Show that theoritically the period of rotation of a satelite which circle a planet at negligible distance from its surface depend only on the density of the planet
To show theoretically that the period of rotation of a satellite circling a planet at a negligible distance from its surface depends only on the density of the planet, we can start by considering the gravitational force that acts on the satellite.
The gravitational force between the satellite and the planet is given by Newton's law of universal gravitation:
F_gravity = (G * m_s * m_p) / r^2
Where:
- F_gravity is the gravitational force
- G is the gravitational constant
- m_s is the mass of the satellite
- m_p is the mass of the planet
- r is the distance between the satellite and the center of the planet
Now, let's analyze the other forces acting on the satellite. At a negligible distance from the planet's surface, the centripetal force required to keep the satellite in circular motion is provided by the gravitational force:
F_centripetal = (m_s * v^2) / r
Where:
- F_centripetal is the centripetal force
- v is the velocity of the satellite
By equating the gravitational force to the centripetal force, we get:
(G * m_s * m_p) / r^2 = (m_s * v^2) / r
By canceling out the mass of the satellite and rearranging the equation, we have:
v^2 = (G * m_p) / r
Now, the period of rotation (T) can be defined as the time it takes for the satellite to complete one full rotation:
T = (2 * pi * r) / v
Substituting for v from our previous equation:
T = (2 * pi * r) / sqrt((G * m_p) / r)
Simplifying further:
T = 2 * pi * sqrt(r^3 / (G * m_p))
In this equation, we can see that the period of rotation (T) is dependent on the radius (r) and the mass of the planet (m_p). There is no direct dependence on the density of the planet. However, the density (ρ_p) of the planet can be related to its mass and radius as follows:
m_p = ρ_p * V_p
Where V_p is the volume of the planet. Since V_p is directly proportional to r^3, we can substitute for m_p in our equation:
T = 2 * pi * sqrt(r^3 / (G * ρ_p * V_p))
The density (ρ_p) cancels out:
T = 2 * pi * sqrt(r^3 / (G * V_p))
Therefore, we can conclude that the period of rotation of a satellite circling a planet at a negligible distance from its surface depends only on the mass and radius (or equivalently, the volume) of the planet, but not directly on the density of the planet.