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Homework #13 Solutions

Homework #13 Solutions - 13 Problem 7(a If the frequency is...

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§ 13 Problem 7 ( a ): If the frequency is f = 6 . 00 Hz = 6 . 00 oscillations per second, then the period is T = 1 /f = 0 . 167 s. ( b ): The angular frequency is ω = 2 πf = 37 . 7 / s. ( c ): We have ω = radicalbigg k m m = k ω 2 = 120 N / m (37 . 7 / s) 2 = 0 . 0844 kg . § 13 Problem 26 ( a ): The total energy at this point is E = 1 2 mv 2 + 1 2 kx 2 = 1 2 (0 . 150 kg)(0 . 300 m / s) 2 + 1 2 (300 N / m)(0 . 0120 m) 2 = 0 . 0284 J . ( b ): The amplitude A is the maximum value of x , at which point all the energy is potential, so 1 2 kA 2 = E A = radicalbigg 2 E k = 0 . 0137 m . ( c ): When the object is at maximimum speed all the energy is kinetic, so 1 2 mv 2 max = E v max = 2 E m = 0 . 615 m / s . § 13 Problem 36: The moment of inertia is I = 1 2 mR 2 = 4 . 84 × 10 - 7 kg m 2 . For a torsional pendulum we have T = 2 π radicalbigg I κ κ = 4 π 2 I T 2 = 1 . 91 × 10 - 5 N m . § 13 Problem 47: The period of oscillation is T = 1 . 36 s. We have T = 2 π radicalBigg L g g = 4 π 2 L T 2 = 10 . 7 m / s 2 . § 13 Problem 55: We treat both problems as physical pendula, and use T = 2 π radicalBigg I Mgd , where d is the distance from the pivot to the center of mass (which is L for both pendula) and I is the moment of inertia of the pendulum about the pivot. For the small bob we have I = ML 2 , while for the large
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