PHYS 250 Recitation 14-Oscillations and Waves

PHYS 250 Recitation #14: Oscillations & Waves
Any system for which the acceleration is linearly proportional to the position (with a negative
proportionality constant), or
a x = -ω 2 x,
undergoes simple harmonic motion, a form of oscillatory motion. The mathematical solution 1 to
this is:
x(t) = A cos(ωt),
where A is the amplitude and ω = 2πf = 2π/T is the angular frequency (f is the frequency in Hz
and T is the period). For a mass on a spring, ω 2 = k/m; for a simple pendulum, ω 2 = g/L.
A wave is just an oscillation that travels in space, so it oscillates in both space and time, or
y(t) = A cos(kx - ωt),
where the wavenumber k = 2π/λ, where λ is the wavelength. [This is for a wave moving in the
+x direction. A wave moving in the –x direction would be A cos(kx + ωt).]
The speed with which the wave travels through a medium is determined by the properties of the
medium, not the wave itself. The speed of sound depends on pressure and density of the gas
(both of which depend on temperature); we will use 343 m/s for sound in air as a typical value
for this class. For a wave on a string, the wave speed is set by the tension (T s ) and the mass
density (mass/length) µ of the string, or
v 2 = T s /µ.
A medium can support a spectrum of wavelengths and frequencies. For a traveling wave created
by an object oscillating at a frequency f, then the wavelength of the wave is set by the wave
speed and frequency through the relationship:
v = λf.
For a standing wave, the length and boundary conditions set the possible λs that can resonate in
the system, and then the frequencies corresponding to these wavelengths are set by the same
v = λf relationship. (For a string or open-open system, λ 1 = 2L; for open-closed, λ 1 = 4L.)
1 Really, the solution is x(t) = Acos(ωt) + Bsin(ωt), which can also be written as x(t) = Acos(ωt
+ φ), which we have just been using with B = 0 (or φ = 0) for simplicity.

Oscillations Problem. A 0.1 kg mass on a spring with k = 5 N/m is stretched from its
equilibrium position by 15 cm and then released. Sketch the position versus time for this
oscillator for 4 seconds. Be as quantitative and precise as possible and be sure to label your
graph.
Calculate the following quantities for this oscillator (be sure to specify units!)
Amplitude A = ___________________
Period T = _____________________
Frequency f = _____________________
Angular frequency ω = ___________________
Maximum speed v max = ___________________
Maximum acceleration a max = ___________________
Total energy E = _________________________

On your diagram (question 1), mark an X where all the energy of the oscillator is in kinetic
energy. Mark an O where all the energy of the oscillator is in elastic potential energy.
What is the mass’s speed when it is at x = 10 cm (i.e., not at the maximum stretching)?
If you instead initially stretched the spring by 30 cm (instead of 15 cm), what would be the
oscillation frequency and period? Justify your answer.
If the spring had twice the spring constant, how would that change the frequency of the
oscillations?
If the object on the spring was four times as massive, how would that change the frequency of
the oscillations?
What would be the mass & length of a simple pendulum with the same period/frequency as the
original mass & spring system?

Waves Problem. A 0.55-m-long guitar string emits 220 Hz as its fundamental. The mass of the
string that is vibrating is 1 gram.
What is the period of this oscillation?
What is the wavelength of this oscillation?
What is the wave speed along the string?
What is the tension in the string?
What is the linear density µ of the string?
What is the string’s second harmonic? What is the string’s third harmonic?
If you halve the tension in the string, what is the fundamental frequency of the string?
By what factor would you have to change the tension (up or down?) for the fundamental
frequency of the string to go up one octave?
If you pluck the string when your finger is on the 5 th fret (so ¾ of the original length now
vibrates), what fundamental sound is emitted?