9.5

Conservation of Momentum: Introduction

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In a system, any interaction of particles constituting the system tends to change the particles' momentum. Still, the total momentum of the system remains constant in the absence of any external forces.

The requirements for momentum conservation are – the system's total mass remains unchanged during the interaction, and the net external force acting on the system is zero. Such a system is called a closed system.

The law of conservation of momentum states that the change in momentum due to particles' interaction in a closed system is zero.

The momentum before interaction is equal to the momentum after interaction between the particles in a closed system.

In the case of the rolling ball hitting the bowling pins, the initial momentum is only due to the ball as pins are at rest. When the ball hits the pins, they start moving, thus gaining momentum. However, the total momentum of the entire system is conserved as the ball slows down after hitting the pins.

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$$\rightleftharpoonup{xx}$$ $$\longleftharp{xx}$$, $$\longrightharp{xx}$$,

The total momentum of a system consisting of N interacting objects is constant in time or is conserved. A system must meet two requirements for its momentum to be conserved:

The mass of the system must remain constant during the interaction. As the objects interact (apply forces on each other), they may transfer mass from one to another; but any mass one object gains is balanced by the loss of that mass from another. The total mass of the system of objects, therefore, must remain unchanged as time passes.
The net external force on the system must be zero. As the objects collide, or explode, and move around, they exert forces on each other. However, all of these forces are internal to the system, and thus each of these internal forces is balanced by another internal force that is equal in magnitude and opposite in sign. As a result, the change in momentum caused by each internal force is canceled out by another momentum change that is equal in magnitude and opposite in direction. Therefore, internal forces cannot change the total momentum of a system because the changes sum to zero. However, if there is some external force that acts on all of the objects (for example, gravity or friction), then this force changes the momentum of the system as a whole. That is to say, the momentum of the system is changed by the external force. Thus, for the momentum of the system to be conserved, the net external force must be zero.

A system of objects that meets these two requirements is said to be a closed system (or an isolated system), and the total momentum of a closed system is conserved. All experimental evidence supports this statement: from the motions of galactic clusters to the quarks that make up the proton and the neutron, and at every scale in between. Note that there can be external forces acting on the system, but for the system's momentum to remain constant, these external forces must cancel so that the net external force is zero. For instance, billiard balls on a table all have a weight force acting on them, but the weights are balanced (canceled) by the normal forces, so there is no net force.

This text is adapted from Openstax, University Physics Volume 1, Section 9.3: C onservation of Linear Momentum.

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