Linear momentum is a concept, that helps us understand and predict the motion of objects. For a system of particles, linear momentum takes on an even more important role, Which provides the collective behaviour of the system.
What is Linear Momentum?
Linear momentum (p) is defined as the product of the mass (m) and velocity (v) of an object: p = mv
It is a vector quantity, it has both magnitude and direction. The SI unit of linear momentum is kg \ s.
A single particle, the linear momentum describes the motion of that particle. However, when we deal with a system of particles, we must consider the momentum of all particles in the system.
System of Particles
A system of particles is a collection of particles that interact with each other and may also interact with external forces. Examples are a group of moving cars, molecules in a gas, or planets in a solar system.
In such systems, analysing the motion of each individual particle can be complex. Instead, physicists use the concept of the center of mass and the total linear momentum of the system to simplify the analysis.
Linear Momentum of a System of Particles
The total linear momentum of a system of particles is the vector sum of the linear momenta of all the individual particles in the system. If there are N particles, each with mass mi and velocity vi, then: P = I = 1 N pi = I = 1 N i = I mi vi
Here, P represents the total linear momentum of the system.
Center of Mass and Linear Momentum
The motion of a system of particles can often be understood by focusing on its center of mass (COM). The position of the center of mass (R) is given by: R = I = 1N mI ri I = 1Nmi
where ri is the position vector of the ith particle.
The velocity of the center of mass (VCOM) is derived as:
VCOM = I = 1 N mi vi / N I = 1 mi
The total linear momentum of the system can also be expressed in terms of the center of mass:
P = MVCOM Where M = I = 1 N mi is the total mass of the system.
This equation shows that the total linear momentum of the system is equivalent to the linear momentum of a single particle of mass M, moving with the velocity of the center of mass.
Conservation of Linear Momentum
One of the most powerful principles is the law of conservation of linear momentum, which states:
The total linear momentum of an isolated system (a system with no external forces) remains constant.
Mathematically: Pinitial = Pfinal
This principle applies to collisions, explosions, and other interactions within a system. For example:
In a collision between two billiard balls, the total momentum before and after the collision is the same, assuming no external forces act on them.
In a rocket propulsion system, the momentum of the rocket and exhaust gases remains conserved.
Importance of Linear Momentum in a System of Particles
1.Simplifies Analysis: Instead of analysing each particle, we focus on the center of mass and total momentum.
2. Collision Studies: It helps to predict outcomes in elastic and inelastic collisions.
3. Practical Applications: Used in astrophysics, fluid dynamics, engineering, and many other fields.
Relation to Force
Newton’s second law connects force and momentum:
Fnet = dP / dt
For a system of particles, the rate of change of total momentum equals the net external force acting on the system. If the net external force is zero (Fnet = 0), the momentum of the system remains constant.
Key Points :-
Linear momentum is a vector quantity defined as the product of mass and velocity.
The total linear momentum of a system is the sum of the momenta of its individual particles.
The motion of a system can be understood in terms of its center of mass
The rate of change of momentum relates to the net external force on a system.
If the total external force acting on the system is zero, the centre of mass moves with a constant velocity. For example :
(i) A heavy nucleus (Ra) splits into a lighter nucleus (Rn) and an alpha particle (He). The CM of the system is in uniform motion.
(ii) The same splitting of the heavy nucleus (Rn) with the centre of mass at test. The two product particles fly back to back.
Linear momentum (p) is the product of an object’s mass (m) and its velocity (v). It is a vector quantity, it has both magnitude and direction. Mathematically, p = mv. Its SI unit is kg \ s.
The total linear momentum of a system of particles is the sum of the linear momenta of all individual particle in the system. If there are N particles, each with mass mi and velocity vi, the total momentum (P) is: P = I = 1 N mi vi
The total linear momentum of a system can be expressed using the center of mass (COM). If M is the total mass of the system and VCOM is the velocity of the center of mass:
P = M VCOM
The system’s total momentum is the same as the momentum of a single particle with mass M, moving at the velocity of the center of mass.
The law of conservation of linear momentum states that the total momentum of an isolated system (a system with no external forces) remains constant over time. Mathematically: Pinitial = Pfinal
This principle applies to situations like collisions, explosions, and other interactions within the system.
Linear momentum is vital because:
It simplifies the analysis of motion for a system of particles.
It helps to predict outcomes in collisions and explosions.
It is conserved in isolated systems, making it a powerful tool in solving problems.
It connects directly to force through Newton’s second law.
Newton’s second law states that the net external force (Fnet) acting on a system is equal to the rate of change of its linear momentum (P): Fnet = dP / dt
If no external force acts on the system (Fnet = 0), the system’s momentum remains constant.
In a collision, the total linear momentum of the system is conserved if no external forces act on it.
In elastic collisions, both momentum and kinetic energy are conserved.
In inelastic collisions, momentum is conserved, but kinetic energy is not.