A block of mass M rests on a table. It is fastened to the...

A block of mass M rests on a table. It is fastened to the lower end of a light, vertical spring. The upper end of the spring is fastened to a block of mass m. The upper block is pushed down by an additional force $3 m g,$ so the spring compression is $4 m g / k .$ In this configuration, the upper block is released from rest. The spring lifts the lower block off the table. In terms of $m,$ what is the greatest possible value for $M ?$

Key Concepts

Hooke's Law

This principle states that the force exerted by a spring is proportional to its displacement from equilibrium, with the constant of proportionality being the spring constant. It is a foundational concept in understanding how springs store and release energy, and it applies to scenarios where spring compression or elongation yields a force that can accelerate connected masses.

Conservation of Energy

In many mechanical systems, energy is transformed between different forms, such as between potential energy stored in a spring and gravitational potential energy. The conservation of energy principle allows one to equate the work done on or by the system to changes in energy, which is crucial when analyzing the motion and forces in problems involving springs and masses.

Newton's Laws of Motion

Newton’s laws provide the framework for analyzing forces in dynamic systems. They are essential for setting up the equations of motion of each mass in systems where forces such as the weight of the masses and the spring force interact. These laws help determine conditions like when a block will lose contact with a surface or begin accelerating.

Free-Body Diagram Analysis

This technique involves drawing a diagram that shows all the forces acting on each individual object within a system. It is an important method for visualizing the interactions between forces such as gravitational, normal, and spring forces, and it simplifies the process of applying Newton's laws to determine the behavior of the system.

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