Linear Momentum

The linear momentum for a system of particles can be derived from Newton’s Second Law. For a system of particles, this law states that:

Newtons second law for system of particles for derivation of linear momentum

Where:

ΣFext is the vector sum of the external forces acting on the system of particles. This sum is sometimes called the net external force

m is the total mass of the system of particles

aG is the acceleration of the center of mass G of the system of particles, with respect to an inertial reference frame (ground). This acceleration is a vector in the direction of the net external force


Schematic showing system of particles for derivation of linear momentum


Note that there is no restriction in the way the particles are connected. As a result, the above equation will also hold true for a rigid body, a deforming body, a liquid, or a gas system.

Now, we can rewrite the above equation as

Newtons second law for system of particles for derivation of linear momentum 2

where vG is the velocity of the center of mass G of the system of particles, with respect to an inertial reference frame (ground).

The above equation can be expressed as

Newtons second law for system of particles for derivation of linear momentum 3

Integrating both sides from time ti to time tf we have

Newtons second law for system of particles for derivation of linear momentum 4

which becomes

Newtons second law for system of particles for derivation of linear momentum 5

The term

Impulse term for system of particles for derivation of linear momentum

is defined as the external linear impulse acting on the system of particles (between ti and tf), due to the sum of the external forces acting on the particles.





We define the linear momentum for the system of particles as

Definition of linear momentum for system of particles

Therefore, equation (1) can be written as

Newtons second law for system of particles for derivation of linear momentum 6

Note that there is no restriction in the way the system of particles are connected. As a result, the above equation will also hold true for a rigid body, a deforming body, a liquid, or a gas system. To see an example problem involving impulse and linear momentum see Rocket Physics.


If no external forces act on the system of particles, then

If the external forces are zero the linear momentum is conserved

Linear momentum is therefore conserved for the system of particles (between ti and tf).



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