Work Done by a Force
Work is the measure of energy transferred to or from an object when a force is applied over a displacement, providing the link between forces and energy. This concept explains how energy moves in systems and how power and efficiency are calculated in both mechanical and biological contexts.
Use this page to revise the following concepts within work done by a force:
Work done by a force is calculated as the product of force and the displacement in the direction of the force.
\[ \text{Work} = \text{Force} \times \text{displacement} \]
\[ W = Fs \]
Where:
- \(W\) = work \((\mathrm{J})\)
- \(F\) = force \((\mathrm{N})\)
- \(s\) = displacement in the direction of the force \((\mathrm{m})\)
As force and displacement are both vector quantities, work is calculated using their dot product, so work is a scalar quantity.
If the force and displacement are not in the same direction, use the generalised work done equation:
\[ W = Fs\cos(\theta) \]
Where:
- \(\theta\) = angle between the force and the displacement
This generalised equation can be used to calculate the work done for any angle between the force and displacement. When the force and displacement are in the same direction, \(\theta = 0^\circ\), so \(\cos(0^\circ) = 1\).
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When considering the direction of force and displacement:
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Worked example
A worker at a warehouse pushes a heavy crate across the floor. The crate has a mass of \(20 \mathrm{~kg}\), and the worker applies a constant horizontal force of \(50 \mathrm{~N}\). The crate is pushed horizontally over a distance of \(8 \mathrm{~m}\). How much work is done by the worker in moving the crate?
From the question, the variables are:
- \(F = 50 \mathrm{~N}\)
- \(s = 8 \mathrm{~m}\)
- \(\theta = 0^\circ\)
Substitute the values:
\[ W = Fs\cos(\theta) \]
\[ W = (50)(8)(1) = 400 \mathrm{~J} \]
The work done by the worker is \(400 \mathrm{~J}\).
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Force-displacement graphs
Force-displacement graphs illustrate how the force acting on an object changes as the object moves, due to a displacement in the direction of the force. They are often used to calculate the work done by the force, which is determined by finding the area under the graph.
Graphs are illustrated with:
- displacement, \(s\), measured in metres \((\mathrm{~m})\), on the \(x\)-axis; and
- force, \(F\), measured in newtons \((\mathrm{~N})\), on the \(y\)-axis.
Example of a graph:
Work done is the area under the graph of force versus displacement, with the sign of the area indicating whether the work is positive (area is above the \( x \)-axis), or negative (area is below the \( x \)-axis).
The area under the graph can be calculated using simple geometry by dividing the graph into shapes such as triangles, rectangles, and trapeziums, and then summing their areas.
For more complex graphs, calculus can be used to determine the area under the curve, which represents the work done.
Worked example
The graph below shows the force applied to a box as it moves across a surface. Calculate the total work done on the box over the first \(4\) metres of displacement.
Work done over the first \(4 \mathrm{~m}\) of displacement is shown by the region indicated above.
\[ W = \text{area under graph} \]
\[ W = \text{area of trapezium} \]
\[ W = \frac{1}{2}(2 + 4)(4) \]
\[ W = 12 \mathrm{~J} \]
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Work-energy principle
The work-energy principle describes the relationship between work done by a force, and the change in kinetic energy. It can be calculated using the following:
\[ W = \Delta E_k \]
\[ W = \frac{1}{2}mv^2 - \frac{1}{2}mu^2 \]
Where \( \Delta E_k\) = change in kinetic energy \((\mathrm{J})\)