Newton's Laws of Motion

What's inside this article?

Motion is everywhere and we know motion very well in our daily life but everything we know about motion is our common sense which is not enough. In Kinematics we learn motion without knowing cause of motion. The cause of motion is always a force, that is force causes motion to a body at rest or changes the motion of a body already in motion.

It's more accurate to say the force causes the change in motion of a body not simply motion. If a body moves with constant velocity, the body is already in motion but there is no force (you'll see this in Newton's first law of motion), so the force causes the change in motion.

But it's true that force causes motion to a body at rest and we can still say the force causes change in motion from rest to motion as the state of rest can be considered as the special case of motion, that is the motion at zero velocity. You'll see this in Newton's first law of motion later where we also define force using Newton's first law.

Newton's laws require modification for particles of very small sizes such as in subatomic level and motion near the speed of light. Classical mechanics or simply mechanics does not deal with the motion of particles of very small sizes such as within atoms and motion near the speed of light.

The dynamics part of Mechanics, that is the study of motion in relationship with force which causes motion to a body at rest or the change in motion to a body already in motion begins with understanding the Newton's laws of motion.

You may have been through a lot of real world activities involving motion without knowing what causes it. You may have developed your own basic understanding of motion and its cause such as the push or pull. For example, our common sense understands push and pull as forces but what about friction forces? Friction forces are hidden and to a common sense it seems there is no force at all.

Here we take the basic understanding (don't let basic understanding of force fool you) aside and understand the real experimental facts based on Newton's laws of motion. In Kinematics all we consider is motion but in dynamics we relate motion with force.

What is Force?

Newton's laws of motion define force based on facts and experiments. Our basic understanding of force is what our common sense tells us about forces. Push and pull are the forces to our common sense. Of course they are forces but defining force based on common sense only is not a good idea.

For example, the gravitational force is pulling us and we don't even know something is pulling us and normal force is also pushing us resulting zero net force, so we can stay still in our place. It is very common to us and we neglect it entirely and we think there is no force at all. When we push or pull something, then we call it force.

So, just the basic understanding as the push or pull is not sufficient (indeed push and pull are forces - your basic understanding is not wrong but not just sufficient enough to define force). The basic understanding sometimes makes the very simple Newton's laws hard to grasp. So be careful!

Force refers to an interaction to a body by means of another body or between a body and its environment.

Force is an interaction, that is not just a single body is involved in force but a result of mutual involvement two bodies (you'll be clear more on this in Newton's third law) exerting and experiencing force. You'll understand in Newton's third law that forces always come in pairs. We'll define force again using Newton's first law later.

When you kick a ball, you exert force on the ball and you do that with physical contact with the ball. The forces with physical contact are called contact forces. Tension, normal and friction forces are the examples of contact forces.

Another kind of forces are field forces where the interaction takes place in empty space without touching or physical contact such as gravitational force. Another example is the force between two magnets.

All fundamental forces of nature are field forces which are gravitational interactions (between bodies), electromagnetic interactions (between electric charges), strong interactions (between nuclear particles), weak interactions (in certain radioactive decay process).

Superposition of Forces and Net Force

A force is an interaction. If you apply a force on a body, the body is not only playing a role here, you are also involved applying the force. The body gets a force and you apply it. So the mutual interaction between bodies or a body and its environment is called force.

If a body has been applied multiple forces, the resultant or net force on the body is the vector sum of all individual forces. Since force is a vector quantity which is always associated with a direction (if you are applying a force, you must first know in which direction you are applying the force), the sum must be the vector sum.

A net force is the vector sum of all forces (interactions) on a body. If the net force is not zero, the motion of a body must change.

If $\vec F_1$, $\vec F_2$ and $\vec F_3$ are acting on a body, the net force due to all individual forces acting on the body $\sum \vec F$ is

\[\sum \vec F = \vec F_1+ \vec F_2 + \vec F_3\]

The resultant force, unbalanced force and total force all mean the net force. We make sure that a body is acted on by no net force in Newton's first law of motion. If you are to understand Newton's laws of motion (especially Newton's first law), you must understand what inertia is.

What is Inertia?

Inertia of a body is the tendency of the body to remain in its original state. If a body is at rest initially, it always remains at rest until a net force is applied to it. The tendency of a body to remain at rest is called inertia of rest.

Likewise, if a body is in motion, it always wants to be in motion until a net force is applied to it. The tendency of a body to remain in motion is called inertia of motion.

