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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 physical pendulum consists of a uniform rod of mass \(M\) and length \(L\) The pendulum is pivoted at a point that is a distance \(x\) from the center of the rod, so the period for oscillation of the pendulum depends on \(x: T(x)\). a) What value of \(x\) gives the maximum value for \(T ?\) b) What value of \(x\) gives the minimum value for \(T ?\)

Imagine you are an astronaut who has landed on another planet and wants to determine the free-fall acceleration on that planet. In one of the experiments you decide to conduct, you use a pendulum \(0.50 \mathrm{~m}\) long and find that the period of oscillation for this pendulum is \(1.50 \mathrm{~s}\). What is the acceleration due to gravity on that planet?

A cylindrical can of diameter \(10.0 \mathrm{~cm}\) contains some ballast so that it floats vertically in water. The mass of can and ballast is \(800.0 \mathrm{~g}\), and the density of water is \(1.00 \mathrm{~g} / \mathrm{cm}^{3}\) The can is lifted \(1.00 \mathrm{~cm}\) from its equilibrium position and released at \(t=0 .\) Find its vertical displacement from equilibrium as a function of time. Determine the period of the motion. Ignore the damping effect due to the viscosity of the water.

A mass \(m\) is attached to a spring with a spring constant of \(k\) and set into simple harmonic motion. When the mass has half of its maximum kinetic energy, how far away from its equilibrium position is it, expressed as a fraction of its maximum displacement?

An \(80.0-\mathrm{kg}\) bungee jumper is enjoying an afternoon of jumps. The jumper's first oscillation has an amplitude of \(10.0 \mathrm{~m}\) and a period of \(5.00 \mathrm{~s}\). Treating the bungee cord as spring with no damping, calculate each of the following: a) the spring constant of the bungee cord. b) the bungee jumper's maximum speed during the ascillation, and c) the time for the amplitude to decrease to \(2.00 \mathrm{~m}\) (with air resistance providing the damping of the oscillations at \(7.50 \mathrm{~kg} / \mathrm{s})\)
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