Linear Momentum

What is Linear Momentum?

Linear momentum is defined as the product of mass of the particle and its velocity, it is a vector quantity.

p=mv\overrightarrow{p}=m\overrightarrow{v}

From newton’s second law of motion, for a fixed mass particle

F=ma=mdvdt=ddt(mv)=dpdt\overrightarrow{F}=m\overrightarrow{a}=m\frac{d\overrightarrow{v}}{dt}=\frac{d}{dt}\left( m\overrightarrow{v} \right)=\frac{d\overrightarrow{p}}{dt}

For a system of n particles with masses m1,m2,m3,mn{{m}_{1}},{{m}_{2}},{{m}_{3}},…{{m}_{n}}and velocities v1,v2,v3,vn\overrightarrow{{{v}_{1}}},\overrightarrow{{{v}_{2}}},\overrightarrow{{{v}_{3}}},…\overrightarrow{{{v}_{n}}} respectively, then the net momentum of the system is

pnet=m1v1+m2v2+m3v3++mnvn=p1+p2+p3++pn\overrightarrow{{{p}_{net}}}={{m}_{1}}\overrightarrow{{{v}_{1}}}+{{m}_{2}}\overrightarrow{{{v}_{2}}}+{{m}_{3}}\overrightarrow{{{v}_{3}}}+…+{{m}_{n}}\overrightarrow{{{v}_{n}}}=\overrightarrow{{{p}_{1}}}+\overrightarrow{{{p}_{2}}}+\overrightarrow{{{p}_{3}}}+…+\overrightarrow{{{p}_{n}}} pnet=MVcm\overrightarrow{{{p}_{net}}}=M\overrightarrow{{{V}_{cm}}}

Differentiating the above expression with respect to time

dpnetdt=MdVcmdt\frac{d\overrightarrow{{{p}_{net}}}}{dt}=M\frac{d\overrightarrow{{{V}_{cm}}}}{dt} Fnet=Macm\overrightarrow{{{F}_{net}}}=M\overrightarrow{{{a}_{cm}}}

And also

Fnet=dpnetdt\overrightarrow{{{F}_{net}}}=\frac{d\overrightarrow{{{p}_{net}}}}{dt}

The magnitude of linear momentum may be expressed in terms of kinetic energy as well

p=mvp=mv p2=m2v2=2m(12mv2)=2mK{{p}^{2}}={{m}^{2}}{{v}^{2}}=2m\left( \frac{1}{2}m{{v}^{2}} \right)=2mK

Law of Conservation of Momentum

If the net force acting on a body is equal to zero, then the momentum of the body remains constant. This is known as law of conservation of momentum.

Fnet=0{{F}_{net}}=0 dpnetdt=0\frac{d{{p}_{net}}}{dt}=0

Therefore, pnet = 0 or pnet = constant

If the velocity of centre of mass is equal to zero, (vcm=0)\left( {{v}_{cm}}=0 \right) then from pnet=MVcm\overrightarrow{{{p}_{net}}}=M\overrightarrow{{{V}_{cm}}} we get, pnet=0.\overrightarrow{{{p}_{net}}}=0.

If the velocity of centre of mass is constant (vcm = constant) then we get pnet\overrightarrow{{{p}_{net}}} = constant.

This is known as the law of conservation of linear momentum of the system of particles.

 

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