Question

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

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

Step-by-step solution

Write down the formula for the period of oscillation

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

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 \(F_{M} = G\frac{m M_{M}}{R_{M}^2}\)

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 \(\frac{\omega_{E}}{\omega_{M}}\): \(\frac{\omega_{E}}{\omega_{M}} = \frac{M_{E} R_{M}^2}{M_{M} R_{E}^2} = \frac{5.98 \cdot 10^{24}\text{kg} \cdot (1.74 \cdot 10^{6}\text{m})^2}{7.36 \cdot 10^{22}\text{kg} \cdot (6.37 \cdot 10^{6}\text{m})^2} \approx 3.678\)

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.

Key concepts

Gravitational Forces

Gravitational forces are the attractive forces that occur between two masses. They are one of the fundamental forces in physics, responsible for the interaction that keeps planets, stars, and galaxies bound together.
The force between two masses is described by Newton's law of universal gravitation, which states that the force (\( F \)) exerted by one mass (\( M \)) on another mass (\( m \)) is proportional to the product of their masses and inversely proportional to the square of the distance (\( R \)) between their centers:

• \[ F = G \frac{m M}{R^2} \], where \( G \) is the gravitational constant.

For the problem in the exercise, we are particularly interested in how this force acts in gravitational fields of planetary bodies, like Earth and Moon. By comparing the gravitational forces on Earth and the Moon, we can understand how different their influences are due to their respective masses and radii. This is crucial for calculating oscillations within tunnels across these celestial bodies.

Period of Oscillation

The period of oscillation refers to the time it takes for an object to complete one full cycle of motion. In scenarios where simple harmonic motion is involved, the period depends on the intrinsic properties of the system.
For the example given, we utilize the formula for the period of oscillation in a tunnel within a planetary body:

• \[ T = \frac{2\pi}{\omega_0} \], where \( \omega_0 \) is the angular frequency.

Specifically, for a tunnel running through the Moon, we know the period is 6485 seconds. To find the period on Earth, we rely on comparing the gravitational forces and mass distributions. This crucial dependency illustrates how periods can drastically differ based on mass and size of the celestial body, making each body's gravitational pull unique in its effect on oscillation.

Physics Problem Solving

Approaching physics problems like these requires a structured method to solve effectively. It typically starts with identifying the key principles involved, such as gravitational forces and simple harmonic motion dynamics in this case.
To solve such problems, one should:

• Understand the basic definitions and use known equations effectively, such as Newton’s law of gravitation.
• Draw parallels or comparisons (like equating gravitational forces on the Moon and Earth) to simplify complex problem solving.
• Use step-by-step solutions to calculate essential values like the period of oscillation, verifying calculations with given or known data.

Moreover, ensuring units are consistent and calculations are checked helps maintain accuracy throughout the process. By following a logical and organized approach, what initially seems daunting becomes clearer and solvable with these basic physics concepts.