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Chapter 14: Problem 19

Pendulum A has a bob of mass \(m\) hung from a string of length \(I_{i}\) pendulum \(B\) is identical to \(A\) except its bob has mass \(2 m\). Compare the frequencies of small oscillations of the two pendulums.

Short Answer

Expert verified
Question: Compare the frequencies of small oscillations of two pendulums: Pendulum A with a mass m and length l, and Pendulum B with a mass of 2m and the same length l. Answer: The frequencies of small oscillations of pendulums A and B are equal.

Step by step solution

01

Formula for frequency of small oscillations

The formula for the frequency of small oscillations of a simple pendulum is given by: \[f = \frac{1}{2\pi}\sqrt{\frac{g}{l}}\] where f is the frequency, g is the acceleration due to gravity, and l is the length of the pendulum.
02

Frequency of Pendulum A

For pendulum A, we have a mass m and length \(l_i\). Using the formula from step 1, the frequency of pendulum A will be: \[f_A = \frac{1}{2\pi}\sqrt{\frac{g}{l_i}}\]
03

Frequency of Pendulum B

Since Pendulum B is identical to Pendulum A except for having a mass of 2m, its length is also \(l_i\), and the frequency of Pendulum B will be given by the same formula: \[f_B = \frac{1}{2\pi}\sqrt{\frac{g}{l_i}}\]
04

Compare the frequencies

Now, we can compare the frequencies of both pendulums. Since both pendulum A and pendulum B have the same length, their frequencies will be the same (since the mass does not affect the frequency in this case) : \[f_A = f_B\] In conclusion, the frequencies of small oscillations of pendulums A and B are equal, despite the difference in mass of their bobs.

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Most popular questions from this chapter

A mass \(M=0.460 \mathrm{~kg}\) moves with an initial speed \(v=3.20 \mathrm{~m} / \mathrm{s}\) on a level frictionless air track. The mass is initially a distance \(D=0.250 \mathrm{~m}\) away from a spring with \(k=\) \(840 \mathrm{~N} / \mathrm{m}\), which is mounted rigidly at one end of the air track. The mass compresses the spring a maximum distance \(d\), before reversing direction. After bouncing off the spring the mass travels with the same speed \(v\), but in the opposite dircction. a) Determine the maximum distance that the spring is compressed. b) Find the total elapsed time until the mass returns to its starting point. (Hint: The mass undergoes a partial cycle of simple harmonic motion while in contact with the spring.)

Object \(A\) is four times heavier than object B. Fach object is attached to a spring, and the springs have equal spring constants. The two objects are then pulled from their equilibrium positions and released from rest. What is the ratio of the periods of the two oscillators if the amplitude of \(A\) is half that of \(B ?\) a) \(T_{A}: T_{\mathrm{g}}=1: 4\) c) \(T_{A}: T_{\mathrm{B}}=2\) : b) \(T_{A}: T_{B}=4: 1\) d) \(T_{A}: T_{B}=1: 2\)

14.10 A pendulum is suspended from the ceiling of an elevator. When the elevator is at rest, the period of the pendulum is \(T\). The elevator accelerates upward, and the period of the pendulum is then a) still T. b) less than \(T_{-}\) c) greater than \(T\).

When the displacement of a mass on a spring is half of the amplitude of its oscillation, what fraction of the mass's energy is kinetic energy?

A shock absorber that provides critical damping with \(\omega_{\gamma}=72.4 \mathrm{~Hz}\) is compressed by \(6.41 \mathrm{~cm} .\) How far from the equilibrium position is it after \(0.0247 \mathrm{~s} ?\)
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