Which one of the following conservation laws is a consequence of the Newton's third law of motion?
Conservation of momentum
Understanding the Connection Between Newton's Third Law and Conservation Laws
The question asks which conservation law is a direct consequence of Newton's third law of motion. Let's break down Newton's third law and its implications for physical systems.
Newton's Third Law Explained
Newton's third law of motion states that:
- For every action, there is an equal and opposite reaction.
- When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.
In simpler terms, forces always occur in pairs, acting on different objects. If object A pushes on object B with force \(\vec{F}_{AB}\), then object B pushes back on object A with force \(\vec{F}_{BA}\), where \(\vec{F}_{AB} = -\vec{F}_{BA}\).
Newton's Third Law and Momentum Conservation
Consider a system of interacting particles or bodies. The forces acting within the system between these particles are called internal forces. According to Newton's third law, for every internal force acting on one particle, there is an equal and opposite internal force acting on another particle within the same system. When we sum up all the internal forces within the system, they cancel out in pairs:
\(\sum \vec{F}_{internal} = 0\)
Now, consider the total momentum of the system, \(\vec{P}_{total}\). According to Newton's second law of motion, the net external force acting on a system is equal to the rate of change of its total momentum:
\(\vec{F}_{net, external} = \frac{d\vec{P}_{total}}{dt}\)
If there are no external forces acting on the system (i.e., the system is isolated or closed, \(\vec{F}_{net, external} = 0\)), then the total net force on the system is just the sum of the internal forces. Since the sum of internal forces is zero due to Newton's third law, the total net force on an isolated system is zero.
\(\vec{F}_{net, external} + \sum \vec{F}_{internal} = 0 + 0 = 0\)
Therefore, for an isolated system:
\(\frac{d\vec{P}_{total}}{dt} = 0\)
This equation means that the total momentum of the system does not change over time; it remains constant. This is the principle of Conservation of Momentum.
Thus, the Conservation of Momentum for an isolated system is a direct consequence of Newton's third law ensuring that internal forces cancel out.
Analyzing Other Options
- Conservation of Energy: While energy conservation is a fundamental principle, it is not a direct consequence derived solely from Newton's third law. It is related to the work done by forces and arises from the time-invariance of the system's Lagrangian or Hamiltonian, which is a deeper principle.
- Conservation of Charge: This is a fundamental law in electromagnetism stating that the total electric charge in an isolated system remains constant. It is not related to Newton's laws of motion.
- Conservation of Mass: In classical mechanics, mass is generally conserved. In relativistic physics, mass-energy is conserved. This conservation law is not a direct consequence of Newton's third law of motion.
Based on the derivation, the conservation law that stems directly from Newton's third law of motion is the Conservation of Momentum.
| Conservation Law | Consequence of |
|---|---|
| Conservation of Momentum | Newton's Third Law (for isolated systems) |
| Conservation of Energy | Time-invariance of laws (related to potential energy and work done by forces) |
| Conservation of Charge | Fundamental principle in electromagnetism |
| Conservation of Mass | Fundamental principle (classical physics); Mass-energy conservation (relativistic physics) |
Revision Table: Newton's Laws and Conservation
| Law | Statement Summary | Related Conservation Law |
|---|---|---|
| Newton's First Law | Inertia: Object in motion stays in motion, object at rest stays at rest unless acted upon by a net external force. | Defines inertial frames, related to momentum concept. |
| Newton's Second Law | Net force equals mass times acceleration: \(\vec{F}_{net} = m\vec{a}\) or \(\vec{F}_{net} = \frac{d\vec{p}}{dt}\). | Relates force to momentum change. Crucial for deriving conservation laws. |
| Newton's Third Law | Action-Reaction: \(\vec{F}_{AB} = -\vec{F}_{BA}\). Forces occur in equal and opposite pairs. | Directly leads to Conservation of Momentum for isolated systems. |
Additional Information: Isolated vs. Closed Systems
In physics, it's important to distinguish between system types when discussing conservation laws:
- Isolated System: A system where no external forces act on it. For such a system, both momentum and energy are conserved (assuming the internal forces are conservative).
- Closed System: A system where no mass enters or leaves. In classical mechanics, mass is conserved in a closed system. A closed system may or may not have external forces acting on it. Momentum is conserved only if it's also isolated (no net external force). Energy is conserved if external forces do no net work AND internal forces are conservative.
The derivation linking Newton's third law to Conservation of Momentum relies on the system being isolated (no net external force), which allows the cancellation of internal forces to be the sole determinant of the total force on the system.
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