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Law of Conservation of Linear Momentum

Law of Conservation of Linear Momentum :- According to the law of conservation of linear momentum, ‘if the total external force acting on a system is zero, then the total linear momentum of the system remains constant’.

According to Newton’s second law, the rate of change of linear momentum of a body is equal to the total external force acting on it, i.e.

$\displaystyle \overrightarrow{F}=\frac{d\overrightarrow{p}}{dt}$

If the total external force $\displaystyle \overrightarrow{F}$ is absent, then $\displaystyle \overrightarrow{F}=0$

$\displaystyle \Rightarrow \frac{d\overrightarrow{p}}{dt}=0$

$\displaystyle \Rightarrow \overrightarrow{p}$ = constant

This is the law of conservation of linear momentum.

Law of Conservation of Linear Momentum for two particles system

Consider a system of two particles A and B :

Let

m_A = mass of particle A, m_B = mass of particle B

$\displaystyle {{\overrightarrow{F}}_{ext1}}$ = External force on particle A, $\displaystyle {{\overrightarrow{F}}_{ext2}}$ = External force on particle B

$\displaystyle {{\overrightarrow{F}}_{AB}}$ = Force on A due to B, $\displaystyle {{\overrightarrow{F}}_{BA}}$ = Force on B due to A

Net force on particle A = $\displaystyle {{\overrightarrow{F}}_{ext1}}+{{\overrightarrow{F}}_{AB}}$ , Net force on particle A = $\displaystyle {{\overrightarrow{F}}_{ext2}}+{{\overrightarrow{F}}_{BA}}$

Applying Newton’s Second Law for particle A :

$\displaystyle {{\overrightarrow{F}}_{ext1}}+{{\overrightarrow{F}}_{AB}}=\frac{d{{\overrightarrow{p}}_{1}}}{dt}$ …..(1)

Similarly for particle B :

$\displaystyle {{\overrightarrow{F}}_{ext2}}+{{\overrightarrow{F}}_{BA}}=\frac{d{{\overrightarrow{p}}_{2}}}{dt}$ …..(2)

Net force acting on the system of particles A and B can be obtained by adding equations (1) and (2)

$\displaystyle {{\overrightarrow{F}}_{ext1}}+{{\overrightarrow{F}}_{AB}}+{{\overrightarrow{F}}_{ext2}}+{{\overrightarrow{F}}_{BA}}=\frac{d{{\overrightarrow{p}}_{1}}}{dt}+\frac{d{{\overrightarrow{p}}_{2}}}{dt}$ …..(3)

But $\displaystyle {{\overrightarrow{F}}_{AB}}$ and $\displaystyle {{\overrightarrow{F}}_{BA}}$ are action reaction forces, so according to Newton’s third law

$\displaystyle {{\overrightarrow{F}}_{AB}}=-{{\overrightarrow{F}}_{BA}}$

$\displaystyle \Rightarrow {{\overrightarrow{F}}_{AB}}+{{\overrightarrow{F}}_{BA}}=0$

Hence from equation (3)

$\displaystyle {{\overrightarrow{F}}_{ext1}}+{{\overrightarrow{F}}_{ext2}}=\frac{d{{\overrightarrow{p}}_{1}}}{dt}+\frac{d{{\overrightarrow{p}}_{2}}}{dt}$ …..(4)

Now if there is no external force acting on the system, then

$\displaystyle {{\overrightarrow{F}}_{ext1}}+{{\overrightarrow{F}}_{ext2}}=0$

In that case equation (4) gives us :

$\displaystyle \frac{d{{\overrightarrow{p}}_{1}}}{dt}+\frac{d{{\overrightarrow{p}}_{2}}}{dt}=0$

$\displaystyle \frac{d}{dt}({{\overrightarrow{p}}_{1}}+{{\overrightarrow{p}}_{2}})=0$

$\displaystyle \Rightarrow {{\overrightarrow{p}}_{1}}+{{\overrightarrow{p}}_{2}}=\text{constant}$

Therefore if the total external force acting on a system is zero then the resultant momentum of that system remains constant.

⇒ Law of Conservation of Linear Momentum is not affected by the presence of the internal forces.

Law of Conservation of Linear Momentum

Law of Conservation of Linear Momentum

Law of Conservation of Linear Momentum for two particles system

Examples of the Law of Conservation of Linear Momentum

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