Gravitational Potential Energy. Kepler Laws

In these revision notes for Gravitational Potential Energy. Kepler Laws, we cover the following key points:

• Express the gravitational potential energy in terms of masses of objects and the distance between them
• Know the meaning of cosmic velocities and what they are used for
• Identify the three Kepler Laws
• Know how to relate Kepler Laws to rotational motion equations
• Know how to use Kepler Laws to study the Universe
• Know the meaning and formula of gravitational potential
• Know how gravitational potential is related to gravitational potential energy

Gravitational Potential Energy. Kepler Laws Revision Notes

Gravitational Potential Energy GPE (or U) represents the work done by gravitational field of one of the objects in the system (usually the largest) to bring the other object from the position r⃗ to zero, i.e. to bring it at the place where the first object is.

The equation of gravitational potential energy is

where M is the mass of the largest object and m is that of the smallest object, we obtain for the gravitational potential energy possessed by an object when it is at a linear distance R from the Earth.

Path independence principle of gravitational field states means the path is not important for the values of gravitational force and gravitational potential energy but only the initial and final positions of the object.

The first cosmic velocity or orbital velocity represents the least velocity required to make an object rotate around the Earth without falling on it. Its equation is

and for objects thrown from Earth surface it is about 7.9 km/s.

The second cosmic velocity or the escape velocity from Earth represents the minimum velocity to send an object away in space but still inside our solar system. Its equation is

and for objects thrown from Earth surface it is about 11.2 km/s.

If we want to launch an object from Earth with such a velocity that it leaves the Solar System and moves freely in deep space, we must apply the third cosmic velocity, or the escape velocity from the solar system. Its equation is

and for objects thrown from Earth surface it is about 41.9 km/s.

Johannes Kepler was the first who formulated a scientific-based theory to explain the planetary motion. This theory is based on three fundamental laws, known as Kepler Laws. They are:

a. First Kepler Law

This law is otherwise known as the Law of Orbits. It states that:

All planets move in elliptical (not circular) orbits, where the Sun is at one of the ellipse foci.

b. Second Kepler Law

This law is known as the Law of Areas and it derives from the principle of conservation of angular momentum. It states that:

Any line that connects a planet to the Sun "wipes out" equal surface areas in equal time intervals.

c. Third Kepler Law

This law is also known as the "Law of Periods". It states that:

The square of period of a planet revolution around the Sun is proportional to the cube of the greater semi axis.

This means that T² ~ a³. More precisely, we have:

where G is the gravitational constant and M is the mass of the Sun.

The quantity

is known as "gravitational potential". It represents the attracting ability of a celestial body. Since G and M are constants, this attracting ability depends only on the distance from the planet. Gravitational potential has the unit of square of velocity, [m²/s²].