Physics: Principles with Applications (7th Edition)

Published by Pearson
ISBN 10: 0-32162-592-7
ISBN 13: 978-0-32162-592-2

Chapter 11 - Oscillations and Waves - Problems - Page 323: 21

Answer

See answer.

Work Step by Step

a.Find the period and frequency using the mass and the spring constant. $$T=2 \pi \sqrt{\frac{m}{k}}=2 \pi \sqrt{\frac{0.885\;kg}{184\;N/m}}\approx 0.436\;s$$ $$f=1/T=2.29\;Hz$$ b. The initial speed is the maximum speed. Find the amplitude. $$v_{max}=A\sqrt{\frac{k}{m}}$$ $$A = v_{max}\sqrt{\frac{m}{k}}= (2.26\;m/s) \sqrt{\frac{0.885\;kg}{184\;N/m}}\approx 0.157\;m$$ c. Find the maximum acceleration from the mass, the spring constant, and the amplitude. $$a_{max}=Ak/m=(0.1567\;m)(184\;N/m)/(0.885\;kg)=32.6\;m/s^2$$ d. The total energy is the PE at maximum distance from equilibrium. $$E=0.5kA^2=0.5(184\;N/m)(0.1567\;m)^2=2.26\;J$$ e. Mechanical energy is conserved. First find the PE at that position. $$PE=0.5k(0.40A)^2=0.5(184\;N/m)((0.40)(0.1567\;m)^2=0.361\;J$$ The KE is the total energy minus that quantity. $$2.26\;J-0.361\;J=1.90\;J$$
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