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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 diaphragm’s 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. , ω 2 a = g, Here, ω = 2 π f is the diaphragm’s 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 π 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 crosssectional area of the body, V its volume, x its submerged depth, ρ its density, and ρ 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 = ρ V s g = ρ A g x. The mass and weight of the body are ρ V and f W = ρ V g, respectively. Hence, the body’s equation of vertical motion is ρ V ¨ x = f W − f B = ρ V g − ρ A g x. In equilibrium, x = x , where x = ρ V ρ A . Let x = x + δx , where  δx  ≪ x . It follows, from expansion of the equation of motion to firstorder in small quantities, that ρ V δ ¨ x = − ρ A g δx....
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 Spring '10
 Fitzpatrick
 Acceleration, Energy, Gravity, Work, Sin, Cos

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