A particle of mass m describes the logarithmic spiral r=r0 e^b θ under a central force F directe

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A particle of mass m describes the logarithmic spiral r=r0...

A particle of mass $m$ describes the logarithmic spiral $r=r_{0} e^{b \theta}$ under a central force $F$ directed toward the center of force $O .$ Using Eq. $(12.35),$ show that $F$ inversely proportional to the cube of the distance $r$ from the particle to $O$.

Key Concepts

Inverse Cube Law

The inverse cube law indicates that the force experienced by the particle is inversely proportional to the cube of the radial distance from the center. In analyzing the motion along a logarithmic spiral under a central force, the form of the trajectory combined with the dynamics leads to the derivation of a force law that scales as 1/r³, highlighting a direct connection between the trajectory's geometry and the force's distance dependence.

Logarithmic Spiral Trajectory

A logarithmic spiral is a curve where the radial distance increases exponentially with the angle, typically expressed as r = r?e^(b?). This type of trajectory implies a particular relationship between the particle's position and the angle, which ultimately leads to a specific form of the acceleration terms in the equations of motion.

Polar Coordinates in Dynamics

Using polar coordinates is essential in problems involving central forces since the motion naturally separates into radial and angular components. The radial distance (r) and angular displacement (?) are used to formulate the equations of motion, and trajectories like spirals have compact representations that facilitate the analysis of how forces vary with distance.

Central Force Motion

Central force motion describes the behavior of particles that are acted upon by forces directed radially towards (or away from) a fixed point. In such cases, the force depends only on the distance from a center and not on the direction, leading to conservation of angular momentum. This property simplifies the analysis of the motion, as expressions in polar coordinates can be derived directly from the radial and angular components.

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