# Force

A net force on a body causes a body to accelerate. The amount of that acceleration depends on the body's inertia (or its tendency to resist changes in motion), which is measured as its mass. When Isaac Newton formulated Newtonian mechanics, he discovered three fundamental laws of motion.

Later, Albert Einstein proved that these laws are just a convenient approximation. These laws, however, greatly simplify calculations and are used when studying objects at velocities that are small compared with the speed of light.

## Friction

It is the force that opposes relative motion or tendency of relative motion between two surfaces in contact represented by f. When two surfaces move relative to each other or they have a tendency to move relative to each other, at the point (or surface) of contact, there appears a force which opposes this relative motion or tendency of relative motion between two surfaces in contact. It acts on both the surfaces in contact with equal magnitude and opposite directions (Newton's 3rd law). Friction force tries to stop relative motion between two surfaces in contact, if it is there, and when two surfaces in contact are at rest relative to each other, the friction force tries to maintain this relative rest. Friction force can assume the magnitude (below a certain maximum magnitude called limiting static friction) required to maintain relative rest between two surfaces in contact. Because of this friction force is called a self adjusting force.

Earlier, it was believed that friction was caused due to the roughness of the two surfaces in contact with each other. However, modern theory stipulates that the cause of friction is the Coulombic force between the atoms present in the surface of the regions in contact with each other.

Formula: Limiting Friction = (Friction Coefficient)(Normal reaction)

Static Friction = the friction force that keeps an object at relative rest.

Kinetic Friction = sliding friction

## Newton's First Law of Motion

(The Law of Inertia)

A static object with no net force acting on it remains at rest or if in movement it will maintain a constant velocity

This means, essentially, that acceleration does not occur without the presence of a force. The object tends to maintain its state of motion. If it is at rest, it remains at rest and if it is moving with a velocity then it keeps moving with the same velocity. This tendency of the object to maintain its state of motion is greater for larger mass. The "mass" is, therefore, a measure of the inertia of the object.

In a state of equilibrium, where the object is at rest or proceeding at a constant velocity, the net force in every direction must be equal to 0.

At a constant velocity (including zero velocity), the sum of forces is 0. If the sum of forces does not equal zero, the object will accelerate (change velocity over time).

It is important to note, that this law is applicable only in non-accelerated coordinate systems. It is so, because the perception of force in accelerated systems are different. A body under balanced force system in one frame of reference, for example a person standing in an accelerating lift, is acted upon by a net force in the earth's frame of reference.

Inertia is the tendency of an object to maintain its velocity i.e. to resist acceleration.

• Inertia is not a force.
• Inertia varies directly with mass.

## Newton's Second Law of Motion

• The time rate of change in momentum is proportional to the applied force and takes place in the direction of the force.
• 'The acceleration of an object is proportional to the force acting upon it.

These two statements mean the same thing, and is represented in the following basic form (the system of measurement is chosen such that constant of proportionality is 1) :

 ${\displaystyle {\vec {F}}={\frac {d}{dt}}(m{\vec {v}})}$

The product of mass and velocity i.e. mv is called the momentum. The net force on a particle is thus equal to rate change of momentum of the particle with time. Generally mass of the object under consideration is constant and thus can be taken out of the derivative.

 ${\displaystyle {\vec {F}}=m{\frac {d{\vec {v}}}{dt}}=m{\vec {a}}}$

Force is equal to mass times acceleration. This version of Newton's Second Law of Motion assumes that the mass of the body does not change with time, and as such, does not represent a general mathematical form of the Law. Consequently, this equation cannot, for example, be applied to the motion of a rocket, which loses its mass (the lost mass is ejected at the rear of the rocket) with the passage of time.

An example: If we want to find out the downward force of gravity on an object on Earth, we can use the following formula:

 ${\displaystyle \|{\vec {F}}\|=m\|{\vec {g}}\|}$

Hence, if we replace m with whatever mass is appropriate, and multiply it by 9.806 65 m/s2, it will give the force in newtons that the earth's gravity has on the object in question (in other words, the body's weight).

## Newton's Third Law of Motion

Forces occur in pairs equal in magnitude and opposite in direction

This means that for every force applied on a body A by a body B, body B receives an equal force in the exact opposite direction. This is because forces can only be applied by a body on another body. It is important to note here that the pair of forces act on two different bodies, affecting their state of motion. This is to emphasize that pair of equal forces do not cancel out.

There are no spontaneous forces.

It is very important to note that the forces in a "Newton 3 pair", described above, can never act on the same body. One acts on A, the other on B. A common error is to imagine that the force of gravity on a stationary object and the "contact force" upwards of the table supporting the object are equal by Newton's third law. This is not true. They may be equal - but because of the second law (their sum must be zero because the object is not accelerating), not because of the third.

The "Newton 3 pair" of the force of gravity (= earth's pull) on the object is the force of the object attracting the earth, pulling it upwards. The "Newton 3 pair" of the table pushing it up is that it, in its turn, pushes the table down.

## Equations

To find Displacement

 ${\displaystyle \Delta x=t{\vec {v}}_{i}+{\frac {t^{2}{\vec {a}}}{2}}}$

To find Final Velocity

 ${\displaystyle {\vec {v}}_{f}={\vec {v}}_{i}+t{\vec {a}}}$

To find Final Velocity

 ${\displaystyle (v_{f})^{2}=(v_{i})^{2}+2a\Delta x}$

To find Force when mass is changing

 ${\displaystyle {\vec {F}}={\frac {d}{dt}}(m{\vec {v}})}$

To find Force when mass is a constant

 ${\displaystyle {\vec {F}}=m{\frac {d{\vec {v}}}{dt}}=m{\vec {a}}}$
Variables
${\displaystyle {\vec {F}}}$ Force (N)
${\displaystyle m\ }$ Mass (kg)
${\displaystyle {\vec {a}}}$ Acceleration (m/s2)
${\displaystyle {\vec {p}}}$ Momentum (kg m/s)
${\displaystyle t\ }$ time (s)
${\displaystyle {\vec {T}}}$ Tension (N)
${\displaystyle {\vec {g}}}$ Acceleration due to gravity near the earth's surface (${\displaystyle \|{\vec {g}}\|=9.806\ 65\mathrm {m/s^{2}} }$ see Physics Constants)
Definitions
Force (F): Force is equal to rate change of momentum with time. (Newton’s second law). A vector. Units: newtons (N)
The newton (N): defined as the force it takes to accelerate one kilogram one metre per second squared (U.S. meter per second squared), that is, the push it takes to speed up one kilogram from rest to a velocity of 1 m/s in 1 second ${\displaystyle 1\mathrm {N} =1\mathrm {kg} \cdot \mathrm {m} /\mathrm {s} ^{2}}$
Mass (m) : Also called inertia. The tendency of an object to resist change in its motion, and its response to a gravitational field. A scalar. Units: kilograms (kg)
Acceleration (a): Change in velocity (Δv) divided by time (t). A vector. Units: meters per second squared (U.S. meters per seconds squared) (m/s2)
Momentum (p): Mass times velocity. Expresses the motion of a body and its resistance to changing that motion. A vector. Units: kg m/s