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DEPARTMENT OF
PHY10012
OSCILLATIONS AND WAVES
2007 Exam paper
PART A Attempt ALL questions. Tick the box by the answer you judge to be correct. Marks will not be deducted for incorrect answers. A1. A harmonic oscillator has maximum kinetic energy when the displacement is [ ] maximum (+A) [ ] minimum (A) [ ] 0 [ ] ± A / √2 [1] A2. In a critically damped system a decrease in the damping constant will result in the system [ ] becoming overdamped [ ] becoming underdamped [ ] becoming critically damped [ ] becoming overdamped or underdamped depending on the oscillator’s mass [1] A3. The de Broglie wavelength of an electron with a velocity 4 x 106 m s1 is [ ] 5.6 x 109 m [ ] 1.8 x 1010 m [ ] 7.3 x 104 m [ ] 2.0 x 1015 m [1] A4. In an underdamped oscillating system the damping constant γ, oscillator mass and natural frequency ω0 satisfy [ ] γ < 2mω0 [ ] γ = 2mω0 [ ] γ > 2mω0 [ ] γ > mω0 [1] A5. The kinetic and potential energies in a simple harmonic oscillator are equal when the displacement equals [ ] ± A [ ] ± A / √2 [ ] 0 [ ] ± ½ A [1] PART B attempt ALL questions
BI. Two sine waves with frequencies of 600 Hz and 650 Hz are travelling along a line. What are
the highest and lowest frequencies in the resulting waveform. [3]
B2. A 0.5 kg mass attached to a spring oscillates at an angular frequency of 10 rads".
Determine the spring constant. [3]
B3. An oscillator’s displacement is given by x = 0.5 cos(at - 5) where @ = 5 rad s' and
5 = 7/4 rad. Calculate the oscillator’s velocity as it passes through the equilibrium position.
(3]
B4. The displacement x of an oscillating 2 kg mass, attached to a spring, is described by the equation of motion 2 d2 x dt2 4 dx dt 10 x=0 Determine the damping constant γ, the spring constant k and the undamped natural frequency ω0. [3] B5. Two waves E1 and E2 have equal amplitude A0 and angular frequency ω, but a phase difference ϕ. Show that the sum of the waves is a wave with amplitude A = 2 A0 cos(½ϕ). [3] PART C Answer TWO questions from this section. C1. At equilibrium, a float of mass M and crosssectional area A projects above the surface of a liquid with density ρ. Show that if the equation of motion of the float is M d2 y dt2 =− Ag y [15] Hence show that the float undergoes simple harmonic motion with angular frequency [10] Given that ρ = 1.2 x 103 kg m3, A = 5 x 105 m2 and M = 2 x 103 kg determine the oscillation frequency and the period. [5] C2. The displacement of a mass m attached to a spring is given by x = Asin(ωt). Derive expressions for the kinetic energy Ek and potential energy Ep (do NOT derive the expression for potential energy U = ½kx2). [10] Hence show that (a) the total energy of the mass is E = ½kA2 where k is the spring constant [10] (b) the average potential energy Ū = ½E [10] C3. A particle of mass m is trapped in an infinite potential well, width L, such that V = ∞ for ∞< x < 0 V = 0 for 0 < x < L V = ∞ for L < x < ∞ a) Show that solutions of the Schrödinger Equation −h2 2 m ∂2 ∂ x2 V =E possible only if the energy E is quantised, b) Show that the normalised wave function is n= 2L sin n xL [15] ω = √(ρAg / M) C4. Two waves y, = Asin(kx-@t) and y2 = Asin(kx+@t) are travelling along a wire. Show that their
sum is a standing wave y = 2Acos(@t)sin(kx). [8]
Sketch the resulting waveform indicating nodes and antinodes, and direction of travel ifany. [7]
Two signals travel in a medium such that the phase velocity v, = v/k. If the phase velocity of the
medium is 0.5c find the wavelength of a 10 kHz and 15 kHz signal in the medium. [8]
Hence show that the group delay v, = Av/Ak = 2.5 x 10’ms". [7]
