The period of oscillation of an object in a frictionless...

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 $\left(R_{\mathrm{I}}=6.37 \cdot 10^{6} \mathrm{~m} ; R_{\mathrm{M}}=1.74 \cdot 10^{6} \mathrm{~m} ; M_{\mathrm{E}}=5.98 \cdot 10^{24} \mathrm{~kg}\right.$ $\left.M_{u}=7.36 \cdot 10^{22} \mathbf{k g}\right) ?$

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 $\left(R_{\mathrm{I}}=6.37 \cdot 10^{6} \mathrm{~m} ; R_{\mathrm{M}}=1.74 \cdot 10^{6} \mathrm{~m} ; M_{\mathrm{E}}=5.98 \cdot 10^{24} \mathrm{~kg}\right.$
$\left.M_{u}=7.36 \cdot 10^{22} \mathbf{k g}\right) ?$

Key Concepts

Mass Distribution and Enclosed Mass

The gravitational influence inside a spherical object is determined by the mass enclosed within a given radius. This principle is central to deriving the effective restoring force that governs the motion of an object in a tunnel, as only the mass at radii smaller than the object's current distance from the center contributes to the gravitational pull experienced at that point.

Scaling Laws in Gravitational Systems

Newton's law of universal gravitation governs the interactions in gravitational systems by linking the force to the masses involved and the distances between them. Scaling laws allow one to relate the mass and radius of different bodies to the characteristics of the gravitational field within them, which is crucial for calculating the period of oscillation in different planetary tunnels.

Simple Harmonic Motion

This concept describes the oscillatory motion that occurs when the net force acting on an object is directly proportional to its displacement from an equilibrium position and acts towards that position. In the context of a tunnel through a spherically symmetric body, the gravitational force inside the body leads to simple harmonic motion, resulting in a consistent oscillation period that is independent of the amplitude of the motion under ideal conditions.

Gravitational Field in a Spherical Body

In a spherically symmetric mass distribution, the gravitational force experienced by an object inside the body depends only on the mass enclosed within the radius at which the object is located. This implies that within such an object, the gravitational force varies linearly with distance from the center, which is the key condition needed to produce simple harmonic motion for an object oscillating through a tunnel.

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