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Unformatted text preview: Physics 315: Oscillations and Waves Homework 1: Solutions 1. The body will fly off the diaphragm whenever the diaphragms downward acceleration exceeds the acceleration, g , due to gravity. Hence, as the fre- quency increases, the body will first fly off the diaphragm when the maxi- mum downward acceleration becomes equal to g : i.e. , 2 a = g, Here, = 2 f is the diaphragms angular frequency of oscillation, f is the same frequency in hertz, and a is the amplitude of the oscillation. It follows that f = 1 2 radicalbigg g a . However, a = 1 10- 5 m and g = 9 . 8 m s- 1 , so f = 157 . 6 Hz. 2. Let A be the cross-sectional area of the body, V its volume, x its submerged depth, its density, and the density of the liquid. The submerged volume is V s = x A . According to Archimedes Principle, the buoyancy force is the weight of the displaced fluid. Hence, f B = V s g = A g x. The mass and weight of the body are V and f W = V g, respectively. Hence, the bodys equation of vertical motion is V x = f W f B = V g A g x. In equilibrium, x = x , where x = V A . Let x = x + x , where | x | x . It follows, from expansion of the equation of motion to first-order in small quantities, that V x = A g x....
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