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.
