Video Transcript
A body moves along the 𝑥-axis under the action of a force 𝐹. Given that 𝐹 is equal to two over 𝑥 newtons, where 𝑥 meters is the displacement from the origin, determine the work done on the body by 𝐹 when the body moves from 𝑥 equals two to 𝑥 equals three. Approximate your answer to the nearest decimal place.
We begin by recalling that the work done by a force on an object as the object moves along a path parallel to the force is given by 𝑊 is equal to the integral of 𝐹 with respect to 𝑥, where 𝑊 is the work done and 𝐹 is the magnitude of the force that acts on the object. In this question, we are told that the force 𝐹 is equal to two over 𝑥 newtons. This means that to find an expression for the work done, we need to integrate two over 𝑥 with respect to 𝑥. As we need to calculate the work done between 𝑥 equals two and 𝑥 equals three, we will have a definite integral with lower limit two and upper limit three.
We can integrate our expression by recalling one of our standard integrals. The integral of one over 𝑥 with respect to 𝑥 is equal to the natural logarithm of 𝑥 plus the constant of integration 𝐶. This means that integrating two over 𝑥 gives us two multiplied by the natural algorithm of 𝑥. As this is a definite integral, there will be no constant of integration. Next, we need to substitute in our limits and then find the difference between these values. This gives us two multiplied by the natural logarithm of three minus two multiplied by the natural logarithm of two.
We can simplify this expression by recalling some of the laws of logarithms. Firstly, 𝑛 multiplied by log base 𝑎 of 𝑥 is equal to log base 𝑎 of 𝑥 to the power of 𝑛. This means that our expression simplifies to the natural logarithm of three squared minus the natural logarithm of two squared, noting that the natural logarithm is log to the base 𝑒. Three squared is equal to nine, and two squared is four. So our expression becomes the natural logarithm of nine minus the natural logarithm of four. Since log base 𝑎 of 𝑥 minus log base 𝑎 of 𝑦 is equal to log base 𝑎 of 𝑥 divided by 𝑦, we can simplify our expression one stage further. It is equal to the natural logarithm of nine over four.
We are asked to give our answer to the nearest decimal place. So typing our expression into our calculator gives us 0.8109 and so on, which rounds down to 0.8. We can therefore conclude that the work done on the body by 𝐹 when the body moves from 𝑥 equals two to 𝑥 equals three is 0.8 joules to one decimal place.
