A \(500-\mathrm{g}\) block on a frictionless, horizontal surface is attached to a rather limp spring with \(k=8.7 \mathrm{N} / \mathrm{m}\). A second block rests on the first, and the whole system executes simple harmonic motion with period \(1.8 \mathrm{s}\). When the amplitude of the motion is increased to \(35 \mathrm{cm},\) the upper block just begins to slip. What's the coefficient of static friction between the blocks?

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It can be visualized as the motion that occurs when an object is released from rest from a displaced position and then allowed to oscillate back and forth over the same path, such as a mass on a spring.

In the context of the block and spring system described in the exercise, the SHM occurs as the blocks move back and forth due to the restoring force of the spring. The key point to grasp here is that the amplitude of this motion—how far from the equilibrium position the system displaces—plays a significant role in not only the speed of the oscillation but also the likelihood of the upper block slipping in relation to the static friction.

Angular frequency, often denoted as \( \omega \), is a measure of how fast something rotates or oscillates. It is akin to the concept of linear frequency, which counts the number of cycles per second, but here it specifically refers to the angular displacement per unit of time. For SHM, \( \omega \) is calculated by the formula \( \omega = \frac{2\pi}{T} \), where \( T \) is the period of the motion.

The exercise provided shows how to calculate the angular frequency for the block-spring system based on its period. Understanding \( \omega \) is crucial because it can help us determine other characteristics of the motion, such as the maximum acceleration the system can experience, which brings us to the concept of maximum acceleration in the context of SHM.

Maximum acceleration in SHM is the greatest rate of change of velocity that the oscillating object experiences, and it occurs when the object is at the maximum displacement from equilibrium, which is at the amplitude of the motion. The formula for the maximum acceleration (\( a_{\text{max}} \)) is \( a_{\text{max}} = \omega^2 \times A \), where \( A \) is the amplitude and \( \omega \) is the angular frequency.

As illustrated in the solution steps for the exercise, by finding the maximum acceleration experienced by the system, we can calculate the maximum force of static friction exerted by the lower block before slipping occurs. This information is vital for determining the coefficient of static friction between the blocks, which indicates how much force is needed to start moving one block on top of the other when initially at rest.