# Dynamics

How do the objects we see in everyday life move the way they do? For instance, have you looked up in the sky and wondered how planes and birds fly across effortlessly? Or perhaps considered why a ball thrown from one end of a football pitch to another, follows a curved trajectory. The field of dynamics in physics will explore these situations and explain the answers to these questions for us in terms of vectors and mathematical equations. Forces are responsible for every change in motion, and the principles of dynamics help us understand precisely how this happens.

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To understand dynamics, we need to understand what a force is and the rules of motion. We'll discuss these below, as well as free-body diagrams and how to use them with Newton's Laws to solve dynamics problems.

## Dynamics and Forces

Before we dive into Newton's laws of motion and free-body diagrams, let's cover some essential knowledge about dynamics and forces to build a strong foundation.

### Dynamics and Forces Definition

As we mentioned earlier, dynamics is the study of the motion of objects we see around us. We can specifically define it as the following.

Dynamics is the study of the relationship between forces and the motion of bodies.

A force is a push or a pull due to an interaction between two or more objects which can cause the interacting objects' motion to change.

An example of a force being exerted on you is when you're being pushed by your friend while sitting on a swing. This pushing force in combination with the tension in the string between the seat and the bar creates a rotational force that allows you to swing backward and forwards. On the other hand, an example of a pulling force could be when you are walking up a snowy hill and pulling your sled behind you. By pulling the string attached to the sled, you are exerting a force on the sled to work against gravity.

An object can't exert a force on itself; a force requires at least two objects (including things like surfaces and fluids) to occur.

There are many different types of forces. The examples above are applied forces, which result from someone or something applying an external force to an object. Contact forces result from objects touching: such as the friction force, spring force, buoyant force, drag force, and normal force. There are also long-range forceswhere the interacting objects don't have to touch to exert forcessuch as gravitational, electric, and magnetic forces. The image below depicts these forces.

Fig. 2 - A hot air balloon floating in the sky is an example of an upward buoyancy force as the helium gas in the balloon is lighter than the surrounding air.

### Dynamic Equilibrium Definition

Any time an object starts moving, stops, slows down, speeds up, or changes direction, a force causes the change. These changes are examples of acceleration. Therefore, whenever an object changes acceleration, we know that something has applied a force to it.

Forces are vectors, meaning they have magnitude and direction. Therefore, they determine how strongly an object is pulled and how it moves. The magnitude and direction of the force are directly related to the magnitude and direction of the resulting change in acceleration. Forces can also act in opposition to each other and effectively cancel each other out, so even if an object isn't moving, forces are still acting on it.

Since force causes acceleration, a force is measured by how much acceleration it creates. The SI unit of force is a newton $$\text N$$, where one newton is equivalent to $$1 \, \mathrm{N} = 1 \, \mathrm{\frac{\text{kg}\,\text{m}}{\text s^2}}$$. We can visualize this unit by thinking of a $$1\,\mathrm{kg}$$ weight. If we push it across a table with an acceleration of $$1 \, \mathrm{\frac{\text{m}}{\text s^2}}$$, with no other forces acting on it, we would be applying $$1\,\mathrm{N}$$ of force on the weight in the direction of our push.

The term dynamic implies movement. Therefore, dynamic equilibrium refers to something that is moving, but yet still in equilibrium. How does that work? How can something be moving, and yet be in equilibrium? The answer lies in forces and acceleration.

When an object is in dynamic equilibrium, that object is not accelerating and has a net force of zero acting on it.

For example, in the absence of any outside forces, kicking a ball would cause it to accelerate. However, once that ball leaves your foot, no net forces will be acting on it. Since the acceleration of an object requires a nonzero net force, that ball will be in dynamic equilibrium; it will keep moving but have a constant velocity: no acceleration.

Another example is objects falling at terminal velocity. At first, when skydivers jump out of a plane, they accelerate rapidly. However, as the drag force of the air builds and builds, their acceleration decreases until the force of gravity equalizes with the drag force of the air acting upward on them. This causes the net forces acting on them to be balanced. Therefore, they stop accelerating, have a constant velocity, and are at dynamic equilibrium.

### Force Equations

The friction force, spring force, and gravitational force each have equations we can use to calculate them. However, to solve for any of the other forces relevant to AP Physics 1 (tension and normal forces), we have to use the other forces acting on the object to solve for them. To do this, we use Newton's Laws of Motion and free-body diagrams, which will be discussed in later sections.

We calculate the friction force using the equation

$F_{\text{f}} = \mu F_{\text{N}} ,$

where $$F_{\text{f}}$$ is the frictional force measured in newtons $$\mathrm{N}$$, $$\mu$$ is the coefficient of friction, and $$F_{\text{N}}$$ is the normal force measured in units of newtons $$\mathrm{N}$$. It is important to note that the coefficient of friction $$\mu$$ is dimensionless and is unique to the material of each surface. The normal force between an object and the surface is the component of the object's weight perpendicular to the surface.

