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Unformatted text preview: CHAPTER 14 Oscillations 1* Deezo the Clown slept in again. As he roller-skates out the door at breakneck speed on his way to a lunchtime birthday party, his superelastic suspenders catch on a fence post, and he flies back and forth, oscillating with an amplitude A . What distance does he move in one period? What is his displacement over one period? In one period, he moves a distance 4 A . Since he returns to his initial position, his displacement is zero. 2 A neighbor takes a picture of the oscillating Deezo (from Problem 1) at a moment when his speed is zero. What is his displacement from the fence post at that time? His displacement is then a maximum. 3 What is the magnitude of the acceleration of an oscillator of amplitude A and frequency f when its speed is maximum? When its displacement is maximum? When v = v max , a = 0; when x = x max , a = w 2 A = 4 p 2 f 2 A . 4 Can the acceleration and the displacement of a simple harmonic oscillator ever be in the same direction? The acceleration and the velocity? The velocity and the displacement? Explain. Acceleration and displacement are always oppositely directed; F = - kx. v and a can be in the same direction, as can v and x ; see Equs. 14-4, 14-5, and 14-6. 5* True or false: ( a ) In simple harmonic motion, the period is proportional to the square of the amplitude. ( b ) In simple harmonic motion, the frequency does not depend on the amplitude. ( c ) If the acceleration of a particle is proportional to the displacement and oppositely directed, the motion is simple harmonic. ( a ) False ( b ) True ( c ) True 6 The position of a particle is given by x = (7 cm) cos 6 p t , where t is in seconds. What is ( a ) the frequency, ( b ) the period, and ( c ) the amplitude of the particles motion? ( d ) What is the first time after t = 0 that the particle is at its equilibrium position? In what direction is it moving at that time? ( a ) Compare the expression with Equ. 14-4 ( b ) T = 1/ f ( c ) See Equ. 14-4 ( d ) x = 0 when cos w t = 0 f = w /2 p = 3 Hz T = 0.33 s A = 7 cm w t = p /2; t = p /12 p = 1/12 s; v is then negative Chapter 14 Oscillations 7 ( a ) What is the maximum speed of the particle in Problem 6? ( b ) What is its maximum acceleration? ( a ) v max = A w = 42 p cm/s = 1.32 m/s. ( b ) a max = A w 2 = 252 p 2 cm/s 2 = 24.9 m/s 2 . 8 What is the phase constant d in Equation 14-4 if the position of the oscillating particle at time t = 0 is ( a ) 0, ( b ) - A , ( c ) A , ( d ) A /2? Compare to Equ. 14-4: ( a ) cos d = 0; d = p /2, 3 p /2. ( b ) cos d = - 1; d = p . ( c ) cos d = 1; d = 0. ( d ) cos d = 1/2; d = p /3. 9* A particle of mass m begins at rest from x = +25 cm and oscillates about its equilibrium position at x = 0 with a period of 1.5 s. Write equations for ( a ) the position x versus the time t , ( b ) the velocity v versus t , and ( c ) the acceleration a versus t ....
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- Spring '08