area under the curve, we will need to integrate the function 𝑓 of 𝑥 with respect to 𝑥. This suggests that the statement might be true.

However, we are interested in the work done to move the object from 𝑥 equals 𝑎 to 𝑥 equals 𝑏. And in order to calculate the shaded area, we will have a definite integral with lower limit 𝑎 and upper limit 𝑏. The limits in the expression in the question are the opposite way round. This expression would give us the work done to move the object from 𝑥 equals 𝑏 to 𝑥 equals 𝑎. And we can therefore conclude that the statement is false.

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