A block with mass M rests on a frictionless surface and is...

A block with mass $M$ rests on a frictionless surface and is connected to a horizontal spring of force constant $k$ . The other end of the spring is attached to a wall (Fig. 13.36$) .$ A second block with mass $m$ rests on top of the first block. The coefficient of static friction between the blocks is $\mu_{s}$ Find the maximum amplitude of oscillation such that the top block will not slip on the bottom block.

Key Concepts

Simple Harmonic Motion

This concept describes the oscillatory motion of a system where the restoring force is proportional to the displacement from equilibrium. In such systems, parameters like amplitude, angular frequency, and maximum acceleration are directly linked by predictable sinusoidal functions. The idea is crucial for understanding how the oscillating block accelerates and how its motion is quantified.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two contacting surfaces. It has an upper limit, determined by the product of the coefficient of static friction and the normal force. In this problem, ensuring that the maximum inertial force does not exceed this limit prevents slipping between the blocks.

Relative Motion in Accelerated Systems

This concept involves analyzing the motion of objects from a non-inertial reference frame. Here, the top block rests on the accelerating lower block and experiences apparent forces due to the acceleration. Understanding relative motion is key to setting up the condition that the frictional force must be large enough to accelerate the top block along with the base block without slipping.

Force Analysis and Maximum Acceleration

A detailed force analysis considers the interplay between the spring force of the oscillating system and the frictional force needed to keep the top block stationary relative to the oscillating base. Recognizing that the maximum acceleration of the oscillating block occurs at the maximum displacement allows one to formulate an inequality that ensures the static friction is sufficient to prevent slipping.

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