A collection of forces is said to be concurrent, if the lines of forces act at a common point. Concurrent forces need not be in the same plane. If they are in the same plane, they are concurrent as well as coplanar forces.

A body, under the action of concurrent forces, is said to be in equilibrium, when there is no change in the state of rest or of uniform motion along a straight line.
The necessary condition for the equilibrium of a body under the action of concurrent forces is that the vector sum of all the forces acting on the body must be zero.
Lami's Theorem: For concurrent forces \(\frac { { F_{ 1 } } }{ sin\ \alpha } =\frac { { F_{ 2 } } }{ sin \ \beta } =\frac { { F_{ 3 } } }{ sin\ \gamma } \). If a system of three concurrent and coplanar forces is in equilibrium, then Lami's theorem states that the magnitude of each force of the system is proportional to sine of the angle between the other two forces. The constant of proportionality is same for all three forces.

Inherent property of all the bodies by virtue of which they cannot change their state of rest or uniform motion along a straight line by their own is called inertia.
Inertia of rest: It is the inability of a body to change by itself, its state of rest. This means a body at rest remains at rest and cannot start moving by its own.
(i) When a horse starts suddenly, the rider tends to fall backward on account of inertia of rest of upper part of the body as explained above.
(ii) Passengers experience a backward push due to inertia of rest.
Inertia of motion: It is the inability of a body to change itself its state of uniform motion i.e., a body in uniform motion can neither accelerate nor retard by its own.
(i) Passengers experience a forward push due to inertia of motion.
(ii) A person jumping out of a moving train may fall forward.
Inertia of direction: It is the inability of a body to change by itself direction of motion.
(i) A stone moves tangential to circle due to inertia of direction.
(ii) When a car goes round a curve suddenly, the person sitting inside is thrown outwards.

No, a single isolated force cannot exist in nature because it violates the Newton's third law.

(i) Due to excessive polishing, more molecules of the two surfaces come closer. Thus they form more bonds and offer great resistances.
(ii) Because friction arises due to molecular adhesion i.e. due to electrostatic forces.

(i) According to Newton's third law when one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.
(ii) When you push a car from inside, the reaction force of your pushing is balanced out by your body moving backward and eventually the seat behind you pushes against to bring things to static equilibrium.

One Newton is that force which produces an acceleration of 1 m/s^-2 in a body of mass 1 kilogram.
∴ 1 Newton = 1 kg m/s²

The rate of change of linear momentum of a body is directly proportional to the external force applied on the body and this change takes place always in the direction of the applied force.
\(F =\frac{\vec{d} p}{d t}=\frac{d(m \vec{v})}{d t}=m \frac{d \vec{v}}{d t}=m \vec{a}\left(\text { as } a=\frac{d \vec{v}}{d t}\right) \)
\(\vec{F} =m \vec{a} \Rightarrow \text { Force }=\text { mass } \times \text { acceleration }\)

Conservation of linear momentum is equivalent to Newton's third law of motion. If no external force acts on a system (called isolated) of constant mass, the total momentum of the system remains constant with time.
According to this law for a system of particles \(\vec{F}=\frac{d \vec{p}}{d t}\)
In the absence of external force F = 0 and \(\vec{p}\) = constant
i.e.,  \(p_{1}+\vec{p}_{2}+\vec{p}_{3}+\dots=\) constant
or \(m_{1} \vec{v}_{1}+m_{2} \vec{v}_{2}+m_{3} \vec{v}_{3}+\dots=\) constant
This equation shows that in absence of external force for a closed system the linear momentum of individual particles may change but their sum remains unchanged with time. For a system of two particles in absence of external force by law of conservation of linear momentum.
\(\vec{p}_{1}+\vec{p}_{2} =\text { constant }  \)
\(m_{1} \vec{v}_{1}+m_{2} \vec{v}_{2} =\text { constant }  \)
\(m_{1} \frac{d \vec{v}_{1}}{d t}+m_{2} \frac{d \vec{v}_{2}}{d t} =0 \Rightarrow m_{1} \vec{a}_{1}+m_{2} \vec{a}_{2}=0 \Rightarrow \vec{F}_{1}+\vec{F}_{2}=0  \)
\(\therefore \vec{F}_{2} =-\vec{F}_{1}\)
For every action there is equal and opposite reaction which is Newton's third law of motion.
Recoiling of a gun: For bullet and gun system, the force exerted by trigger will be internal so the momentum of the system remains unaffected.
Let m_G= mass of gun, m_B= mass of bullet, v_G velocity of gun, v_B= velocity of bullet Initial momentum of system =0
Final momentum of system \(=m_{G} \vec{v}_{G}+m_{B} \vec{v}_{B}\)
By the law of conservation linear momentum \(m_{G} \vec{v}_{G}+m_{B} \vec{v}_{B}=0\)
So recoil velocity \(v_{G}=-\frac{m_{B}}{m_{G}} \vec{v}_{B}\)
Here negative sign indicates that the velocity of recoil \(\vec{v}_{G}\) is opposite to the velocity of the bullet.
\(v_{G} \propto \frac{1}{m_{G}}\) i.e. higher the mass of gun, lesser the velocity of recoil of gun.
While firing the gun must be held tightly to the shoulder, this would save hurting the shoulder because in this condition the body of the shooter and the gun behave as one body. Total mass become large and recoil velocity becomes too small.
\(\vec{v}_{G} \propto \frac{1}{m_{G}+m_{\operatorname{man}}}\)