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Chapter 14: Problem 18

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, A and B, where pendulum A has a bob of mass m and string length L, and pendulum B has a bob of mass 2m and the same string length L as pendulum A. Answer: The frequencies of small oscillations of pendulums A and B are the same.

Step by step solution

01

Recall the formula for the frequency of a simple pendulum

The formula for the frequency 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

Find the frequency of pendulum A

Since pendulum A has a length of \(L_i\) and a bob of mass \(m\), we can substitute these values into the frequency formula to find its frequency: $$ f_A=\frac{1}{2\pi}\sqrt{\frac{g}{L_i}} $$
03

Find the frequency of pendulum B

Pendulum B has the same length as pendulum A, \(L_i\), but with a bob of mass \(2m\). Since the mass of the bob does not affect the frequency of a simple pendulum, the formula for the frequency of pendulum B is the same as for pendulum A. Hence, the frequency of pendulum B is: $$ f_B=\frac{1}{2\pi}\sqrt{\frac{g}{L_i}} $$
04

Compare the frequencies of small oscillations of the two pendulums

We can now compare the frequencies \(f_A\) and \(f_B\): Since both pendulums have the same length \(L_i\) and the mass of the bob does not affect their frequencies, their frequencies will be the same. $$ f_A=f_B $$ In conclusion, the frequencies of small oscillations of pendulums A and B are the same, regardless of the mass of their bobs.

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Most popular questions from this chapter

The period of a pendulum is \(0.24 \mathrm{~s}\) on Earth. The period of the same pendulum is found to be 0.48 s on planet \(X,\) whose mass is equal to that of Earth. (a) Calculate the gravitational acceleration at the surface of planet \(X\). (b) Find the radius of planet \(\mathrm{X}\) in terms of that of Earth.

A block of wood of mass \(55.0 \mathrm{~g}\) floats in a swimming pool, oscillating up and down in simple harmonic motion with a frequency of \(3.00 \mathrm{~Hz}\). a) What is the value of the effective spring constant of the water? b) A partially filled water bottle of almost the same size and shape as the block of wood but with mass \(250 . g\) is placed on the water's surface. At what frequency will the bottle bob up and down?

With the right choice of parameters, a damped and driven physical pendulum can show chaotic motion, which is sensitively dependent on the initial conditions. Which statement about such a pendulum is true? a) Its long-term behavior can be predicted. b) Its long-term behavior is not predictable. c) Its long-term behavior is like that of a simple pendulum of equivalent length. d) Its long-term behavior is like that of a conical pendulum. e) None of the above is true.

Identical blocks oscillate on the end of a vertical spring one on Earth and one on the Moon. Where is the period of the oscillations greater? a) on Earth d) cannot be determined b) on the Moon from the information given c) same on both Farth and Moon

Mass-spring systems and pendulum systems can both be used in mechanical timing devices. What are the advantages of using one type of system rather than the othes in a device designed to generate reproducible time measurements over an extended period of time?
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