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Homework Problems

 

HW#37   Ch 15: Oscillations

Instructions: Be sure to write formulas and show substitutions to earn full credit.

1.    A Moravian lion is attached to an oscillating horizontal spring situated on a frictionless horizontal surface. Its position is given by \(x(t) = \left( {8cm} \right)\sin \left( \dfrac{2\pi t}{5} + \dfrac{3\pi}{2}\right)\) , where t is in seconds. (a) What is the period of oscillation? (b) What is the frequency of oscillation? (c) What is its speed and direction (positive/negative) at \(t = \dfrac{T}{4}\)? (d) What is its speed and direction at \(t = \dfrac{3T}{4}\)?

2.    A particle undergoes simple harmonic motion given by the equation \(x = Acos(\omega t + \delta)\). Suppose that the particle's position is at \(x = \) 33.01 cm to the right of the origin at \(t\) = 1 second, with a speed of \(v = \) 40.65 cm/s towards the left at the same time. The angular frequency is \( \omega = 1.8 \) rad/s. What is (a) the phase constant δ and (b) the amplitude of oscillation?

3.    A bag of jellybeans oscillates on a spring with a period of π/2 seconds. If it is moving at 30 cm/s when it is at the equilibrium position and it reaches its maximum amplitude at π/3 seconds, write the equation governing its position as a function of time.

4.    A chocolate rabbit of mass 165 g is resting on an 80 gram platform that is attached to a vertical spring with spring constant k = 12 N/m. The rabbit is pushed downward against the vertical spring so that it oscillates about its equilibrium position. (a) What is the period of oscillation? (b) What is the maximum amplitude of oscillation so that the rabbit does not fly off the spring? (c) What is the maximum speed in example 4b)?

5.    Peter Cottontail, who has a mass M, decides to improve his hopping ability by applying springs to his feet. At some point along the Bunny Trail, the springs get caught on a vine, and Peter starts to oscillate up and down with amplitude A. If the combined spring constant of both springs is k, what is his speed if his kinetic energy is three times his elastic potential energy, given in terms of k, M and A?