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10-203-hwk-Ch13

# 10-203-hwk-Ch13 - Chapter 13 Oscillations About Equilibrium...

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Chapter 13: Oscillations About Equilibrium Solutions to Problems 2. Picture the Problem : The rocking chair completes one full cycle or oscillation each time it returns back to its original position. Strategy: The period is the time for one cycle. The frequency is the inverse of the period, or the number of cycles per second. Solution: 1 . Divide the total time by the number of cycles to determine the period: 21 s 1.75 s 12 cycles t T n = = = 2 . Invert the period to determine the frequency: 1 1 0.57 Hz 1.75 s f T = = = Insight: Since period and frequency are inverses of each other, when the period is greater than a second, the frequency will be less than a hertz. 8. Picture the Problem : The position of the mass oscillating on a spring is given by the equation of motion. Strategy: The oscillation period can be obtained directly from the argument of the cosine function. The mass is at one extreme of its motion at t = 0, when the cosine is a maximum. It then moves toward the center as the cosine approaches zero. The first zero crossing will occur when the cosine function first equals zero, that is, after one-quarter period. Solution: 1. (a) Identify T with the time 0.58 s: Since 2 2 cos cos 0.58 t t T s π π = , therefore T = 0.58 s. 2. (b) Multiply the period by one-quarter to find the first zero crossing: ( ) 1 0.58 s 0.15 s . 4 t = = Insight: A cosine function is zero at ¼ and ¾ of a period. It has its greatest magnitude at 0 and ½ of a period. 10. Picture the Problem : The mass is attached to the spring and pulled down from equilibrium and released. The stiffness of the spring causes the mass to oscillate at the given frequency. Strategy: Since the mass starts at x = A at time t = 0, this is a cosine function given by ( ) cos x A t ω = . From the data given you need to identify the constants A and ω . Solution: 1. Identify the amplitude as A : A = 6.4 cm 2. Calculate ω from the period: 2 2 0.83 s T π π ω = = 3 . Substitute A and ω into the cosine equation: ( ) 2 6.4 cm cos 0.83 s x t π = Insight: A cosine function has a maximum amplitude at t

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