Superposition of Forces

Imagine ten people all lifting a grand piano at the same time. How could we know the total amount of force exerted on the piano if the people all exert individual upward forces? The answer is the principle of superposition of forces. This principle allows us to add each person’s individual force together to get the total force and understand that the piano would act the same under one total force as it does with the ten individual forces. In this article, we’ll discuss the definition of the principle of superposition of forces, the superposition of forces formula, and some examples of how to apply this important concept.

The Principle of Superposition of Forces

In everyday life, rarely does a single force influence the motion of systems. We’ve learned that forces work in pairs, and recognize that there are often many interactions at play that cause and shape motion, such as the force of friction opposing translational motion from an applied force, or the force of gravity working against the tension in a hanging object. Solving problems involving multiple acting forces involves the principle of superposition of forces.

The net force, or resultant force, means the vector sum of individual forces. Returning to our piano lifting example, this means that the combined net force of the group lifting feels the same as each individual’s lift added together. Each of the ten people could lift fifty pounds together, or one person could lift five hundred pounds alone—the piano lifts the same amount either way. In a group effort, each person standing beneath the piano applies an upward pushing force across the surface of the piano, creating one net upward push altogether.

Superposition of Forces Formula

We can express the principle of superposition of forces using the following formula:

\begin{align*} R=\Sigma F=F_1+F_2+F_3... \end{align*}

where \(R\) is the resultant force or net force equal to \(\Sigma F\), which means we’ve taken the sum of all the forces added together. The amount of individual forces \(F_n\) we add together depends on the problem at hand.

Forces are vectors, which means they have magnitude and direction. Because forces are vectors, the addition of forces follows the rules of vector addition. We can’t simply add the magnitudes of two forces together to get a net force—the direction of the vectors has to be taken into account. The image below shows an example of vector addition.

Vector addition of two components on the x-y plane, Wikimedia Commons

\(a\) is the vector \(\langle 3,3\rangle\) and \(b\) is the vector \(\langle 2,1\rangle\). If we add the \(x\) components together and the \(y\) components together, we get the resultant vector in red of \(\langle 5,2\rangle\). In this way, we can add force vectors to find one net force vector. We can also notate these vectors in terms of their magnitude and direction rather than vector notation, as force vectors are generally written.

Superposition of Forces and Newton’s Second Law

Typically, the principle of superposition of forces is most applicable to contact forces when we want to apply Newton’s Second Law of Motion to a particular situation. This law states that the sum of the forces acting on an object is dependent on the mass of the object and the acceleration of the object. It results in the following equation:

\begin{align*} \Sigma F=ma \end{align*}

Force and acceleration have to apply in the same direction for this equation to be accurate. Forces that occur along the \(x\)-axis contribute to accelerations in the \(x\)-direction, and forces that occur along the \(y\)-axis contribute to accelerations in the \(y\)-direction. For example, if you push a hockey puck only horizontally across a field of ice, it will gain speed to move across the ice; it won’t randomly gain speed upwards.

We use the superposition principle with Newton’s Second Law equation to take multiple forces and add them together, creating one single force that relates to the acceleration of the object.

What if both a horizontal force and a diagonal force act on an object? If we have forces acting in different directions, we can’t just add their magnitudes together. Instead, we have to break the forces into components and look at the \(x\) and \(y\)-directions independently. Using the principle of superposition, we can take a single force and split it into two, making the vector addition between multiple forces easier.

This concept is shown in the image below. In this image, we have a friction force acting to the left, a normal force acting upwards, a gravitational force (or weight) acting downwards, and a tension force acting diagonally upwards and to the right at an angle \(\theta\).

Free body diagram with a diagonal force split into x and y components, StudySmarter Originals

If we want to add all the forces in the \(x\)-direction (to relate the \(x\)-direction force with the \(x\)-direction acceleration), we need to split the tension force into an \(x\) component and a \(y\) component. This can be seen in red—instead of having one diagonal tension force, we’ve created a tension force acting to the right and a tension force acting upwards that vector adds to equal the total tension force. In order to split the force into its components, we need to remember a little trigonometry:

\begin{align*} T_x=T\mathrm{cos\theta\,\,and\,\,}T_y=T\mathrm{sin\theta} \end{align*}

Once we split the tension into \(x\) and \(y\) components, we can add them to the other \(x\) and \(y\) forces respectively, and use them in Newton’s Second Law equation to relate them to acceleration. The equation for the \(x\)-direction would look like the following:

\begin{align*} T\mathrm{cos\theta+F_{fric}}=ma_x \end{align*}

In the \(y\)-direction we have:

\begin{align*} T\mathrm{sin\theta+F_N-W}=ma_y \end{align*}

The superposition principle allows us to add all the forces into one single force, but this method of adding the forces into two resultant forces (one in the \(x\)-direction and one in the \(y\)-direction) is often more practical in solving physics problems.

Superposition of Forces Examples

Let’s go over a couple of examples of using the principle of superposition of forces, starting with the classic example of equal and opposite forces.

Let’s go through one more example, this time adding multiple applied forces into one net force.

Superposition of Gravitational Forces

In this article, we discussed the principle of superposition mainly with respect to contact forces, but this principle applies to all forces. Superposition even applies to subjects besides forces, such as waves, electric fields, and beyond.

The principle of superposition works the same way for gravitational forces as it does for contact forces—the net gravitational force acting on an object has the same effect as the sum of the individual gravitational forces. The vectors of gravitational force add in the same manner as we showed above.