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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

If you kick a harmonic oscillator sharply, you impart to it an initial velocity but no initial displacement. For a weakly damped oscillator with mass \(m\), spring constant \(k\). and damping force \(F_{y}=-b v,\) find \(x(t),\) if the total impulse delivered by the kick is \(J_{0}\).

A mass \(m=5.00 \mathrm{~kg}\) is suspended from a spring and oscillates according to the equation of motion \(x(t)=0.5 \cos (5 t+\pi / 4) .\) What is the spring constant?

An object in simple harmonic motion is isochronous, meaning that the period of its oscillations is independent of their amplitude. (Contrary to a common assertion, the operation of a pendulum clock is not based on this principle. A pendulum clock operates at fixed, finite amplitude. The gearing of the clock compensates for the anharmonicity of the pendulum.) Consider an oscillator of mass \(m\) in one-dimensional motion, with a restoring force \(F(x)=-c x^{3}\) where \(x\) is the displacement from equilibrium and \(c\) is a constant with appropriate units. The motion of this ascillator is periodic but not isochronous. a) Write an expression for the period of the undamped oscillations of this oscillator. If your expression involves an integral, it should be a definite integral. You do not need to evaluate the expression. b) Using the expression of part (a), determine the dependence of the period of oscillation on the amplitude. c) Generalize the results of parts (a) and (b) to an oscillator of mass \(m\) in one-dimensional motion with a restoring force corresponding to the potential energy \(U(x)=\gamma|x| / \alpha\), where \(\alpha\) is any positive value and \(\gamma\) is a constant

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A massive object of \(m=5.00 \mathrm{~kg}\) oscillates with simple harmonic motion. Its position as a function of time varies according to the equation \(x(t)=2 \sin ([\pi / 2] t+\pi / 6)\). a) What is the position, velocity, and acceleration of the object at \(t=0 \mathrm{~s}^{2}\) b) What is the kinetic energy of the object as a function of time? c) At which time after \(t=0 \mathrm{~s}\) is the kinetic energy first at a maximum?
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