Video: Calculating the Amount of Work Done by a Force given the Force Expression

Video Transcript

A body moves along the 𝑥-axis under the action of a force, 𝐹. Given that 𝐹 is equal to eight 𝑠 plus 12 newtons, where 𝑠 metres is the displacement from the origin, determine the work done on the body by 𝐹 when the body moves from 𝑠 equal seven metres to 𝑠 equals eight metres.

We know that when applying a constant force, work done is equal to the force multiplied by the displacement. The work done will be measured in joules, the force will be measured in newtons, and the displacement in metres. In this question, however, the force is not constant. It is a function in terms of 𝑠, the displacement. We will, therefore, calculate the work done using integration. The work done is equal to the definite integral of 𝑓 of 𝑠 between the two limits, 𝑎 and 𝑏. Our function 𝐹 of 𝑠 is equal to eight 𝑠 plus 12. We need to integrate this with respect to 𝑠. We need to calculate the work done between 𝑠 equals seven metres and 𝑠 equals eight metres.

Therefore, our limits are seven and eight. Integrating eight 𝑠 gives us eight 𝑠 squared over two. We increase the power by one and divide by the new power. This can be simplified to four 𝑠 squared. Integrating the constant 12 with respect to 𝑠 gives us 12𝑠. We need to evaluate this between the limits, eight and seven. We, firstly, substitute eight into the expression. This gives us four multiplied by eight squared plus 12 multiplied by eight. This is equal to 352. Our next step is to substitute in 𝑠 equals seven. This gives us four multiplied by seven squared plus 12 multiplied by seven. This is equal to 280. 352 minus 280 is equal to 72. The work done under the action of the force 𝐹 is 72 joules.