Does the period of a satellite in a circular orbit increase or decrease as its distance from earth increases?
Thanks :)
The period increases with distance.
P^2/R^3 remains constant.
The period of a satellite in a circular orbit increases as its distance from Earth increases. This can be understood by considering the basic principles of circular motion.
The period of an object in circular motion refers to the time it takes for the object to complete one full revolution around its orbit. In the case of a satellite orbiting Earth, the gravitational force between the satellite and Earth provides the centripetal force necessary for the satellite to stay in orbit.
The centripetal force required for circular motion is given by the equation:
F = (mv²) / r
Where:
F is the centripetal force,
m is the mass of the satellite,
v is the velocity of the satellite, and
r is the distance between the center of Earth and the satellite.
From this equation, it can be seen that the centripetal force is inversely proportional to the distance (r) between the satellite and Earth's center. As the distance increases, the centripetal force decreases.
Now, considering Newton's second law of motion (F = ma), we can equate the centripetal force to the gravitational force:
F = G(mM) / r²
Where:
G is the gravitational constant,
M is the mass of Earth, and
m is the mass of the satellite.
Combining these two equations, we get:
G(mM) / r² = (mv²) / r
Simplifying the equation, we find:
v² = (GM) / r
Now, we can see that the square of the orbital velocity (v²) is inversely proportional to the distance (r) between the satellite and Earth's center. As the distance increases, the orbital velocity decreases.
Finally, the period (T) of a satellite in circular orbit can be calculated using the relation:
T = (2πr) / v
Substituting the expression for v, we get:
T = (2πr) / √((GM) / r)
Simplifying further, we obtain:
T = 2π√((r³) / GM)
From this equation, it is evident that the period (T) is directly proportional to the square root of the cube of the distance (r) between the satellite and Earth's center. As the distance increases, the period of the satellite also increases.
In conclusion, as the distance from Earth increases, the period of a satellite in a circular orbit also increases.