Force and Laws of Motion

Force can be defined intuitively as push or pull. In physics the force on a body means the product of its mass $m$ and acceleration $\vec a$.

\[\vec{F}=m\vec{a}\]

A body accelerates in the direction of force. The unit of force is $kg\cdot m/{{s}^{2}}$. The unit of force is also expressed as a single letter $N$ (Newton) that is, $1N=1kg\cdot m/{{s}^{2}}$.

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 Laws of Motion

There are three well known Newton's laws of motion and they are the most important laws in Physics. And I talk about these laws of motion here.

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 $\vec{w}=m\vec{g}=-\vec{n}$.

Tension, Normal and Friction

All forces have the same general nature which is push or pull. In Figure 3 a box is being pulled by a rope. The force exerted by a rope on a body is called tension $T$. And the force a surface exerts perpendicular to the surface is normal force. A book of weight $\vec{w}$ is on a surface of a table as shown in Figure 2. The book exerts a force equal to its weight on the surface of the table and in return the surface also applies the same magnitude of force on the book. Here the force exerted by the book is its weight but the force exerted by the surface is normal force $\vec{n}$.

Figure 3 A loader pulls a box with a rope.
Figure 4 A box on an inclined plane. The net force on the box is zero, i.e. the box is in equilibrium condition.

In Figure 4 a box is on an inclined plane. The weight of the box $\vec{w}=m\vec{g}$ is vertically downwards and the normal force $\vec{n}$ the surface exerts on the box is perpendicular to the inclined plane. In this situation the weight has two components; one is perpendicular to the inclined plane and another is parallel to the inclined plane. The perpendicular component ${{w}_{\bot }}=w\cos \phi $ is balanced by the normal force $\vec{n}$ and the parallel component ${{w}_{\parallel }}=w\sin \phi $ is balanced by the friction between the surfaces. Thus, the total net force acting on the box is zero and the box is in equilibrium condition.

You may be quiet familiar with friction in everyday life. It has both advantages and disadvantages. For example you can walk on the ground due to the friction between the surfaces of your feet and floor. But different parts of machines may not work properly due to friction which you can fix by lucubrating oils such as grease or something else. But still you can not imagine the world without friction. Here we discuss about static and kinetic friction forces.

In Figure 3 a loader pulls a box with a rope. The box does not move initially which is due to the friction between the surfaces of the box and the floor. And as he goes on increasing the force on the rope or the tension $T$, the box just starts to move at a particular point called critical point. The friction before the box in motion is called static friction. The maximum value of static friction is the maximum force at which the box just starts to move.

The total force applied by the loader until the box just starts to move is balanced by the corresponding static friction (maximum static friction) between the surfaces of box and the floor. Experiments have shown that the static friction is directly proportional to the normal force on the box, that is ${{f}_{s}}\propto n$ where ${{f}_{s}}$ is the static friction and $n$ is the normal force. The maximum value of the static friction is

\[{{f}_{s-\max }}={{\mu}_{s}}n \tag{1} \label{1}\]

In Eq. \eqref{1} ${{\mu}_{s}}$ is a constant called coefficient of static friction. The value of static friction can vary from zero to a maximum value and ${{f}_{s}}$ represents any value from zero to maximum. But \eqref{1} is valid only for maximum value of static friction otherwise the coefficient of static friction will be no longer be constant as indicated by ${{\mu }_{s}}=\frac{{{f}_{s}}}{n}$.

After the maximum static friction is overcome by the applied force, the box starts to move. Note that it's easier to pull the box when it is in motion. So the friction decreases when the box is in motion and the friction when the box is in motion is called kinetic friction denoted by ${{f}_{k}}$. The kinetic friction is lesser than the static friction. Again the experiments show that kinetic friction is directly proportional to the normal force on the box, that is, ${{f}_{k}}\propto n$ and,

\[{{f}_{k}}={{\mu}_{k}}n \tag{2} \label{2}\]

In Eq. \eqref{2}, ${\mu}_{k}$ is a constant called the coefficient of kinetic friction. Here ${f}_{k}$ is approximately constant throughout the relative motion between the surfaces.

When two surfaces come in contact, the molecules on the surfaces form bonds with one another. Smother the surfaces, greater the number of molecules come in contact and more bonds are formed which increases the friction. The kinetic friction is approximately constant (not exactly constant) because the bonds between the surfaces form and break during the motion.

Mechanics

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