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A body moves along the 𝑥-axis under the action of a force, 𝐹.
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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.
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We know that when applying a constant force, work done is equal to the force multiplied by the displacement.
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The work done will be measured in joules, the force will be measured in newtons, and the displacement in metres.
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In this question, however, the force is not constant.
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It is a function in terms of 𝑠, the displacement.
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We will, therefore, calculate the work done using integration.
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The work done is equal to the definite integral of 𝑓 of 𝑠 between the two limits, 𝑎 and 𝑏.
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Our function 𝐹 of 𝑠 is equal to eight 𝑠 plus 12.
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We need to integrate this with respect to 𝑠.
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We need to calculate the work done between 𝑠 equals seven metres and 𝑠 equals eight metres.
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Therefore, our limits are seven and eight.
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Integrating eight 𝑠 gives us eight 𝑠 squared over two.
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We increase the power by one and divide by the new power.
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This can be simplified to four 𝑠 squared.
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Integrating the constant 12 with respect to 𝑠 gives us 12𝑠.
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We need to evaluate this between the limits, eight and seven.
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We, firstly, substitute eight into the expression.
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This gives us four multiplied by eight squared plus 12 multiplied by eight.
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This is equal to 352.
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Our next step is to substitute in 𝑠 equals seven.
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This gives us four multiplied by seven squared plus 12 multiplied by seven.
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This is equal to 280.
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352 minus 280 is equal to 72.
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The work done under the action of the force 𝐹 is 72 joules.