Video Transcript
A body moves along the 𝑥-axis under the action of a force 𝐹. Given that 𝐹 is equal to two root 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 one to 𝑥 equals four.
We recall 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, in this question, two root 𝑥 newtons. It is the work done on the body by 𝐹 that we are trying to calculate, and we can find an expression for this by integrating two root 𝑥 with respect to 𝑥. As we’re trying to calculate the work done when the body moves from 𝑥 equals one to 𝑥 equals four, we will have a definite integral with lower and upper limits equal to one and four, respectively. These values of 𝑥 are the displacement of the body from the origin in meters.
We recall from our rules of exponents or indices that the square root of 𝑥 can be written as 𝑥 to the power of one-half. This means that we need to integrate two 𝑥 to the power of a half with respect to 𝑥 between the limits one and four. The power rule of integration tells us that the integral of 𝑥 to the power of 𝑛 with respect to 𝑥 is equal to 𝑥 to the power of 𝑛 plus one over 𝑛 plus one plus the constant of integration 𝐶, where 𝑛 cannot be equal to negative one. Adding one to a half gives us three over two. So the integral of two 𝑥 to the power of a half is two 𝑥 to the power of three over two divided by three over two. As we are dealing with a definite integral, there will be no constant of integration.
Dividing by three over two is the same as multiplying by two-thirds. So our expression simplifies to four-thirds 𝑥 to the power of three over two. Our next step is to substitute in the two limits and find the difference between our two answers. We can calculate four to the power of three over two by square rooting four and then cubing our answer. This simplifies to two cubed, which is equal to eight. In the same way, one raised to the power of three over two is the square root of one cubed, and this is equal to one. Our expression simplifies to four-thirds multiplied by eight minus four-thirds multiplied by one. This is the same as 32 over three minus four over three, which gives us a final answer of 28 over three.
The work done on the body by 𝐹 when the body moves from 𝑥 equals one to 𝑥 equals four is 28 over three joules.
