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solutions1 - Homework Set 1 Due Thursday, 06/30 Problem 1:...

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Homework Set 1 Due Thursday, 06/30 Problem 1 : A spring has a relaxed length of 15.0cm. When a 100g mass is hung from the spring, the spring is stretched to 17.5cm. When an unknown mass M is hung from the spring, the spring stretches to 18.0cm. (a) What is the mass of M? (b) What is the frequency of oscillations with the 100g mass and with mass M? (c) If a 250g mass is attached to the spring while the spring is relaxed, and then dropped from that position, what will be the amplitude of the oscillations? Solution: (a) First, calculate the spring constant. From mg = kz 0 , where z 0 is the stretching of the spring beyond the relaxed point, we get Now we can use the spring constant to calculate the second object's mass: (b) The frequency is given by (c) When a 250g mass is added to the relaxed spring, the equilibrium position shifts down by Since the mass is released from rest 6.25cm from the new equilibrium, that distance will be the amplitude of the oscillations. So, the amplitude is 6.25cm.
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Problem 2 : The position of an oscillator is given by (a) What is the amplitude, angular frequency, frequency and period of the oscillations? (b) What is the speed of the oscillator at t = 0 ? (c) If the oscillator has a mass of 0.05kg, what is the spring constant? Solution: (a) The amplitude is the maximum displacement reached by the oscillator. It is 0.0135m, or 1.35 cm. The angular frequency is the factor multiplying the time in the cosine: it is equal to 6.58 s -1 . The frequency is f = ϖ / 2 π = 6.58 s -1 / 2 = 1.05 s -1 . The period is T = 1/f = 0.95 s . (b) Differentiating the position with respect to time, or recalling the kinematic equations of a harmonic oscillator, the speed is given by At t = 0 , the speed is (c) The spring constant is related to the angular frequency by
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Problem 3: A cylindrical barrel with length L = 0.40m, radius R = 0.15m and density ρ = 600 kg/m 3 floats in water (density 1000 kg/m 3 ). The forces acting on the barrel are gravity, pointing down, and the buoyant force, equal to the weight of the displaced water according to Archimedes' law, pointing up. (a) If the barrel is in equilibrium (no net force), how high is the top of the barrel above the surface of the water? (b) Show that the barrel will obey Hooke's law if displaced slightly up or down from equilibrium. What is the effective “spring constant” and the period of oscillations? Solution: (a) Let z be the height of the top of the barrel above the water. The net force on the barrel (up is positive) is equal to Here, the subscript w denotes the mass, volume and density of the displaced water, while the subscript b indicates the same quantities pertaining to the barrel. At the equilibrium position z 0 , the net force is zero: (b) If the top of the barrel is z meters above the water, the net force is given by the first equation in part (a). Now we define
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solutions1 - Homework Set 1 Due Thursday, 06/30 Problem 1:...

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