The gravitational force on a particle of mass m inside the...

The gravitational force on a particle of mass $m$ inside the earth at a distance $r$ from the center ( $r<$ the radius of the earth $R$ ) is $F=-m g r / R$ (Chapter 6 , Section 8, Problem 21). Show that a particle placed in an evacuated tube through the center of the earth would execute simple harmonic motion. Find the period of this motion.

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) occurs when the net force acting on a particle is directly proportional to its displacement from an equilibrium position and is directed towards that equilibrium position. This results in oscillatory motion characterized by sinusoidal time dependence, constant angular frequency, and a period that is independent of amplitude under ideal conditions.

Gravitational Force inside a Uniform Sphere

Within a uniformly dense sphere, such as an idealized Earth, the gravitational force experienced by a particle is proportional to its distance from the center. This arises from the fact that only the mass enclosed within a sphere of radius r contributes to the gravitational force at that point. For a particle at distance r from the center, the gravitational force can be expressed as F = - (mg/R) r, reflecting a linear restoring force analogous to Hooke's law.

Derivation of the Oscillation Period

The period of oscillation in simple harmonic motion is determined by the angular frequency, which for a linear restoring force is given by ? = ?(force constant/mass). In the context of a particle moving inside the Earth, identifying the force constant from F = - (mg/R) r leads to an angular frequency ? = ?(g/R), and the corresponding period T = 2??(R/g). This formulation demonstrates that the period is independent of the particle's mass and amplitude of the motion.

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