## 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 **principle of superposition of forces** states that a single, net, or resultant force has the same effect as the sum of the individual forces acting on an object.

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.

\(a\) 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.

Two people push a \(\mathrm{10\,kg}\) box to the right. One person pushes with a force of \(\mathrm{12\,N}\), while the second person pushes with a force of \(\mathrm{15\,N}\). Let’s add the forces together and find the acceleration of the box:

\begin{align*} \Sigma F=ma \\ \mathrm{12\,N+15\,N=10\,kg}\cdot a \\ a=2.7\,\mathrm{\frac{m}{s^2}} \end{align*}

So, the box accelerates at a rate of \(\mathrm{2.7\,\frac{m}{s^2}}\) in the direction of pushing across the floor. In this simple problem, we can take this motion to be in the positive \(x\)-direction, so the sign of our final answer is positive.

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 \(\theta\).

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.

If we had a box where one person pushes with \(\mathrm{100\,N}\) to the right, and another person pushes on the opposite side with \(\mathrm{100\,N}\) to the left, what is the resultant force?

In this problem, we want to use the principle of superposition of forces. Let’s add our forces together, taking the direction of each force into account:

\begin{align*} R=F_1+F_2 \\ R=\mathrm{100\,N-100\,N} \\ R=0\,\mathrm{N} \end{align*}

The net force acting on the box is \(\mathrm{0\,N}\), meaning the box does not accelerate. This makes sense logically—if two people pushed on opposite sides of a box with the same amount of force, the box wouldn’t move in either direction. Being able to add forces in this manner, with the superposition principle, helps us understand that when forces are equal and opposite, they add to zero and the object won’t accelerate. The reverse is also true—when something isn’t accelerating, the forces acting on it will add to equal zero.

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

Consider three forces acting on a box: \(F_1=\mathrm{20.0\,N}\) at an angle of \(\theta_1=45^{\circ}\) measured from the horizontal, \(F_2=\mathrm{5.00\,N}\) at an angle of \(\theta_2=60^{\circ}\), and the weight \(W=\mathrm{15.0\,N}\) pulling directly down in the \(y\)-direction. What is the net force acting on the box?

We’ll want to add all the vectors using vector addition. You can do this using any method you are comfortable with; we’ll go over one way to solve this. Choosing left as negative and right and positive for our coordinate system, we’ll find the x-components of each force and add them together:

\begin{align*} F_x=-F_{1x}+F_{2x}+W \\ F_x=\mathrm{(-20\,N\cdot cos(45^{\circ}))+(5\,N\cdot cos(60^{\circ}))+0\,N} \\F_x=\mathrm{-11.6\,N} \end{align*}

Then, choosing down to be the negative \(y\)-direction and up to be the positive \(y\)-direction, we can find the \(y\) components of each force and add them together:

\begin{align*} F_y=F_{1y}+F_{2y}+W \\ F_y=\mathrm{(20\,N\cdot sin(45^{\circ}))+(5\,N\cdot cos(60^{\circ}))-15\,N} \\ F_y=3.47\,\mathrm{N} \end{align*}

To find the single net force, we can put the \(x\) and \(y\) components back together. In the next diagram, we’ve shown the \(x\) and \(y\) components of the resultant force acting on the box in red. We’ll use these components to find the total force magnitude and direction.

To find the magnitude, we calculate the hypotenuse of the triangle we created:

\begin{align*} F=\sqrt{\mathrm{(3.47\,N)^2+(-11.6\,N)^2}} \\ F=\mathrm{12.1\,N} \end{align*}

Then, we can use trigonometry to find the angle of the force from the horizontal:

\begin{align*} \theta=\mathrm{tan^{-1}(\frac{3.47\,N}{11.6\,N})} \\ \theta=16.5^{\circ}\end{align*}

Now we have the total magnitude and direction of all the forces acting on the box.

## 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.

## Superposition of Forces - Key takeaways

- The principle of superposition of forces states that the net force acting on an object has the same effect as the sum of the individual forces.
- Forces are vectors and follow vector addition rules.
- We use the principle of superposition when adding forces to relate them to the acceleration they cause.
- The superposition principle helps us understand that we can break diagonal forces down into x and y components to add them to other x and y forces, respectively.

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##### Frequently Asked Questions about Superposition of Forces

What is superposition of forces?

The principle of superposition of forces states that a single, net, or resultant force has the same effect as the sum of the individual forces acting on an object.

What is an example of superposition of forces?

An example of the superposition of forces is if two people push towards the right on a box, one with 12 N and the other with 15 N, the box would feel a total force of 27 N to the right. The box would act with the same result if just one person pushed with 27 N to the right.

How do you solve superposition of forces?

You can solve superposition of forces problems by using vector addition to add each force vector acting on an object.

How does superposition work in physics?

In physics, superposition of forces works by allowing one to add force vectors together to find a single net force.

What is the principle of superposition in mechanics?

In mechanics, the principle of superposition states that a single, net, or resultant force has the same effect as the sum of the individual forces acting on an object.

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