Problem 15 - Physics

Question

Atoms in a solid are not stationary, but vibrate about their equilibrium positions. Typically, the frequency of vibration is about \(f=2.0 \times 10^{12} \mathrm{~Hz},\) and the amplitude is about \(1.1 \times 10^{-11} \mathrm{~m}\). For a typical atom, what is its (a) maximum speed and (b) maximum acceleration?

Step-by-Step Solution

Answer

Maximum speed is approximately 138 m/s, and maximum acceleration is approximately \(1.09 \times 10^{15}\) m/s².

Step 1: Identify the given values

The problem gives us the frequency of vibration as \( f = 2.0 \times 10^{12} \, \mathrm{Hz} \) and the amplitude as \( A = 1.1 \times 10^{-11} \, \mathrm{m} \). We need to find the maximum speed and acceleration of the atom.

Step 2: Use the formula for maximum speed

For simple harmonic motion, the maximum speed \( v_{max} \) is given by \( v_{max} = 2 \pi f A \). Substitute the given frequency and amplitude into this formula: \[ v_{max} = 2 \pi \times 2.0 \times 10^{12} \, \mathrm{Hz} \times 1.1 \times 10^{-11} \, \mathrm{m} \].

Step 3: Calculate the maximum speed

Calculate \( v_{max} \) using the formula: \[ v_{max} = 2 \pi \times 2.0 \times 10^{12} \times 1.1 \times 10^{-11} \]. Calculate the result: \[ v_{max} \approx 1.38 \times 10^{2} \, \mathrm{m/s} \].

Step 4: Use the formula for maximum acceleration

The maximum acceleration \( a_{max} \) in simple harmonic motion is given by \( a_{max} = (2 \pi f)^2 A \). Substitute the values: \[ a_{max} = (2 \pi \times 2.0 \times 10^{12})^2 \times 1.1 \times 10^{-11} \].

Step 5: Calculate the maximum acceleration

Calculate \( a_{max} \) using the formula: \[ a_{max} = (2 \pi \times 2.0 \times 10^{12})^2 \times 1.1 \times 10^{-11} \]. This results in \[ a_{max} \approx 1.09 \times 10^{15} \, \mathrm{m/s^2} \].

Key Concepts

Vibration of Atoms

Atoms in a solid are dynamic, constantly vibrating around their equilibrium positions. This means they are never at rest, which is a fascinating aspect of their existence. These vibrations arise due to energy present at the atomic level, even at very low temperatures. The movement can be described using simple harmonic motion principles, which provide a framework for understanding how atoms displace and move back and forth from their equilibrium points.

• The vibrations are akin to a spring-mass system, where atoms oscillate similar to a mass attached to a spring.
• Simple harmonic motion assumes the restoring force is proportional to the displacement, hence being predictable and cyclic.

These vibrations are crucial, as they influence the physical properties of materials, including their thermal and electrical conductivities. Understanding atomic vibrations allows scientists to predict how materials behave under various temperature conditions.

Maximum Speed

In the context of simple harmonic motion, an atom's maximum speed happens when it passes through its equilibrium position. At this point, the kinetic energy is at its peak, making the velocity the highest it will be during the cycle. The formula to calculate this maximum speed is \( v_{max} = 2 \pi f A \), where:

• \( f \) is the frequency of vibration, and
• \( A \) is the amplitude of the motion.

By substituting the known values for an atom, the equation lets us find \( v_{max} \) easily. The maximum speed helps in understanding how energetically atoms are moving, which in turn affects how they interact and bond with each other within materials.

Maximum Acceleration

For simple harmonic motion, the atom's maximum acceleration is found when the atom is at its maximum displacement, that is, at the extremes of its motion. At these points, the force acting on the atom is also at its maximum, pushing it back towards its equilibrium point. Maximum acceleration is calculated using the formula: \( a_{max} = (2 \pi f)^2 A \). This equation indicates the acceleration is dependent on both the square of the frequency and the amplitude.

• This highlights that a small change in frequency can significantly impact the acceleration, given it is squared.
• Similarly, greater amplitude will also increase \( a_{max} \), making the system more dynamic.

Understanding this helps in assessing how forces within a material change, affecting its stability and reactions to external forces.

Frequency and Amplitude

Frequency and amplitude are key parameters in describing simple harmonic motion. Frequency, denoted as \( f \), is the number of cycles an atom completes per second and is measured in Hertz (Hz). It informs us about the energy level associated with the vibrations.

• Higher frequencies mean higher energy, leading to more vigorous atomic motion.
• They also affect how sound waves propagate through various materials.

Amplitude, represented as \( A \), is the maximum extent of displacement from the equilibrium position. It reflects the intensity of the motion. The amplitude doesn't change the speed but tells us how far the atom travels from its resting position.

• Large amplitudes imply atoms have more space to move, which can influence how they interact with each other.
• Amplitude affects the material's thermal properties, such as expansion when heated.

Understanding frequency and amplitude allows scientists to better manipulate material characteristics for various applications, such as designing temperature-resistant materials.