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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 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?

Consider two identical oscillators, each with spring constant \(k\) and mass \(m\), in simple harmonic motion. One oscillator is started with initial conditions \(x_{0}\) and \(v_{j}\) the other starts with slightly different conditions, \(x_{0}+\delta x\) and \(v_{0}+\delta v_{1}\) a) Find the difference in the oscillators' positions, \(x_{1}(t)-x_{2}(t)\) for all t. b) This difference is bounded; that is, there exists a constant \(C\) independent of time, for which \(\left|x_{1}(t)-x_{2}(t)\right| \leq C\) holds for all \(t\). Find an expression for \(C\). What is the best bound, that is, the smallest value of \(C\) that works? (Note: An important characteristic of chaotic systems is exponential sensitivity to initial conditions; the difference in position of two such systems with slightly different initial conditions grows exponentially with time. You have just shown that an oscillator in simple harmonic motion is not a chaotic system.)

What is the period of a simple pendulum that is \(1.00 \mathrm{~m}\) long in each situation? a) in the physics lab b) in an clevator accelerating at \(2.10 \mathrm{~m} / \mathrm{s}^{2}\) upward c) in an elevator accelerating \(2.10 \mathrm{~m} / \mathrm{s}^{2}\) downward d) in an elevator that is in free fall

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) \(T_{A}: T_{B}=1: 2\)

A \(3.00-\mathrm{kg}\) mass attached to a spring with \(k=140 . \mathrm{N} / \mathrm{m}\) is oscillating in a vat of oil, which damps the oscillations. a) If the damping constant of the oil is \(b=10.0 \mathrm{~kg} / \mathrm{s}\), how long will it take the amplitude of the oscillations to decrease to \(1.00 \%\) of its original value? b) What should the damping constant be to reduce the amplitude of the oscillations by \(99.0 \%\) in 1.00 s?
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