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Free-body diagrams are visual tools used to illustrate the relative magnitude and direction of all forces acting upon a single object. These diagrams make it easier to understand the dynamics of how objects interact. In our scenario, the focus is on crates A and B, positioned side by side on a frictionless surface. Each crate is influenced by several forces, including the applied horizontal force and gravitational force.

For crate A, the free-body diagram consists of:

• The applied force \( \vec{F} \) to the right, moving the system.
• A contact force \( \vec{F}_{AB} \) exerted by crate B to the left.
• The normal force \( \vec{N}_A \) directed upwards balancing the gravitational pull.
• The weight force \( \vec{W}_A = m_Ag \) acting downwards due to gravity.

For crate B, the main components are:

• A contact force \( \vec{F}_{BA} \) exerted by crate A, pointing rightward.
• Its own normal force \( \vec{N}_B \) pushing upwards against gravity.
• The weight force \( \vec{W}_B = m_Bg \), which also acts downward.

Such diagrams help determine net forces, predicting acceleration, and understanding interactions, making them a cornerstone in physics problem-solving.

Newton's Third Law of Motion is famously summarized as "For every action, there is an equal and opposite reaction." This principle explains how forces operate in pairs. In our example with crates A and B, the contact forces between them form an action-reaction pair.

When crate A pushes on crate B with the force \( \vec{F}_{BA} \), crate B pushes back equally and oppositely with force \( \vec{F}_{AB} \). These forces are equal in magnitude but opposite in direction (\( \vec{F}_{AB} = -\vec{F}_{BA} \)).

This action-reaction pair exhibits how forces between interacting bodies are mutual, leading to every pushed or pulled object reciprocating the force. This insight helps us understand why both crates will indeed accelerate in the same direction when a force is applied. They move as a single system because the internal forces between them balance each other out, allowing the external force \( \vec{F} \) to dictate their motion.

A frictionless surface is an idealized concept where there is no friction acting against an object's motion. In real-world scenarios, frictional forces would oppose motion, but on a frictionless surface, objects can move unimpeded. This concept simplifies calculations as the absence of friction means we only consider external forces, like applied or normal forces.

In our example with the crates, because the surface is frictionless, there is no resistance to the applied horizontal force \( \vec{F} \). This results in any net horizontal force, no matter how small, causing the crates to accelerate. It underscores an important point: even if the applied force \( \vec{F} \) is less than the combined weight of the crates, it can still cause them to move, as long as it overcomes any other horizontal forces—in this ideal case, there are none.

This scenario reinforces Newton's first law, the principle of inertia, where objects in motion remain in motion unless acted on by an external force—in this case, since there's nothing to stop them, any applied force results in movement.