PHYS 214 Final, Spring 2020

Phys.214 Final Exam (two hours) 1- Consider three wave sources S1, S2, and S3. The source S1 emits a wave of frequency 262 Hz, while S2 emits a wave of frequency 266 Hz. When the wave emitted by the source S3 interferes with that emitted by S1, a beat frequency of 1 Hz is produced. When the wave emitted by the source S3 interferes with that emitted by S2, a beat frequency of 3Hz is produced. What is the frequency of the wave emitted by the source S3? (10 pts) 2- Two vibrations at right angles to one another are described by the equations 𝑥 = 10cos⁡(5𝜋𝑡) and 𝑦 = 10𝑐𝑜𝑠(10𝜋𝑡 + 𝜋 3 ⁄ ). Construct the Lissajous figure of the combined motion. Note that the Lissajous figure is a graph describing the superposition of the motions. (15 pts) 3- Demonstrate that 𝑥 = 𝐴𝑒−𝛼𝑡𝑐𝑜𝑠(𝜔𝑡)⁡is a possible solution of the equation of a damped harmonic oscillator, which is given by 𝑑2𝑥 𝑑𝑡2 + 𝛾 𝑑𝑥 𝑑𝑡 ⁄ + 𝜔0 2𝑥 = 0 ⁄ and find 𝛼 and 𝜔 in terms of 𝛾 and 𝜔0. (20 pts) 4- Describe the TRANSIENT behavior of a damped oscillator when the periodic driving force is off resonance and when the periodic driving force is at exact resonance. (20 pts) 5- Derive the frequencies of the transverse and longitudinal normal modes of N coupled oscillators, and provide the total number of normal modes (transverse + longitudinal) in the system. (20 pts) 6- Two stretched strings, each of length 1.2 m, support standing waves. On the first string (string I), there is a second harmonic standing wave of frequency 660 Hz. However, the same frequency of 660Hz is the third harmonic on the second string (string II). Find the speed at which individual waves travel on each string. (15 pts) 7- A tube open at one end is cut into two tubes (of nonequal lengths). the piece that is open at both ends has a fundamental frequency of 475 Hz, while the piece that is open at one end only has a fundamental frequency of 675 Hz. What is the fundamental frequency of the original tube? (15 pts)