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Chapter 14: Problem 71

The period of oscillation of an object in a frictionless tunnel running through the center of the Moon is \(T=2 \pi / \omega_{0}\) \(=6485 \mathrm{~s}\), as shown in Fxample 142 . What is the period of oscillation of an object in a similar tunnel through the Earth \(\left(R_{\mathrm{I}}=6.37 \cdot 10^{6} \mathrm{~m} ; R_{\mathrm{M}}=1.74 \cdot 10^{6} \mathrm{~m} ; M_{\mathrm{E}}=5.98 \cdot 10^{24} \mathrm{~kg}\right.\) \(\left.M_{u}=7.36 \cdot 10^{22} \mathbf{k g}\right) ?\)

Short Answer

Expert verified
Answer: The period of oscillation of an object in a similar tunnel through Earth is approximately 23855 seconds.

Step by step solution

01

Write down the formula for the period of oscillation

The formula for the period of oscillation is given by \(T=2 \pi / \omega_{0}\).
02

Write down Earth's and Moon's gravitational force formulas.

The gravitational forces on Earth and Moon are given by: \(F_{E} = G\frac{m M_{E}}{R_{E}^2}\) and \(F_{M} = G\frac{m M_{M}}{R_{M}^2}\)
03

Equate the gravitational force equations and solve for the ratio \(\frac{\omega_{E}}{\omega_{M}}\)

Equate the expressions for the gravitational forces: \(\frac{F_{E}}{F_{M}} = \frac{G\frac{m M_{E}}{R_{E}^2}}{G\frac{m M_{M}}{R_{M}^2}}\) Solve for the ratio \(\frac{\omega_{E}}{\omega_{M}}\): \(\frac{\omega_{E}}{\omega_{M}} = \frac{M_{E} R_{M}^2}{M_{M} R_{E}^2} = \frac{5.98 \cdot 10^{24}\text{kg} \cdot (1.74 \cdot 10^{6}\text{m})^2}{7.36 \cdot 10^{22}\text{kg} \cdot (6.37 \cdot 10^{6}\text{m})^2} \approx 3.678\)
04

Find Earth's period of oscillation

Use the found ratio \(\frac{\omega_{E}}{\omega_{M}}\) to find Earth's period of oscillation: \(T_{E} = \frac{2\pi}{\omega_{E}} = \frac{2\pi}{\frac{\omega_{M}}{3.678}} = 3.678 \cdot \frac{2\pi}{\omega_{M}} = 3.678 \cdot T_{M}\) Substitute the given value of \(T_{M}\): \(T_{E} = 3.678 \cdot 6485\text{s} \approx 23855\text{s}\) The period of oscillation of an object in a similar tunnel through Earth is approximately 23855 seconds.

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Most popular questions from this chapter

A mass, \(M=1.6 \mathrm{~kg}\), is attached to a wall by a spring with \(k=578 \mathrm{~N} / \mathrm{m}\). The mass slides on a frictionless floor. The spring and mass are immersed in a fluid with a damping constant of \(6.4 \mathrm{~kg} / \mathrm{s}\). A horizontal force, \(\mathrm{F}(\mathrm{t})=\mathrm{F}_{\mathrm{d}} \cos \left(\omega_{\mathrm{d}} \mathrm{t}\right)\) where \(F_{d}=52 \mathrm{~N},\) is applied to the mass through a knob, caus. ing the mass to oscillate back and forth. Neglect the mass of the spring and of the knob and rod. At approximately what frequency will the amplitude of the mass's oscillation be greatest, and what is the maximum amplitude? If the driving frequency is reduced slightly (but the driving amplitude remains the same), at what frecuency will the amplitude of the mass's ascillation be half of the maximum amplitude?

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 \(T ?\)

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 \(J_{0}\).

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?

Cars have shock absorbers to damp the oscillations that would otherwise occur when the springs that attach the wheels to the car's frame are compressed or stretched. Ideally, the shock absorbers provide critical damping. If the shock absorbers fail, they provide less damping, resulting in an underdamped motion. You can perform a simple test of your shock absorbers by pushing down on one corner of your car and then quickly releasing it If this results in an up-and- down oscillation of the car, you know that your shock absorbers need changing. The spring on each wheel of a car has a spring constant of \(4005 \mathrm{~N} / \mathrm{m}\), and the car has a mass of \(851 \mathrm{~kg}\), equally distributed over all four wheels. Its shock absorbers have gone bad and provide only \(60.7 \%\) of the damping they were initially designed to provide. What will the period of the underdamped oscillation of this car be if the pushing-down shock absorber test is performed?
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