Newton's Laws of Motion

The dynamics part of Mechanics, that is the study of motion in relationship with force which causes the 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 causes.

Here we take the basic understanding 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?

The basic understanding of force is the push or pull. This definition however is incomplete as a force is the interaction that one body exerts on another.

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).

If you are here 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 an external 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 an external force is applied to it. And the tendency of a body to remain in motion is called inertia of motion. In other words if no force is applied to the body at rest, the body will always be in the state of rest and if no force is applied to the body in motion, it'll always move with constant velocity (never stops until a force is applied to it).

Consider a coin 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.

Consider a bus 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, the bus stops and 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 moved backwards. Initially the passengers were at rest and when the bus accelerates the passengers still want to be in rest due to the inertia of rest and move backwards.

Newton's First Law

The Newton's first law can be stated as a body initially at rest always remains at rest and a body initially in motion always keeps moving with constant velocity provided that no external force is applied on it. It means if no external force is applied 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.

The first law is quiet straightforward which we learnt in the property of inertia above. Newton's first is the equilibrium condition where the force applied to the body is zero that is acceleration $a$ is always zero and $F = 0$.

Newton's Second Law

Newton's second law reveals the relationship between the force and acceleration. It can be stated as if an external net force is applied on a body, the body accelerates in the direction of force.

The alternative statement can be, if there is a net force acting on a body, the body should accelerate. Greater the magnitude of force, greater the magnitude of acceleration. In this case the net force applied to a body is not zero and the body accelerates under the action of net force. The magnitude of net force applied is the product of mass and acceleration of the body that is, $F = ma$.

Newton's Third Law

Newton's third law is based on the interaction of forces. If you apply a force on a body, the body applies the force with the same magnitude on you. Newton's third law can be stated as every action has equal and opposite reaction.

It means if a body ${{B}_{1}}$ exerts a force on another body ${{B}_{2}}$, the body ${{B}_{2}}$ exerts the force of the same magnitude on body ${{B}_{1}}$. If the force is greater and the body $B_2$ is accelerated, the body $B_2$ still exerts the same magnitude of force on body $B_1$ in opposite direction.

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

If you apply a force ${{\vec{F}}_{\text{you on wall}}}$ on a wall as shown in Figure 1, the wall also exerts the same magnitude of force on you. Therefore,

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

Here ${{\vec{F}}_{\text{you on wall}}}$ is the force you applied on the wall, the wall now in return exerts the force of the same magnitude but in opposite direction.

A book of mass $m$ is on a table as shown in Figure 2. The force on the book due to gravitational force is $w=mg$ which is vertically downwards also called weight.

According to Newton's third law the table also exerts equal amount of force on the book in opposite direction (vertically upwards). The force a surface exerts perpendicular to the surface is normal force (discussed later) $\vec{n}$. In Figure 2 the weight is $\vec{w}=m\vec{g}$. So $\vec{w}=m\vec{g}=-\vec{n}$.

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