Principle of Conservation of Linear Momentum and its Applications Physics Notes
Principle of Conservation of Linear Momentum and its Applications:
If the net external force acting on a system of bodies is zero, then the momentum of the system remains constant. This is the basic principle of conservation of linear momentum.
According to Newton’s second law
Considering external force on the particle (or a body)= zero
we have \(\vec{F}=\frac{d \vec{p}}{d t}\) = 0
⇒ \(\vec{p}\) = constant …(1)
If net force (or the vector sum of all forces) on system of particle is equal to zero, the vector sum of linear momentum of all particles remains conserved.
Consider a system of two bodies on which no external force acts. The bodies can mutually interact with each other. Due to the mutual interaction of the bodies, the momentum of the individual bodies may increase or decrease according to the situation, but the momentum of the system will always be conserved, as long as there is no external net force acting on it.
Thus, if \(\vec{p}_{1}\) and \(\vec{p}_{2}\) are momentum of the two bodies at any instant, then in absence of external force
\(\vec{p}_{1}+\vec{p_{2}}\) = constant …….(3)
If due to mutual interaction, the momentum of two bodies becomes p{ and p2 respectively, then according to the principle of conservation of momentum
Where \(\vec{u}_{1}\) and \(\vec{u}_{1}\) are initial velocities of the two bodies of masses m1 and m2 and v1 and v2 are their final velocities.
Therefore, the principle of conservation of linear momentum may also be stated as follows:
For an isolated system (a system on which no external force acts), the initial momentum of the syster A is equal to the final momentum of the system.
Practical Applications of the Principle of Conservation of Momentum:
1. Recoiling a Gun: Let’s consider the gun and bullet in its barrel as an isolated system. In the beginning when the bullet is not fired both the gun and the bullet are at rest. So the momentum before firing is zero
or \(\overrightarrow{p_{c}}\) = 0
Now when the bullet is fired, it moves, in the forward direction and gun recoils back in the opposite direction.
Let mb be the mass and vb be the velocity of the bullet and mg and vg be the mass and velocity of the gun after firing.
Total momentum of the system after the firing would be
\(\overrightarrow{p_{f}}=m_{b} \overrightarrow{v_{b}}+m_{g} \overrightarrow{v_{g}}\)
Since, no external forces are acting on the system, we can apply the law of conservation of linear momentum therefore,
Total momentum of gun and bullet before firing = Total momentum of gun and bullet after firing
0 = mb \(\overrightarrow{v_{b}}\) + mg \(\overrightarrow{v_{g}}\)
or
\(\overrightarrow{v_{g}}=-\frac{m_{b} \overrightarrow{v_{b}}}{m_{g}}\)
The negative sign shows that vg and vb are in opposite directions i.e., as the bullet moves forward, then the gun will move in the backward direction. The backward motion of the gun is called the recoil of the gun.
2. While firing a bullet, the gun must be held tight to the shoulder: This would save hurting the shoulder of the man who fires the gun as the recoil velocity of the gun. If the gun is held tight to the shoulder, then the gun and the body of the man recoil as one system. As the total mass is quite large, the recoil velocity will be very small and the shoulder of the man will not get hurt.
3. Rockets work is on the principle of conservation of momentum: The rocket’s fuel burns and pushes the exhaust gases downwards, due to this the rocket gets pushed upwards. Motorboats also work on the same principle, it pushes the water backwards and gets pushed forward in reaction to conserve momentum.