# 14.5 Conservation of linear momentum

 Page 1 / 3
The linear momentum can not change in a closed system of particles.

We have briefly defined linear momentum, while describing Newton's second law of motion. The law defines force as the time rate of linear momentum of a particle. It directly provides a measurable basis for the measurement of force in terms of mass and acceleration of a single particle. As such, the concept of linear momentum is not elaborated or emphasized for a single particle. However, we shall see in this module that linear momentum becomes a convenient tool to analyze motion of a system of particles - particularly with reference to internal forces acting inside the system.

It will soon emerge that Newton's second law of motion is more suited for the analysis of the motion of a particle like objects, whereas concept of linear momentum is more suited when we deal with the dynamics of a system of particles. Nevertheless, we must understand that these two approaches are interlinked and equivalent. Preference to a particular approach is basically a question of suitability to analysis situation.

Let us now recapitulate main points about linear momentum as described earlier :

(i) It is defined for a particle as a vector in terms of the product of mass and velocity.

$\begin{array}{l}\mathbf{p}=m\mathbf{v}\end{array}$

The small " p " is used to denote linear momentum of a particle and capital " P " is used for linear momentum of the system of particles. Further, these symbols distinguish linear momentum from angular momentum ( L ) as applicable in the case of rotational motion. By convention, a simple reference to "momentum" means "linear momentum".

(ii) Since mass is a positive scalar quantity, the directions of linear momentum and velocity are same.

(iii) In physical sense, linear momentum is said to signify the "quantum or quantity of motion". It is so because a particle with higher momentum generates greater impact, when stopped.

(iv) The first differentiation of linear momentum with respect to time is equal to external force on the single particle.

$\begin{array}{l}{\mathbf{F}}_{\mathrm{Ext.}}=\frac{d\mathbf{p}}{dt}=m\mathbf{a}\end{array}$

## Momentum of a system of particles

The concept of linear momentum for a particle is extended to a system of particles by summing the momentum of individual particles. However, this sum is a vector sum of momentums. We need to either employ vector addition or equivalent component summation with appropriate sign convention as discussed earlier. Linear momentum of a system of particles is, thus, defined as :

Momentum of a system of particles
The linear momentum of a system of particles is the vector sum of linear momentums of individual particles.

$\begin{array}{l}\mathbf{p}={m}_{1}{\mathbf{v}}_{1}+{m}_{2}{\mathbf{v}}_{2}+................+{m}_{n}{\mathbf{v}}_{n}\end{array}$

$\begin{array}{l}⇒\mathbf{p}=\sum {m}_{i}{\mathbf{v}}_{i}\end{array}$

From the concept of "center of mass", we know that :

$\begin{array}{l}M{\mathbf{v}}_{\mathrm{COM}}={m}_{1}{\mathbf{v}}_{1}+{m}_{2}{\mathbf{v}}_{2}+................+{m}_{n}{\mathbf{v}}_{n}\end{array}$

Comparing two equations,

$\begin{array}{l}\mathbf{P}=M{\mathbf{v}}_{\mathrm{COM}}\end{array}$

The linear momentum of a system of momentum is, therefore, equal to the product of total mass and the velocity of the COM of the system of particles.

## External force in terms of momentum of the system

Just like the case for a single particle, the first differentiation of the total linear momentum gives the external force on the system of particles :

what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
can anyone tell who founded equations of motion !?
n=a+b/T² find the linear express
Quiklyyy
Moment of inertia of a bar in terms of perpendicular axis theorem
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
A balloon is released from the ground which rises vertically up with acceleration 1.4m/sec^2.a ball is released from the balloon 20 second after the balloon has left the ground. The maximum height reached by the ball from the ground is
work done by frictional force formula
Torque
Why are we takingspherical surface area in case of solid sphere
In all situatuons, what can I generalize?
the body travels the distance of d=( 14+- 0.2)m in t=( 4.0 +- 0.3) s calculate it's velocity with error limit find Percentage error
Explain it ?Fy=?sN?mg=0?N=mg?s