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Understanding Newton's laws of motion is fundamental when analyzing the dynamics of objects. These laws provide a framework for relating the motion of an object to the forces acting on it, which is essential for solving many physics problems for exams like the JEE MAIN.
Newton's first law, also known as the law of inertia, states that an object will remain at rest or in uniform motion in a straight line unless acted upon by a net external force. This concept is witnessed in the exercise where both blocks remain at rest until a force is applied.

Newton’s Second Law

The second law, the centerpiece for many physics problems, states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This is represented by the formula:
\[ F = ma \]
where \( F \) is the net force, \( m \) is the mass of the object, and \( a \) is the acceleration. In the exercise, this law is applied to relate the force acting on block M to its acceleration, and separate equations are formed for each mass based on their unique forces and accelerations.

Newton’s Third Law

The third law states that for every action, there is an equal and opposite reaction. This implies that if the spring exerts a force on block \( M \), then block \( M \) exerts an equal and opposite force on the spring. And since the spring is attached to block \( m \), the spring exerts the same force on block \( m \), which accelerates it. These interactions are crucial when analyzing forces within systems, as seen in the step-by-step solution.

The spring constant, denoted as \( k \), is a measure of the stiffness of a spring. It is defined as the force required to extend or compress a spring by unit length. Hooke's law gives the relationship between the force \( F \) exerted by a spring and its displacement \( x \) from the equilibrium position as:
\[ F = -kx \]
A higher spring constant means a stiffer spring, requiring more force to achieve the same displacement, and vice versa. In the provided exercise, the spring's constant plays a critical role in determining the force experienced by block \( m \). When the force \( F \) acts on block \( M \), the spring stretches, creating a restoring force that is proportional to the spring constant and the displacement. It is this force that accelerates block \( m \).
Understanding the concept of the spring constant is crucial for students aiming to excel in physics examinations, as it frequently appears in problems involving oscillatory motion and mechanics.

The direct relationship between force and acceleration is a key concept in physics and is integral to problems involving object dynamics. Newton's second law of motion, as previously mentioned, illustrates this relationship and allows us to solve for either force or acceleration when given the other and the mass of the object.
In the context of the JEE MAIN physics problem, applying a constant force to one block results in the acceleration of both blocks. Calculating the acceleration requires an understanding that the net force on an object is equal to the mass of the object multiplied by its acceleration (\( F = ma \)).
By manipulating the provided equations and applying Newton’s second law, the solution reveals that the acceleration of block \( m \) is determined by the combined mass of both blocks and the applied force. The result showcases how changes in force and mass directly affect acceleration, an important concept when resolving complicated physics problems and a vital part of JEE MAIN preparation.