Two identical planets, each of mass M, have their centers a distance D apart. Halfway between them, at gravitational equilibrium, lies a satellite of mass m. The satellite is then displaced a small distance and released, after which it undergoes SHM.
a) In what direction must the satellite be displaced: along the line joining the two planets, or perpendicular to that line?
b) What is the period of the satellite’s SHM?
a) perpendicular. The equilibrium is unstable in the other direction.
b) P = 2 pi sqrt(k/m) for SHM
k is the restoring force per unit perpendicular displacement. m is the satellite's mass
For a small displacement y (<<D), the restoring force is
F = 2*GMm/D^2*[y/(D/2)], so
k = F/y = 4GMm/D^3
P = 4 pi sqrt(GM/D^3)
M is the planet's mass and G is the universal constant of gravity.
a) To determine the direction in which the satellite must be displaced, we need to consider the stability of equilibrium. In this case, the equilibrium position of the satellite is halfway between the two planets, where the gravitational forces from both planets on the satellite cancel out. If the satellite is displaced perpendicular to the line joining the two planets, it will experience a net force towards the equilibrium position, resulting in a stable equilibrium. On the other hand, if the satellite is displaced along the line joining the two planets, it will experience a net force away from the equilibrium position, leading to an unstable equilibrium. Therefore, the satellite must be displaced perpendicular to the line joining the two planets.
b) To determine the period of the satellite's Simple Harmonic Motion (SHM), we can use the concept of gravitational force and apply Hooke's law.
1. First, let's determine the force acting on the satellite when it is displaced from the equilibrium position. The gravitational force between the satellite and each planet can be calculated using Newton's law of universal gravitation:
F1 = G * (m * M) / (r1^2) (force due to the first planet)
F2 = G * (m * M) / (r2^2) (force due to the second planet)
Where G is the gravitational constant, m is the mass of the satellite, M is the mass of each planet, r1 is the distance of the satellite from the first planet, and r2 is the distance of the satellite from the second planet.
2. Since the two forces acting on the satellite are equal in magnitude, we have:
F1 = F2
G * (m * M) / (r1^2) = G * (m * M) / (r2^2)
r1^2 = r2^2
r1 = r2
This implies that the distances r1 and r2 are equal, which means the satellite is equidistant from both planets.
3. Using this result, we can derive an expression for the period of SHM. Since the motion is along a straight line (perpendicular to the line joining the two planets), it can be approximated as simple harmonic motion.
The force acting on the satellite can be treated as a restoring force, given by Hooke's law:
F = -k * x
Where k is the force constant and x is the displacement of the satellite from the equilibrium position.
4. Equating the gravitational force to the restoring force:
G * (m * M) / (r1^2) = -k * x
G * (m * M) / (r^2) = -k * x (since r1 = r2 = r)
x = -((G * m * M) / (k * r^2))
5. The period of SHM can be calculated using the formula:
T = 2π * sqrt(m / k)
Substituting the value of x:
T = 2π * sqrt((m * r^2) / (G * M))
Therefore, the period of the satellite's SHM is given by 2π * sqrt((m * r^2) / (G * M)).