PHYS102 Study Guide

Unit 1: Mechanical Vibrations and Waves in Extended Objects

1a. describe the properties of simple harmonic motion and provide examples

  1. What kind of force causes the objects to undergo simple harmonic motion?
  2. How do position, velocity, and acceleration depend on time in SHM? What is their relationship?

Simple harmonic motion (SHM) is a unique kind of motion in which the position is a sinusoidal function of time:

 x=x_\mathrm{max}\cos(\omega t+\phi)

Here,  x_\mathrm{max} is amplitude,  \omega is angular frequency, and  \phi is the phase of the oscillations. The period of the oscillations can be calculated as  T=\frac{2\pi}{\omega} . In general,  x does not have to be a position; it could be any variable describing a system.

Mechanical vibrations and waves are similar to electromagnetic waves. Both mechanical and electromagnetic waves begin with an oscillatory system where vibrations propagate in space. The most basic of all oscillatory phenomena is simple harmonic motion. It can occur in a variety of mechanical systems, as discussed in Vibrations.

Harmonic Motion covers the mathematical description and properties of the simple harmonic motion as applied to a system of a mass on a spring.

 

1b. define the following terms related to wave motion: frequency, wavelength, diffraction, and interference

  1. What are the properties of wave motion?
  2. How is wave motion different from particle motion?
  3. What is superposition?
  4. What is the relationship between frequency, wavelength, and velocity of a periodic wave?

Waves surrounding us have such a variety of manifestations – ripples on a surface of water, sound, or light – that it is not easy to realize that they all have the same underlying properties. The main property that distinguishes wave motion from the motion of matter particles is that waves can pass through each other without affecting each other's motion. When more than one wave is present at the same place, they combine. This interference of waves is known as superposition. Waves also demonstrate diffraction, which is the ability to bend around an obstacle. When a wave encounters a boundary between two media, it undergoes reflection (traveling backward) and transmission (also referred to as refraction).

Periodic waves are characterized by their wavelength, which is the distance the wave travels during one period - the time it takes for the oscillation to go through a full cycle. Thus, the relationship between the wavelength and period is  \lambda = vT , where  v is the speed of the wave propagation. Alternatively, wavelength can be related to frequency. Frequency is a reciprocal of period, or the number of cycles the wave goes through in one second:  f=\frac{1}{T} . Thus,  \lambda = \frac{v}{f} .

Read Free Waves to review the properties and mathematical description of wave motion.

 

1c. state Hooke's Law

  1. How can Hooke's Law be used to analyze the motion of a system?
  2. How can the magnitude and direction of a restoring force be found?

Simple harmonic motion arises when a restoring force appears in a system when that system is disturbed from equilibrium, and when the force is proportional to and has the opposite direction of the displacement. A typical example of such a system is a point mass attached to the end of a massless spring. In this case, Hooke's law states that the restorative force is  F_x=-kx , where  x is the displacement of the mass and is equal to the length by which the spring is stretched or compressed.  k is a spring constant, determined by the material of the spring and how tightly it is wound. The negative sign indicates that the force is in the direction opposite to the displacement.

Springs and Hooke's Law discusses this system in detail.

 

1d. solve problems using simple harmonic motion

  1. What are the necessary conditions for a system to undergo simple harmonic motion?
  2. What quantities are necessary to calculate the angular frequency, frequency, and period of oscillations?
  3. What quantities are necessary to calculate the total energy of oscillations at a given time
  4. What is the relationship between displacement, velocity, and acceleration of a system at a given time? What is the relationship between the maximum values of these variables?

Application problems involving simple harmonic motion typically focus on the relationship between the maximum displacement (amplitude) and maximum velocity of a system, and on calculating angular velocity, period, and energy of the oscillations. These are the formulas you should know that pertain to the situations involving SHM:

  • Equations of motion:  x=x_\mathrm{max}\cos(\omega t + \phi) ,  v=-x_\mathrm{max}\omega\sin(\omega t + \phi) . From here, it can be seen that maximum velocity is  v_\mathrm{max}=\omega x_\mathrm{max} . Typically, the displacement is maximum at  t = 0 . Then,  \phi = 0 and  x=x_\mathrm{max}\cos(\omega t) .
  • The acceleration at any time is proportional to displacement:  a_x=-\omega^2 x .
  • The angular frequency depends on the system:
    For a mass m on the spring with spring constant  k ,  \omega = \sqrt{\frac{k}{m}} .
    For a mathematical pendulum (point mass on the string of length  l ),  \omega = \sqrt{\frac{g}{l}} .
  • The frequency of the oscillations is  f=\frac{\omega}{2\pi} and the period is  T=\frac{1}{f}=\frac{2\pi}{\omega} .
  • The energy of oscillations is conserved and equals the sum of potential and kinetic energy. For a mass-and-spring system the energy is  E=\frac{kx^2}{2}+\frac{mv^2}{2} .

See the solved examples of application problems in Oscillatory Motion.

 

Unit 1 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

  • Acceleration
  • Amplitude
  • Angular frequency
  • Displacement
  • Energy, potential and kinetic
  • Equilibrium
  • Force constant (same as spring constant)
  • Frequency
  • Hooke's Law
  • Oscillation
  • Period
  • Phase
  • Restorative (or restoring) Force
  • Simple Harmonic Motion
  • Superposition
  • Velocity
  • Vibration
  • Wave
  • Wavelength