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Unformatted text preview: 1347 Chapter 14 Oscillations Conceptual Problems 1 True or false: ( a ) For a simple harmonic oscillator, the period is proportional to the square of the amplitude. ( b ) For a simple harmonic oscillator, the frequency does not depend on the amplitude. ( c ) If the net force on a particle undergoing one-dimensional motion is proportional to, and oppositely directed from, the displacement from equilibrium, the motion is simple harmonic. ( 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. This is the condition for simple harmonic motion 2 If the amplitude of a simple harmonic oscillator is tripled, by what factor is the energy changed? 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. 3 [SSM] An object attached to a spring exhibits simple harmonic motion with an amplitude of 4.0 cm. When the object is 2.0 cm from the equilibrium position, what percentage of its total mechanical energy is in the form of potential energy? ( a ) One-quarter. ( b ) One-third. ( c ) One-half. ( d ) Two-thirds. ( e ) Three-quarters. 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 force 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 = Express the ratio of the potential energy of the object when it is 2.0 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 = = Chapter 14 1348 Evaluate this ratio for x = 2.0 cm and A = 4.0 cm: ( ) ( ) ( ) 4 1 cm 4.0 cm 2.0 cm 2 2 2 tot = = E U and ) ( a is correct. 4 An object attached to a spring exhibits simple harmonic motion with an amplitude of 10.0 cm. How far from equilibrium will the object be when the systems potential energy is equal to its kinetic energy? ( a ) 5.00 cm. ( b ) 7.07 cm. ( c ) 9.00 cm. ( d ) The distance cant be determined from the data given. Determine the Concept Because the objects total energy is the sum of its kinetic and potential energies, when its potential energy equals its kinetic energy, its potential energy (and its kinetic energy) equals one-half its total energy. Equate the objects potential energy to one-half its total energy: total 2 1 s E U = Substituting for U s and E total yields: ( ) 2 2 1 2 1 2 2 1 kA kx = 2 A x = Substitute the numerical value of A and evaluate x to obtain: cm 07 . 7 2 cm . 10 = = x and ( ) b is correct. 5 Two identical systems each consist of a spring with one end attached to a block and the other end attached to a wall. The springs are horizontal, and the blocks are supported from below by a frictionless horizontal table. The blocks are oscillating in simple harmonic motions such that the amplitude of the motion of block A is four times as large as the amplitude of the motion of block B. How do their maximum speeds compare? ( their maximum speeds compare?...
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