The quartz crystal in a watch executes simple harmonic motion at \(32,768 \mathrm{Hz}\) (This is \(2^{15} \mathrm{Hz}\), chosen so that 15 divisions by 2 give a signal at \(1.00000 \mathrm{Hz}\) ) If each face of the crystal undergoes a maximum displacement of \(100 \mathrm{nm}\), find the maximum velocity and acceleration of the crystal faces.

When studying simple harmonic motion (SHM), knowing the concept of 'angular frequency' is vital. It is closely linked to the frequency of the motion, but involves the inherent geometry of circular movements, hence the term 'angular'.

Angular frequency, denoted as \(\textbf{\(\omega\)}\), is a measure describing how fast something goes through its cycle. It is given by the equation \(\omega = 2\pi f\), where \(f\) is the frequency, and \(2\pi\) is a full revolution in radians. If we think of SHM as a projection of circular motion, this measure gives us an idea of how quickly we are 'sweeping' through the circle.

In our case, the quartz crystal in a watch vibrates at a frequency of \(32,768 \mathrm{Hz}\), an incredibly rapid back-and-forth movement. Calculating its angular frequency involves simply multiplying by \(2\pi\), giving us a much larger number that speaks volumes about the speed of the oscillation. This is especially important when determining other aspects of SHM, such as velocity and acceleration.

Maximum velocity is a key concept in simple harmonic motion. It refers to the highest speed achieved by the oscillating object as it moves back and forth. In SHM, this maximum velocity occurs exactly when the object passes through the equilibrium point, where no restoring force is acting upon it.

The formula to calculate maximum velocity in SHM is \(v_{\text{max}} = \omega \times A\), with \(\omega\) as the angular frequency and \(A\) as the amplitude, or the maximum displacement from the equilibrium position.

For a quartz crystal in a watch, we'd use the angular frequency we calculated earlier and multiply it by the maximum displacement, which is given as \(100 \text{nm}\). Despite the tiny size of the displacement, the resulting maximum velocity is quite fast, demonstrating the efficiency and precision needed in timekeeping. This concept ensures the watch's timekeeping elements move quickly enough to keep accurate time, but not so fast that they wear out the mechanism.

Maximum acceleration is yet another crucial aspect to grasp when dealing with simple harmonic motion. The object undergoing SHM experiences the greatest acceleration at the points of maximum displacement, where the restoring force is at its peak.

The mathematical representation for this is \(a_{\text{max}} = \omega^2 \times A\), with \(\omega\) being the angular frequency and \(A\) representing the amplitude. For our quartz crystal, this equation allows us to find how quickly the crystal's facing accelerates back toward the equilibrium position once it has reached its furthest point in the oscillation cycle.

This maximum acceleration is pivotal for understanding the forces involved in SHM and ensuring the structural integrity of the oscillating system. It's an intrinsic measure of how 'forceful' the motion is and relates directly to the stresses the material of the crystal faces during its operation.