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Hw14_Solutions - Physics 7A Homework 14 December 2008 All...

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Physics 7A Homework 14 December 2008 All problems are from Giancoli’s Physics for Scientists and Engineers. 1. Problem 14.20 A mass m = 1 . 25 kg stretches a vertical spring a distance x 0 = . 210 m. If it is stretched x 1 = 1 . 30 m and released, how long will it take to reach equilibrium? How long will it take to return to this new equilibrium? First, find the spring constant. We know - kx = mg k = mg x 0 The mass returns to the equilibrium position after one quarter period (passes it on the way up). So, 1 4 T = 1 4 2 π ω = π 2 r m k = . 23 s. 2. Problem 14.35 A mass resting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. It takes E = 3 . 6 J of work to compress the spring by x 1 = 0 . 10 m. If the spring is compressed, and the mass is released from rest, it experiences a maximum acceleration of 10 meters per second squared. Find the value of the spring constant and the mass. For the spring constant, recall the energy stored in a spring is U spring = 1 2 kx 2 . Here, this is supplied by the work we do on the spring. Then U spring = 1 2 kx 2 1 k = U spring 1 2 x 2 1 = 720N/m After release the only force acting on the mass is the force due to the compressed spring. By Newton’s second law Σ F = ma = - kx 1 . Then ma = - kx 1 m = - kx 1 a = 7 . 2 kg 1
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3. Problem 14.54 An aluminum disk, d = 11 . 0 cm in diameter and m = 380 g in mass, is mounted on a vertical shaft with very low friction. One end of a flat coil spring is attached to the disk, the other end to the base of the apparatus. The disk is set into rotational oscillation and the frequency is f = 0 . 370 Hz. What is the torsional spring constant K ( τ = - )? Recalling Σ τ = , r = d 2 , T = 1 f = 2 π ω , and I = 1 2 Mr 2 , we have I d 2 θ dt 2 = - ω 2 = K/I Since ω = 2 π T = 2 πf we have K/I = ω 2 K = 1 2 M d 2 4 (2 πf ) 2 = . 00311 Remember to convert units.
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