In other words if no net force is applied to a body at rest, the body will always be in the state of rest and if no net force is applied to a body in motion, it'll always move with constant velocity (never stops until a net force acts on it).

Inertia of a body is the tendency of that body to resist the change in its current state either at rest or in motion.

A coin is over a paper. When you suddenly pull the paper towards you with high speed, the coin falls back to the original place where it was initially; the coin didn't come with the paper towards you but instead it wanted to remain in its original place at rest. This indicates that the coin resists to come in motion with the paper. This tendency of the coin at rest to resist to come in motion is the inertia of rest.

A bus example is very familiar and easy to understand inertia. A bus is moving with constant velocity with some passengers. Initially the passengers are also moving with the same velocity in the bus but when driver applies breaks to stop the bus, the passengers move forward.

The reason behind this is, the passengers were initially in motion and when the bus stops the passengers tend to be in motion due to the inertia of motion and move forward. Now the bus is stopped and the passengers and the bus are both at rest. When the driver accelerates the bus, the passengers move backwards. Initially the passengers were at rest and when the bus accelerates the passengers still want to be at rest due to the inertia of rest and move backwards.

The forward or backward movement of your body becomes more clear when you understand inertial frame of reference. When viewed from inside the bus, it may seem weird that you are moving without any force applied on you. Inertia is the heart of Newton's first law of motion and we also call the Newton's first law of motion as the law of inertia.

Newton's First Law

A nonzero net force causes the change in motion of a body. What happens when there is no net force on a body? The answer of this question is given by Newton's first law. Newton's first law determines what happens to a body if there is no net force on the body (you'll see later how we change the statement slightly to define force using Newton's first law).

NEWTON'S FIRST LAW: A body initially at rest always remains at rest and a body initially in motion always keeps moving with constant velocity if no net external force acts on the body.

If there is no net force on a body, the body always remains in its current state either at rest if it was initially at rest or moving if it was initially in motion. We just stated Newton's first law of motion where we said "body at rest" which is not necessary as a body with zero velocity is the same thing as being at rest. An alternative statement can be

NEWTON'S FIRST LAW: A body has constant velocity (can be zero) if no net external force acts on it.

Here we said "no net force" which is the same as no forces at all, that is in the absence of any forces or multiple forces cancel out to zero. To be able to define force only using Newton's first law we must consider an isolated system where no forces act on a body at all, that is the same as the absence of all forces.

NEWTON'S FIRST LAW: A body has constant velocity in the absence of any external forces.

The statement just stated transfers the case what happens if net force is zero to Newton's second law. In other words this statement lets Newton's second law to handle the situation of multiple forces acting on a body cancel to zero focusing completely on an isolated system without any forces at all (absence of all forces).

Newton's second law can handle the situation of zero net force (there are multiple forces but they cancel to zero) but you'll see it'll be converted back to Newton's first law, so we stick to the Newton's first law of motion for zero net force as well.

It's obvious from the above statement of Newton's first law that only a force can cause the change in motion of a body.

FORCE: A force is an interaction which causes the change in motion of a body.

The isolated system of no forces at all we considered in the statement of Newton's first law above is equivalent to zero net force, that is multiple forces exist but they cancel to zero. In real life situations where multiple forces on a body cancel to zero, only a nonzero net force can cause the change in motion and zero net force is equivalent to no forces at all.

Applying zero net force, that is multiple forces cancel to zero is equally valid in Newton's first law of motion and zero or no net force is equivalent to the absence of all forces on a body.

If a body has a constant velocity (can be zero), the body is said to be in equilibrium. A constant velocity tells everything, sure it can be zero which determines the state of rest and nonzero constant velocity means uniform motion. This eventually means a body initially at rest remains at rest and a body initially in motion remains in motion if there is no net force on the body and the body is said to be in equilibrium.

Newton's first law is the equilibrium condition of a body. A body can be said to be in equilibrium if no forces act on the body at all or the vector sum of all forces on the body is zero (zero net force).

Remember the constant velocity means constant speed along straight line and the change in motion is the change of velocity. Velocity is a vector quantity and if it changes there must be a force. Newton's first law does not tell that a body in uniform circular motion (motion with constant speed in a circle) is acted on by no force. The direction of velocity always changes, that is the motion changes and there must be a force.

