Variable forces occur when the magnitude of the force varies as position changes. Unlike constant forces, the computation for total work will not be straightforward in these instances.

In this post, we expand on computing work with a variable force.

What is Variable Force?

Let's say we are pushing a box continuously, as shown in the figure. At first, we will have full strength in pushing it; Eventually, there will be a point where we'll get tired. From this example, the force we applied at the start differs from when we felt tired.

Consider another example, say we are to compress a spring as much as possible. At first, it's easy to push it; however, as we squeeze it even further, it will take more force to flatten it.

All of these situations represent variable forces - the magnitude of the force varies as position changes. We usually represent this force in terms of a function.

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To illustrate the difference between constant and variable, recall that we can graphically illustrate work using a force-position graph \(F-\Delta\). Let's investigate two cases:

The first graph represents work performed with constant force

The other one represents work for the variable force.

Work is the area under the curve for these situations.

In the case of the first graph, the curve is a straight line (or a constant function); hence, work is the area of the rectangle:

\(W=F \cdot \Delta_x\)

It corresponds to the formula for work in the beginning.

Moving to the second graph, we can observe that the force can be any math function. Since the curve varies, to find the area, we use one of the key applications of definite integration:

\(W=\int_{x_1}^{x_2} F(x) \cdot d x\)

\(W\) is the total work done

\(x_1\) is the lower limit (the initial position)

\(x_2\) is the upper limit (the final position)

\(F(x)\) is the variable force as a function of position \(x\)

\(dx\) is the differential position

This equation is the general work equation that would encompass all instances, including the constant force situation. If we try to evaluate this equation with a constant function, we have:

\(W=\int_{x_1}^{x_2} F \cdot d x\)

\(W=F (x_2-x_1)\)

\(W=F \Delta_x\)

This result corresponds to the earlier equation.

Summary

Variable forces occur when the magnitude of the force varies as position changes. We usually represent this force in terms of a function.

To find the total work with varying forces, we have \(W=\int_{x_1}^{x_2} F(x) \cdot d x\). Variables \(W\) is the total work done, \(x_1\) is the lower limit (the initial position), \(x_2\) is the upper limit (the final position), \(F(x)\) is the variable force as a function of position \(x\), and \(dx\) is the differential position

This equation is the general work equation that would encompass all instances, including the constant force situation.