Furthermore, we can also define the spring force equation as

$F_{\text{s}} = k x ,$

where $$F_{\text{s}}$$ is the spring force measured in newtons $$\mathrm{N}$$, $$k$$ is the spring constant of the spring measured in units of $$\mathrm{\frac{N}{m}}$$, and $$x$$ is the displacement of the spring from its resting position measured in meters $$\mathrm{m}$$. This law is also referred to as Hooke's law and showcases that the spring force is proportional to the spring displacement. It is important to note that the spring can be both compressed and extended, thus the displacement vector $$x$$ can take positive as well as negative values.

Finally, we have the gravitational force, also referred to as the weight force. We can define the gravitational force as

$F_{\text{g}} = mg ,$

where $$F_{\text{g}}$$ is the gravitational force measured in newtons $$\mathrm{N}$$, $$m$$ is the mass of the object measured in kilograms $$\mathrm{kg}$$, and $$g$$ is the gravitational acceleration. On the surface of Earth, the gravitational acceleration is given by a value of $$g = 9.81 \, \mathrm{\frac{m}{s^2}}$$.

It's important to remember that mass is not the same thing as weight. Mass is measured in $$\mathrm{kilograms}$$ and doesn't change based on location, whereas weight is a force (measured in $$\mathrm{newtons}$$) equal to mass times gravity, meaning it changes depending on the gravitational field it is in.

## Newton's Laws of Motion

Newton's laws of motion explain the relationship between an object's motion and the forces acting on it. Newton's three Laws of motion are the following:

1. Newton's First Law of motionObjects remain at rest or constant velocity unless acted on by a net external force.

2. Newton's Second Law of motionAn object's acceleration depends on its amount of mass, and the amount of force applied. This law of motion results in the equation $$\sum\vec{F}=m\vec{a}$$.

3. Newton's Third Law of motionIf an object exerts a force on a second object, the second object exerts a force with equal magnitude and opposite direction on the first one.

## Free-Body Diagrams

To represent external forces acting on an object, we draw free-body diagrams. Free-body diagrams allow us to visualize forces exerted on an object, which helps us write the equations representing the physical situation. We draw arrows representing forces in the direction of those forces, with lengths typically relating to the strengths of the forces. The image below is an example of a free-body diagram.

Fig. 3 - Free body diagram.

The object in the image has four forces acting on ita normal force $$F_\mathrm{n}$$ acting upwards, a gravitational force $$F_g$$ acting downwards, a friction force $$F_\mathrm{f}$$ acting to the left, and a tension force $$T$$ acting to the right.

When drawing free-body diagrams, remember that the gravitational force acts straight down, and the normal force always acts perpendicularly away from the surface.

We can choose which object, or group of objects (called a system), to analyze based on which information we want to obtain. If we choose a system to analyze, we can group it into the same free-body diagram and act like the group is one object.

## Fluid Dynamics

We have now covered the field of dynamics in a general sense, but a more specific study of dynamics is the field of fluid dynamics. In this field, we specifically look at the dynamics of liquids and gases. For instance, fluid dynamics would allow us to explain the lift an airplane experiences due to the flow of air underneath its wings. It also is an important foundation in the study of atmospheric physics and how different winds affect the weather across the globe. Fluid dynamics is a much more advanced field of study than what is covered in AP 1 , and will begin to be introduced in university physics.

## Dynamics Physics Examples

Once we draw a free-body diagram, we can use Newton's laws to understand the motion of an object. Newton's Second Law of Motion is particularly useful because of the following equation:

$$\sum\vec{F}=m\vec{a}\mathrm{.}$$

Force $$F$$ is measured in $$\text N$$, mass $$m$$ in $$\text{kg}$$, and acceleration $$a$$ in $$\frac{\text m}{\text s^2}$$. This equation means that the vector sum of the forces (also known as a net or resultant force) acting on an object equals its mass multiplied by its acceleration.

The force and acceleration are both vectors, as indicated by the arrows above the variables. The direction of the net force determines the direction of the object's acceleration; this means we can only use the equation along individual directions (for example, the sum of the forces in the $$x$$-direction equals the object's mass times the acceleration in the $$x$$-direction alone). We can use the principle of superposition of forces to add the forces as vectors or to break a diagonal force into $$x$$ and $$y$$ components.

The direction of the force is the same as the acceleration, but that does not mean that the direction of the force is the same as the direction of the velocity. So, for example, if an object currently moving to the right is pushed to the left, the resulting acceleration acts to the left, but the object may continue moving to the right at a slower speed.

