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ism_chapter_14 - Chapter 14 Oscillations Conceptual...

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1039 Chapter 14 Oscillations Conceptual Problems 1 Determine the Concept The acceleration of an oscillator of amplitude A and frequency f is zero when it is passing through its equilibrium position and is a maximum when it is at its turning points. When v = v max : 0 = a When x = x max : A f A a 2 2 2 4 π ω = = 2 Determine the Concept The condition for simple harmonic motion is that there be a linear restoring force; i.e., that F = kx. Thus, the acceleration and displacement (when they are not zero) are always oppositely directed. v and a can be in the same direction, as can v and x . 3 ( a ) False. In simple harmonic motion, the period is independent of the amplitude. ( b ) True. In simple harmonic motion, the frequency is the reciprocal of the period which, in turn, is independent of the amplitude. ( c ) True. The condition that the acceleration of a particle is proportional to the displacement and oppositely directed is equivalent to requiring that there be a linear restoring force; i.e., F = kx ma = kx or a = ( k/m ) x. *4 Determine the Concept The energy of a simple harmonic oscillator varies as the square of the amplitude of its motion. Hence, tripling the amplitude increases the energy by a factor of 9. 5 •• Picture the Problem The total energy of an object undergoing simple harmonic motion is given by , 2 2 1 tot kA E = where k is the stiffness constant and A is the amplitude of the motion. The potential energy of the oscillator when it is a distance x from its equilibrium position is () . 2 2 1 kx x U =
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Chapter 14 1040 Express the ratio of the potential energy of the object when it is 2 cm from the equilibrium position to its total energy: ( ) 2 2 2 2 1 2 2 1 tot A x kA kx E x U = = Evaluate this ratio for x = 2 cm and A = 4 cm: ( ) ( ) () 4 1 cm 4 cm 2 cm 2 2 2 tot = = E U ( ) ( ) 4 1 cm 4 cm 2 cm 2 2 2 tot = = E U and correct. is ) ( a 6 ( a ) True. The factors determining the period of the object, i.e., its mass and the spring constant, are independent of the oscillator’s orientation. ( b ) True. The factors determining the maximum speed of the object, i.e., its amplitude and angular frequency, are independent of the oscillator’s orientation. 7 False. In order for a simple pendulum to execute simple harmonic motion, the restoring force must be linear. This condition is satisfied, at least approximately, for small initial angular displacements. 8 True. In order for a simple pendulum to execute periodic motion, the restoring force must be linear. This condition is satisfied for any initial angular displacement. *9 •• Determine the Concept Assume that the first cart is given an initial velocity v by the blow. After the initial blow, there are no external forces acting on the carts, so their center of mass moves at a constant velocity v /2. The two carts will oscillate about their center of mass in simple harmonic motion where the amplitude of their velocity is v /2.
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ism_chapter_14 - Chapter 14 Oscillations Conceptual...

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