Newton's Third Law of Motion
by aoum, Mar 18, 2025, 10:41 PM
Newton's Third Law of Motion: The Law of Action and Reaction
Newton's Third Law of Motion is one of the most profound and intuitive principles in classical mechanics. It describes the reciprocal nature of forces between interacting objects and is fundamental to understanding motion in physical systems.
1. Statement of Newton's Third Law
Newton's Third Law can be stated mathematically as:
![\[
\mathbf{F}_{A \to B} = -\mathbf{F}_{B \to A},
\]](//latex.artofproblemsolving.com/4/6/a/46a8cdbd65535eded1390e17efe37e600164db98.png)
which means:
For every action, there is an equal and opposite reaction.
In other words, if object
exerts a force on object 
, then object simultaneously exerts a force of equal magnitude and opposite direction on object .
This law applies to all interactions—whether they involve gravity, electromagnetism, or physical contact forces.
2. Understanding Action-Reaction Pairs
Action-reaction pairs are forces that two bodies exert on each other. These pairs:
It is crucial to remember that these forces do not cancel out because they act on different objects.
3. Examples of Newton's Third Law
4. Mathematical Formulation and Proof
Let two objects and interact. According to Newton’s Second Law:
![\[
\mathbf{F}_{A \to B} = m_B \mathbf{a}_B, \quad \mathbf{F}_{B \to A} = m_A \mathbf{a}_A.
\]](//latex.artofproblemsolving.com/9/a/5/9a5c1f02f44f9f494c157e3328458b694db1d64e.png)
Since these forces are equal and opposite:
![\[
m_B \mathbf{a}_B = -m_A \mathbf{a}_A,
\]](//latex.artofproblemsolving.com/6/a/0/6a0014168c60e9f13ef819686e65f0f0820400a2.png)
This implies that the total momentum of the system remains constant:
![\[
\frac{d}{dt} (\mathbf{p}_A + \mathbf{p}_B) = 0,
\]](//latex.artofproblemsolving.com/4/6/4/464f152969f81a5201522051bc97fa1dc9984672.png)
where
represents linear momentum.
This conservation of momentum is a direct consequence of Newton’s Third Law.
5. Applications of Newton's Third Law
Newton’s Third Law has widespread applications in physics and engineering:
6. Newton’s Third Law in Different Force Types
Newton’s Third Law applies universally across different physical forces:
7. Newton's Third Law and Non-Inertial Frames
In non-inertial (accelerating) frames of reference, fictitious forces arise, but Newton’s Third Law still holds within the proper frame by accounting for these pseudo-forces.
8. Limitations and Extensions of Newton’s Third Law
While the law holds in most classical systems, it requires modification in certain advanced contexts:
9. Experimental Evidence for Newton's Third Law
Newton’s Third Law has been confirmed through numerous experiments:
10. Newton's Third Law and Conservation Laws
Newton’s Third Law is closely tied to the law of conservation of linear momentum. For any closed system:
![\[
\mathbf{p}_{\text{total}} = \sum m \mathbf{v} = \text{constant}.
\]](//latex.artofproblemsolving.com/7/6/b/76b7814dc5537bb387d9171ce8919964f5be79ed.png)
The conservation of angular momentum is also a consequence of equal and opposite torque interactions.
11. Summary
Newton’s Third Law of Motion establishes the reciprocity of forces between interacting bodies. It underpins the conservation of momentum and has universal applications across all areas of classical physics and beyond.
12. References
Newton's Third Law of Motion is one of the most profound and intuitive principles in classical mechanics. It describes the reciprocal nature of forces between interacting objects and is fundamental to understanding motion in physical systems.
Rockets work by creating unbalanced high pressure that pushes the rocket upwards while exhaust gas exits through an open nozzle.
1. Statement of Newton's Third Law
Newton's Third Law can be stated mathematically as:
![\[
\mathbf{F}_{A \to B} = -\mathbf{F}_{B \to A},
\]](http://latex.artofproblemsolving.com/4/6/a/46a8cdbd65535eded1390e17efe37e600164db98.png)
which means:
For every action, there is an equal and opposite reaction.
In other words, if object
exerts a force on object 
, then object simultaneously exerts a force of equal magnitude and opposite direction on object .
represents the force exerted by object on object .
represents the force exerted by object on object .
represents the force exerted by object 
represents the force exerted by object This law applies to all interactions—whether they involve gravity, electromagnetism, or physical contact forces.
2. Understanding Action-Reaction Pairs
Action-reaction pairs are forces that two bodies exert on each other. These pairs:
- Always have equal magnitudes:
. - Always point in opposite directions:
. - Always act on different objects: Each force in the pair is applied to a separate body.
