Problem 69 - College Physics

Question

A quartz watch contains a crystal oscillator in the form of a block of quartz that vibrates by contracting and expanding. Two opposite faces of the block, \(7.05 \mathrm{~mm}\) apart, are antinodes, moving alternately toward and away from each other. The plane halfway between these two faces is a node of the vibration. The speed of sound in quartz is \(3.70 \times 10^{3} \mathrm{~m} / \mathrm{s}\). Find the frequency of the vibration. An oscillating electric voltage accompanies the mechanical oscillation, so the quartz is described as piezoelectric. An electric circuit feeds in energy to maintain the oscillation and also counts the voltage pulses to keep time.

Step-by-Step Solution

Answer

The frequency of the vibration in the quartz watch is achieved by performing the calculation in step 3.

Step 1: Understand Nodes and Antinodes

In a wave motion like in this quartz block, nodes are points that always remain at rest. They represent points of minimum amplitude. On the other hand, antinodes are points that oscillate between a maximum and minimum amplitude, representing points of maximum disturbance in the medium. In the problem, the point half-way between the two antinodes is said to be a node.

Step 2: Calculate the Wavelength

In general, the distance between two nodes or two antinodes or two identical points in a wave is equal to the wavelength (\(\lambda\)) of the wave. However, in this case, we are given the distance between one node and an antinode, which is half of the wavelength. Thus, the full wavelength is twice the given distance, so, \(\lambda = 2 \times 7.05\) mm, which converts to \( \lambda = 2 \times 7.05 \times 10^{-3}\) m.

Step 3: Calculate the Frequency

Frequency (\(f\)) is calculated using the formula: \(f = v / \lambda\) where \(v\) is the speed of sound in the medium and \(\lambda\) is the wavelength. Substituting known values, we get \( f = 3.70 \times 10^3 m/s / (2 \times 7.05 \times 10^{-3} m)\).

Key Concepts

Nodes and Antinodes in Wave Motion

Understanding the behavior of waves is essential for numerous applications, including the operation of quartz crystal oscillators found in devices like watches. In the context of wave motion, nodes and antinodes are two fundamental concepts.

A node is a specific point along a medium where the wave has minimal amplitude; it remains stationary over time. Think of a node as the 'calm eye' in a storm of motion, where disturbances are at their lowest. In contrast, an antinode is a point where the wave exhibits maximum amplitude. This is where the medium's particles undergo the most significant movement, akin to the peaks of a sea wave.

An easy way to visualize this is by imagining a jump rope held by two people. If one person shakes the rope, you'll see points where the rope barely moves (nodes) and points where the rope moves violently (antinodes). These coherent patterns of nodes and antinodes are crucial characteristics of standing waves, which are waves that remain in a constant position.

Speed of Sound in Quartz

The speed of sound is a measure of how quickly vibrational energy can travel through a medium, whether it's air, water, or, as in our case, quartz crystal. This speed is vital to understanding the oscillation frequency in a quartz oscillator.

In quartz, sound waves travel at a speed of approximately 3.70 x 10^3 meters per second, substantially faster than in the air. This speed comes into play when calculating wave-based phenomena like the frequency of a crystal oscillator. High speed in solid mediums like quartz is due to the closely packed atoms that transmit vibrational energy more rapidly compared to gases or liquids.

One might wonder why the speed of sound matters in a watch. Well, in a quartz oscillator, this speed, along with the dimensions of the quartz piece, determines how fast the crystal vibrates, which in turn defines the precise frequency for timekeeping purposes. As we use the knowledge of nodes and antinodes along with the speed of sound in quartz, we can calculate the frequency of the oscillations that govern the accuracy of a quartz timepiece.

Piezoelectric Effect

The piezoelectric effect is a fascinating phenomenon where certain materials can generate an electric charge in response to applied mechanical stress. Conversely, these materials can deform when exposed to an electric field.

This behavior is central to the function of a quartz oscillator in a watch. When the quartz crystal vibrates, it creates an oscillating voltage due to the piezoelectric effect, and this electrical signal can be measured to determine the oscillation frequency. In simple terms, piezoelectric materials like quartz are natural pressure-to-electricity converters.

Here's an analogy to make it easier: imagine pressing a sponge and having it produce light. Press it faster, and it flashes more quickly. Quartz’s ability to convert pressure to an electrical signal in a similar way makes it indispensable in precision devices. It's used not only in watches but also in microphones, speakers, and sensors, showcasing the piezoelectric effect’s versatility.