13.5: Central-Force Motion
The central force system operates by exerting a force on an object directed towards a fixed point, typically the origin, with the force magnitude determined by the object's distance from this fixed point. In the context of an object with mass 'm,' polar coordinates are employed to express the equation of motion. Notably, the azimuthal component of force is nonexistent in this system. A comprehensive rewrite and integration of this equation reveal that the product of the squared radial distance and angular velocity remains constant.
When the object undergoes angular displacement represented by dθ, it traces an area dA, signifying a constant areal velocity. Utilizing the chain rule of differentiation and considering the object's areal velocity, the radial component's first and second time derivatives can be expressed. Introducing a new dependent variable facilitates the simplification of the radial and angular components in the equation of motion.
By substituting the radial and angular velocity components into the equation of motion, a new formulation emerges, describing the trajectory of the object under the influence of the central force. This refined representation provides a more accessible understanding of the dynamics governing the motion of objects subjected to central forces.