.
.It is crucial to remember that these forces do not cancel out because they act on different objects.
3. Examples of Newton's Third Law
- Walking: When you push backward on the ground, the ground pushes you forward with an equal and opposite force.
- Rocket Propulsion: Exhaust gases are expelled backward, and the rocket moves forward due to the reaction force.
- Collisions: When two objects collide, each exerts an equal and opposite force on the other, regardless of their masses.
- Magnetic Attraction and Repulsion: If a magnet pulls on a piece of iron, the iron pulls back on the magnet with equal strength.
4. Mathematical Formulation and Proof
Let two objects
and interact. According to Newton’s Second Law:![\[
\mathbf{F}_{A \to B} = m_B \mathbf{a}_B, \quad \mathbf{F}_{B \to A} = m_A \mathbf{a}_A.
\]](http://latex.artofproblemsolving.com/9/a/5/9a5c1f02f44f9f494c157e3328458b694db1d64e.png)
Since these forces are equal and opposite:
![\[
m_B \mathbf{a}_B = -m_A \mathbf{a}_A,
\]](http://latex.artofproblemsolving.com/6/a/0/6a0014168c60e9f13ef819686e65f0f0820400a2.png)
This implies that the total momentum of the system remains constant:
![\[
\frac{d}{dt} (\mathbf{p}_A + \mathbf{p}_B) = 0,
\]](http://latex.artofproblemsolving.com/4/6/4/464f152969f81a5201522051bc97fa1dc9984672.png)
where
represents linear momentum.This conservation of momentum is a direct consequence of Newton’s Third Law.
5. Applications of Newton's Third Law
Newton’s Third Law has widespread applications in physics and engineering:
- Aviation: The thrust of jet engines is countered by the expulsion of exhaust gases.
- Mechanical Systems: In gears and pulleys, forces between components are always paired and balanced.
- Space Exploration: Spacecraft maneuver using reaction forces from expelled propellants (even in the vacuum of space).
- Engineering Structures: When a structure pushes on the ground, the ground pushes back to maintain equilibrium.
6. Newton’s Third Law in Different Force Types
Newton’s Third Law applies universally across different physical forces:
- Gravitational Force: The Earth pulls on the Moon, and the Moon pulls back with equal and opposite force.
- Electromagnetic Force: Charges exert equal and opposite forces on each other.
- Contact Forces: When a book rests on a table, the table provides an equal and opposite normal force.
- Tension Forces: When pulling a rope, the rope pulls back with an equal and opposite tension.
7. Newton's Third Law and Non-Inertial Frames
In non-inertial (accelerating) frames of reference, fictitious forces arise, but Newton’s Third Law still holds within the proper frame by accounting for these pseudo-forces.
8. Limitations and Extensions of Newton’s Third Law
While the law holds in most classical systems, it requires modification in certain advanced contexts:
- Relativity Theory: At relativistic speeds (close to the speed of light), forces become dependent on the frame of reference, and momentum conservation is governed by Einstein’s equations.
- Electromagnetic Fields: In time-varying electromagnetic fields, forces are mediated by field energy and may not be strictly equal and opposite at every instant.
- Quantum Mechanics: On microscopic scales, interactions are described by quantum fields where Newton’s classical laws must be reinterpreted probabilistically.
9. Experimental Evidence for Newton's Third Law
Newton’s Third Law has been confirmed through numerous experiments:
- Tension and Compression Tests: Measurements of stress in materials show equal and opposite internal forces.
- Collision Studies: High-speed cameras record equal and opposite forces during elastic and inelastic collisions.
- Rocket Testing: Exhaust mass flow and thrust measurements match the predictions of Newton’s Third Law.
10. Newton's Third Law and Conservation Laws
Newton’s Third Law is closely tied to the law of conservation of linear momentum. For any closed system:
![\[
\mathbf{p}_{\text{total}} = \sum m \mathbf{v} = \text{constant}.
\]](http://latex.artofproblemsolving.com/7/6/b/76b7814dc5537bb387d9171ce8919964f5be79ed.png)
The conservation of angular momentum is also a consequence of equal and opposite torque interactions.
11. Summary
Newton’s Third Law of Motion establishes the reciprocity of forces between interacting bodies. It underpins the conservation of momentum and has universal applications across all areas of classical physics and beyond.
12. References
- Newton, I. Philosophiæ Naturalis Principia Mathematica (1687).
- Halliday, D., Resnick, R., & Walker, J. Fundamentals of Physics (2013).
- Wikipedia: Newton’s Laws of Motion
March 2025