Understanding Conservative Forces and Kepler’s Laws of Planetary Motion
Block 1
Conservative Forces
For many types of forces, the work done to move a body between two points depends on the path followed (e.g., frictional forces). However, this is not the case for a particular type of force called conservative forces. A conservative force is capable of returning the work done against it. These forces are characterized by performing work that only depends on the initial and final positions, not the path taken. Consequently, when work is done in a closed path, the net work is zero. This property is exhibited by central forces. The fields in which these forces act are, in turn, called conservative fields. Considering that the performance of work is accompanied by a change in energy, we can define a new type of energy associated with position. This energy, called potential energy, depends only on the coordinates and is such that the work done by the conservative force equals the difference between the initial and final values of this potential energy.
Gravitational Potential Energy
To derive the expression for gravitational potential energy, we calculate the work done by gravity to move an object from infinity to any point in the field at a distance r from the object creating the field:
(The above result is obtained by applying the properties of definite integrals). Substituting, we arrive at the expression: if we set the potential energy source (zero) at infinity:
This is the value of the potential energy of an object of mass m’ located at a distance r from an object of mass m that creates the field.
The potential energy is always negative and increases as one moves away from the surface of the Earth.
Kepler’s Laws of Planetary Motion
Kepler’s First Law
All planets move in elliptical orbits around the Sun, with the Sun located at one focus of the ellipse.
Since the angular momentum vector L is constant in the system, its magnitude, direction, and sense do not change (it is oriented perpendicular to the ecliptic plane, upward, in a counterclockwise direction). Therefore, the plane of the orbit does not change.
Kepler’s Second Law
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. In other words, the areal velocity remains constant.
Proof: In a time dt, a planet travels through a space:
Looking at the drawing, the area of the triangle that is formed is given by:
Remember that the vector product represents the area of the parallelogram.
We can see that the swept area depends on the mass, the magnitude of the angular momentum, and time. The mass is constant during displacement, and the angular momentum L is constant, as mentioned in the first law, since gravitational forces are central forces and there are no external forces. Therefore, for the same period, the swept area will be the same.
This indicates that planets move faster when they are at perihelion (the point in their orbit closest to the Sun) than at aphelion (the point farthest from the Sun) because they must sweep the same area even when the radius is smaller.
Kepler’s Third Law
The squares of the periods of revolution of two planets around the Sun are proportional to the cubes of the semi-major axes of their orbits. T2 = kR3