Inertial Frame of Reference

Newton's first law is incomplete if we don't include inertial frame of reference in our statement of Newton's first law. To understand what inertial frame of reference is we begin with an example.

You are travelling in a bus moving with constant velocity at the back seat and there is a ball on the floor sitting next to you. If the driver applies brakes, the ball moves towards the front of the bus, not only the ball moves you also move forward. Why do you think there are seat belts in a car? Let's move back to the bus example and focus on the ball only for now.

When the ball moved, you didn't see any force applied on the ball but the ball moved why? Initially the ball was in motion with the bus, when the bus tried to stop the ball didn't want to come to rest due to the inertia of motion and you saw the ball moved even without any force applied on it inside the bus.

No force was ever applied on the ball, the force was only applied to stop the bus not the ball but inside the bus you see the ball (which was already in motion with the bus) moving from rest without any force being applied on it which completely violates the Newton's first law of motion.

The Newton's first law of motion is no longer valid when the driver applies breaks inside the bus, any object should not be moving from rest without force. So you can understand that the Newton's first law in invalid in an accelerated system.

The frame of reference in which Newton's first law is valid is called inertial frame of reference.

The frame of reference of the bus at the time when the bus tried to stop (accelerated system) is no longer an inertial frame. If the bus was moving with constant velocity, the bus would be an inertial frame and Newton's first law would be valid.

So an accelerated system is not an inertial frame. The Earth is not completely an inertial frame due to acceleration associated with its rotation but we can say the Earth is approximately an inertial frame.

Newton's first law is valid only in an inertial frame of reference, so the final statement of Newton's first law can stated as

NEWTON'S FIRST LAW: When viewed from inertial frame of reference, a body has constant velocity if no net external force acts on it.

No net force or zero net force mean both multiple forces that cancel to zero and the absence of all forces on a body. In real life situations we find systems where multiple forces cancel to zero which results equilibrium and we describe such situation with Newton's first law.

Newton's Second Law

In Newton's first law of motion we asked what happens when there is no net force on a body (zero net force) but in Newton's second law we ask what happens when there is a nonzero net force on a body.

You'll see this later that if net force is zero in Newton's second law of motion, the motion is described by Newton's first law. It means the statement of Newton's second law is converted back to the Newton's first law of motion (constant velocity or zero acceleration).

We know from many experiments that if a nonzero net force acts on a body, the body must accelerate. It means if a constant nonzero net force acts on a body, the velocity of the body changes continuously in a constant rate. And that rate of change of velocity is called acceleration.

The direction of acceleration is in the same direction of net force. For example, a driver applies brakes of a bus, the directions of force and acceleration are in the opposite direction to the velocity of the bus, that is both force and acceleration are in the same direction, and the bus finally comes to rest.

The statement of Newton's second law based on the relationship between force and acceleration can be stated as

NEWTON'S SECOND LAW: When viewed from inertial frame of reference, a body accelerates in the direction of force if a nonzero net external force acts on it.

The statement of Newton's second law recently stated gives us an idea about the relationship between force and acceleration but not a complete statement of Newton's second law.

Suppose you apply a net force $\vec F$ on a body, the body moves with constant acceleration $\vec a$ in the direction of applied force. We know from experiments that if you double the force ($2\vec F$), the acceleration doubles ($2\vec a$) and if you tipple the force ($3\vec F$), the acceleration triples ($3\vec a$) and so on.

It has been seen from the experiments that the acceleration of a body is directly proportional to net force acting on it. Before stating Newton's second law of motion again we define what mass is.

Mass and Force

Let's start with an example of a tennis ball and a basketball. If you throw both balls with the same force, the tennis ball is accelerated more than the basketball. It means the tennis ball has less resistance to the change in motion than the basketball. Greater the mass of a body, more it "resists" the change in its motion.

We know the property of a body to resist the change in its motion is inertia (tendency of a body to remain in its current state is the same thing as the resistance to the change in motion). The quantitative measure of inertia of a body is the mass of the body.

If the same force is applied to two bodies of masses $m_1$ and $m_2$ and $a_1$ and $a_2$ are the corresponding accelerations, the ratio of $m_1$ to $m_2$ is equal to the ratio of $a_2$ to $a_1$, that is the ratio of masses is equal to the inverse ratio of magnitudes of accelerations.

\[\frac{m_1}{m_2} = \frac{a_2}{a_1}\]

If the mass of one body is known, you can determine the mass of another body using acceleration measurements in above equation. Many observations reveal the fact that the net force on a body is inversely proportional to the mass of the body. We learnt above that the net force on a body is directly proportional to the acceleration of the body.

