Newton's third law Knowledge-Based Study

Study notes

1. Forces Always Occur in Pairs

   In classical mechanics, a force is not an isolated entity that exists on its own; rather, it is an interaction between two objects. Whenever one object exerts a force on another, the second object simultaneously exerts a force back on the first.

   Isaac Newton formalized this fundamental property of nature as his Third Law of Motion:

   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.

   Mathematically, if object A exerts a force on object B, denoted as $\vec{F}_{A \text{ on } B}$, then object B exerts a force on object A, denoted as $\vec{F}_{B \text{ on } A}$:

   $$\vec{F}_{A \text{ on } B} = -\vec{F}_{B \text{ on } A}$$

   These two paired forces are commonly referred to as an action-reaction pair.

   Criteria for Third-Law Pairs

   To correctly identify whether two forces form an action-reaction pair, consider the following four strict rules:

   • Equal Magnitude: Both forces have precisely the same quantitative magnitude, regardless of mass, size, velocity, or internal state of the objects involved.
   • Opposite Directions: The two forces act along the same line of action but point in opposite directions.
   • Different Bodies: The action force acts on one object, while the reaction force acts on the other object. They never act on the same object.
   • Same Nature of Force: Both forces belong to the same fundamental physical interaction. For example, if the action is a gravitational pull, the reaction must also be a gravitational pull. If the action is a normal contact force, the reaction is also a normal contact force.

   Why Action and Reaction Do Not Cancel Each Other

   A frequent point of confusion is why an object can ever accelerate if every force has an equal and opposite reaction force. The critical distinction lies in the system boundary and the specific body being analyzed:

   • To find the acceleration of a single object, we compute the net force acting on that object alone using Newton's Second Law: $$\Sigma \vec{F} = m\vec{a}$$
   • Since an action force acts on object A and the reaction force acts on object B, they never appear on the same free-body diagram. Therefore, they cannot cancel each other out to prevent acceleration.

   Unequal Masses and Acceleration

   Consider a heavy truck colliding with a small compact car. According to Newton's Third Law, the magnitude of the contact force exerted by the truck on the car is exactly equal to the magnitude of the contact force exerted by the car on the truck:

   $$|\vec{F}_{\text{truck on car}}| = |\vec{F}_{\text{car on truck}}| = F$$

   However, because the car has a significantly smaller mass ($m$) compared to the truck's mass ($M$), their resulting accelerations differ markedly according to Newton's Second Law:

   $$a_{\text{car}} = \frac{F}{m} \quad \text{and} \quad a_{\text{truck}} = \frac{F}{M}$$

   Because $m < M$, the car experiences a vastly larger acceleration and therefore suffers far greater sudden kinematic change.

   Everyday Interactions and Non-Contact Forces

   Newton's Third Law governs both contact forces and field forces (action-at-a-distance):

   • Walking: When you walk forward, your foot exerts a backward horizontal friction force on the ground. Simultaneously, the ground exerts a forward horizontal friction force of equal magnitude on your foot, propelling you forward.
   • Rocket Propulsion: A rocket accelerates in space not by pushing against ambient air, but by forcefully expelling combustion gases downward. As the rocket pushes exhaust gas backward, the expanding gas pushes the rocket forward with an equal and opposite force.
   • Universal Gravitation: Earth exerts a downward gravitational force on you (your weight). In turn, you exert an upward gravitational force of the exact same magnitude on Earth. Earth's immense mass renders its resulting acceleration entirely undetectable in practice, yet the interaction is perfectly symmetric in force.