If a $$25\,\text{kg}$$ box is slipping down an inclined plane where $$\theta=30^\circ$$ and the coefficient of friction is $$0.20$$, what is the acceleration of the box?

First, we want to draw a free-body diagram for the scenario, as shown below:

Fig. 4 - Free-body diagram example problem

We included a normal force directed perpendicularly away from the surface, a friction force acting against the slipping motion, and a gravitational force acting directly down. Since most of the forces act on an axis corresponding to the surface, we chose a coordinate system with $$x$$ parallel to the surface and y perpendicular to the surface, as shown. Since the gravitational force is acting diagonally in this coordinate system, we want to determine the $$x$$ and $$y$$ components of the force, shown in red. We will use trigonometry to find these force components ($$F_{gx}=F_g\sin\theta$$ and $$F_{gy}=F_g\cos\theta$$).

To find the acceleration of the box, we can write Newton's Second Law equation in the $$x$$-direction:

$$-F_\mathrm{f}+F_{gx}=ma_x\mathrm{.}$$

To find the friction, we use the equation for friction. Since we know the box is slipping, we know the friction is equal to the friction coefficient times the normal force:

$$|F_\mathrm{f}|=\mu|F_\mathrm{n}|\mathrm{.}$$

To know the normal force, we need to look at the forces in the $$y$$-direction. Since the box is not accelerating in the $$y$$-direction, the sum of the forces equals zero

$$F_\mathrm{n}-F_{gy}=0\mathrm{.}$$

Now, rearrange to solve for the normal force and substitute $$mg\cos\theta$$ for $$F_{gy}$$

$$F_\mathrm{n}=mg\cos\theta\mathrm{;}$$

substituting this into the friction equation yields

$$F_\mathrm{f}=\mu\,mg\cos\theta\mathrm{.}$$

We can substitute this into our first equation, and substitute $$mg\sin\theta$$ for $$F_{gx}$$

$$-(\mu\,mg\cos\theta)+mg\sin\theta=ma$$

and rearrange our equation to solve for acceleration by dividing everything by the mass

$$a=g(\sin\theta-\mu\,\cos\theta)\mathrm{.}$$

Then, we can plug in our given numbers:

$$a=9.8\, \tfrac{\text m}{\text s^2}\,(\sin{30^\circ}-0.2\cos{30^\circ})$$

$$a=3.2\, \tfrac{\text m}{\text s^2\mathrm{.}}$$

The box will slide down the incline at an acceleration of $$3.2\, \frac{\text m}{\text s^2}$$.

And there you have it! You now know why things move the way they do and how objects exert forces on other objects. You have the proper foundation to become a physics wizard!

## Dynamics - Key takeaways

• Dynamics is the study of the relationship between force and motion.
• A force is a push or a pull due to an interaction between two or more objects.
• Forces are vectors, meaning they have magnitude and direction.
• Forces cause changes in acceleration. The direction of the net force determines the direction of the acceleration.
• Friction, spring, and gravitational forces have specific equations used to calculate them; tension and normal force don't.
• We use free-body diagrams to visualize forces acting on an object.
• We use free-body diagrams and Newton's Laws of Motion to solve force and acceleration problems.

## References

1. Fig. 1 - Airplane, Wikimedia Commons (https://commons.wikimedia.org/wiki/File:Finnair.a320-200.oh-lxf.arp.jpg) Licensed by Public Domain.
2. Fig. 2- Hot air balloon, Wikimedia Commons (https://commons.wikimedia.org/wiki/File:Hot_air_balloon_and_moon.jpg) Licensed by CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/)
3. Fig. 3 - Free body diagram, StudySmarter Originals.
4. Fig. 4 - Dynamics example free-body diagram, StudySmarter Originals.

#### Flashcards in Dynamics 12

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What does dynamics mean?

Dynamics is the study of the relationship between force and motion.

What are examples of dynamics in physics?

Anything that involves forces and motion is an example of dynamics: a car collision, the earth exerting the force of gravity on a skydiver, dribbling a basketball, the oscillation of a spring, and many more.

What are the two types of dynamics?

Dynamics can either be referring to linear or rotational motion.

What are dynamic forces?

Dynamic forces are forces that obey Newton's Second Law. Therefore, they are the net forces that are the product of an object's mass and acceleration.

What are the importance of dynamics?

Studying and understanding dynamics allows us to know how the universe works. With dynamics, we can predict an object's motion and use that knowledge to our advantage; this allows us to make safer cars, planes, and machinery. It also allows us to track the motion of astronomical bodies and predict the location of objects in space.

## Test your knowledge with multiple choice flashcards

When drawing a free-body diagram (about a scenario on Earth), what is always true?

What is dynamics?

What is a force?

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