MASS: The ratio of net force on a body to the acceleration of the body is called mass of the body.

The quantity of matter in a body is determined by the quantitative measure of inertia of the body, that is how much the body resists the change in its motion, that is if we apply a net force on a body the motion of the body changes continuously in a certain rate and the body continuously resists the changes in its motion and the measure of that resistance is mass. If a body has acceleration $\vec a$ due to net force $\sum \vec F$, the mass $m$ is

\[m = \frac{|\sum \vec F|}{a}\]

The above equation gives the quantitative measure of mass of a body. We could have defined the net force before defining mass but either way we reach the same destination. After analyzing the results of many experiments and observations, it turns out that the net force on a body is equal to the product of its mass and acceleration. We know that the acceleration is directly proportional to the the net force and inversely proportional to the mass. Taking the proportionality constant $1$ we obtain the expression of net force.

\[\sum \vec F = m\vec a\]

In the first statement of Newton's second law of motion we just focused on what happens if there is nonzero net force on a body but now a more complete statement (a summarized statement of the observations of nonzero net force on a body) of the Newton's second law is

NEWTON'S SECOND LAW: When viewed from inertial frame of reference, a body accelerates in the direction of force if a nonzero net external force acts on it. The net force is equal to the product of mass and acceleration of the body.

Newton's second law is valid only in inertial frame of reference. It is not valid in an accelerating car, where you move forwards or backwards without any force at all! Newton's first law is the special case of Newton's second law when net force is zero. If the net force is zero, Newton's second law is converted to the Newton's first law, that is acceleration is zero which means constant velocity. The force is always external net force. Internal forces do not cause the change in motion. If they do, you can pull yourself to reach your ceiling!

Now we come to the interesting situation of the above Newton's second law of motion. What if the mass changes not the velocity and there is still force? You'll occasionally encounter such situations in Physics and the above statement of the Newton's second law is invalid in such situations.

Newton realized that the quantity of motion is better described by the momentum which is the product of mass and velocity of a body. The momentum $\vec p$ of a body of mass $m$ and velocity $\vec v$ is

\[\vec p = m\vec v\]

Now we state the final statement of Newton's second law which is equivalent to the Newton's original statement in terms of rate of change of momentum.

NEWTON'S SECOND LAW: When viewed from inertial frame of reference, the net force on a body is equal to the rate of change of momentum of the body.

The above statement of the Newton's second law is expressed in mathematical form as

\[\sum \vec F = \frac{d\vec p}{dt} \tag{1} \label{1}\]

Consider the mass is constant and we know $\vec p = m\vec v$ and solving above equation we get

\[\sum \vec F = m\frac{d\vec v}{dt} = m\vec a \tag{2} \label{2}\]

Both equations represent the mathematical form of Newton's second law of motion. Newton originally stated the second law in terms of the rate of change of momentum expressed by Equation \eqref{1} but we generally use Equation \eqref{2} as the Newton's second law. Remember the Equation \eqref{2} is the special case when the mass is constant!

The product of mass and acceleration, that is $m\vec a$ is not a force! It may be the point of confusion that you may call $m\vec a$ is a force. The acceleration is the result of force and $m\vec a$ is the value of the force not the force itself. The product $m \vec a$ is the result of net force which is the vector sum of all forces acting on a body. You move backwards when your car suddenly accelerates. There is no force on you but if $m\vec a$ is a force, your acceleration mislead you to think there is a net force on you.

As you know now $m\vec a$ is not a force, we never define force as the product of mass and acceleration of a body, that is we don't say force is $m\vec a$ but we say the net force is equal to $m \vec a$. Note that "is equal to" to is used instead of simply "is".

Newton's Third Law

In Newton's first law we asked what happens when the net force on a body is zero and in second law we asked what happens if there is nonzero net force on a body. Newton's third law simply allows us to identify what forces act on a body.

A force can not appear alone. A force is an interaction between two bodies. It's an interaction which is not possible with a single body. When you kick a ball, the ball gets the force but only the ball is not involved; it's the interaction between you and the ball, that is you are required to exert force on the ball. You can not think about moving a ball by the ball itself!

A force always comes in a pair. If you kick a ball, the ball also kicks back on you with the same magnitude of force in opposite direction. If you pull a door, the door also pulls back on you. The result of any force (action) is the opposite force (reaction) of the same magnitude.

Figure 1 You exert a force on a wall.
Figure 2 A book on a table.

In Figure 1 you exert a force on a wall and the wall exerts the same magnitude of force on you in opposite direction. In Figure 2 a book is on a table. The weight of the book exerts downward force on the table and the table exerts the same magnitude of force on the book in upward direction. Both forces balance each other and the book stays on the table.

\[\begin{align*} {{\vec{F}}_{\text{wall on you}}}&=-{{\vec{F}}_{\text{you on wall}}} \\ \text{or,}\quad {{\vec{F}}_{\text{wall on you}}}+{{\vec{F}}_{\text{you on wall}}}&=0 \end{align*}\]

If you summarize the similar observations, you end up with a Newton's third law of motion which can be stated as

NEWTON'S THIRD LAW: If a body $A$ exerts a force on body $B$ (action), the body $B$ exerts the same magnitude of force on body $A$ (reaction) in opposite direction.

If body $A$ exerts force $\vec F_\text{AB}$ on body $B$, the body $B$ exerts the same magnitude of force $\vec F_\text{BA}$ on body $A$ but in opposite direction. The subscript $\text{AB}$ means the force of body $A$ on body $B$ and similarly $\text{BA}$ means the force of body $B$ on body $A$.

We know that every force in nature has a pair. And this pair includes two forces having the same magnitude but opposite direction. We call that pair action-reaction pair. If you apply a force on a body, you do action on the body and the body applies the same magnitude of force back on you as a reaction.

The statement of Newton's third law above is the modern statement of what Newton originally stated based on action and reaction forces as "for every action there is equal and opposite reaction". Here equal and opposite forces means equal magnitudes of forces interacting in opposite direction. The modern statement includes both action and reaction forces called action-reaction pair or interaction-pair or third-law pair.

Newton's third law always involves two forces and the most important thing to note is that these two forces never act on the same body. Two forces acting on the same body do not form an action-reaction pair. In Figure 2 a book rests on a table. The weight of the book acts on the book vertically downwards and the normal force of table acts on the book vertically upwards. In this case both of these forces act on the same body which is the book, so these two forces do not form an action-reaction pair.

The only action-reaction pair in Figure 2 is the force book exerts on the table due to its weight and the normal force the table exerts on the book. Here the forces are not applied on a single body but applied on the table by the book as an action and on the book by the table as a reaction. These forces act on each other forming an action-reaction pair. You should be careful that forces equal in magnitude and opposite in direction do not always form an action-reaction pair. So, in Figure 2 $m \vec g$ and $\vec n$ are equal in magnitude and opposite in direction but do not form action-reaction pair.

The book exerts force on the table and the table exerts the same magnitude of force on the book in opposite direction. It is not possible to exert force on a body without the body exerting force back in opposite direction.The action and reaction forces are not always contact forces. Newton's third law is also applies for long-range forces such as gravitational force.

Newton's third law involves a pair of forces acting on different bodies but Newton's first and second law involve forces acting on a single body. In Newton's third law, it is very important to determine the correct action-reaction pair. Newton's third law helps us identify the forces acting on a body and Newton's first and second law focus on all forces acting on a body to determine the net force.

Pushing a Car

When you push a stuck car on a way to your adventure, you exert force on the car and the car exerts the same magnitude of force on you in opposite direction. When the car moves, the car still exerts the same magnitude of force back on you. How is this possible? If both forces are equal in magnitude and opposite in direction, how the car can move forward? The answer lies to the correct interaction pair or action-reaction pair.

Remember that these two forces do not act on the same object. One force acts on the car (you exert on car) and the other force acts on you (car exerts on you). If we go to the atomic level, while you exert force on the car, the atoms of your hand and the car interact like very tiny springs and there is always the same magnitude of force at both ends of a compressed or stretched spring.

In Figure 2 the net force on the book is zero but Newton's third law is equally valid even if the net force is not zero. For example, you are pushing to move the car. You exert force on the car and the car exerts force of equal magnitude back on you in opposite direction. Once your force is greater than the force car exerts on you, the car moves. Now your force is greater and car can not provide enough force back on you and to provide that force the car moves. Newton's third law applies whether two bodies are stationary or moving (moving with constant velocity or accelerating).

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