14.7:
Acceleration due to Gravity on Other Planets
Suppose an object released from rest takes t seconds to reach the ground, then its displacement h equals half g times t squared.
Near the Earth's surface, acceleration due to gravity of this object is estimated by measuring the time taken by the object to free-fall through a known height.
However, acceleration due to gravity on any other planet can be estimated by measuring its satellite's orbital period.
Any satellite of mass m orbiting a planet is always in a free-fall motion. Hence, equating the force on the satellite mg with the centripetal force mω2r, g can be expressed as ω2r.
Now, for one complete orbit around the planet, the satellite's angular velocity ω equals 2π divided by its orbital period.
Thus, gravitational acceleration on any other planet equals 4π2 times the distance between the satellite and the planet divided by the square of the satellite's orbital period.
Subsequently, once the value of the acceleration due to gravity of the planet becomes known, its mass mp can be determined.
14.7:
Acceleration due to Gravity on Other Planets
The gravitational acceleration of an object near the Earth's surface is called the acceleration due to gravity. It can be measured by conducting simple experiments on Earth. However, such an experiment is impossible to conduct on the surface of other planets.
Astronomical observations are thus used to measure the acceleration due to gravity on other planets. This can be determined by observing the effect of a planet's gravity on objects close to it. The crucial factor that helps in this calculation is that the acceleration is affected by the planet's gravitation but is independent of the object's mass.
Astronomical observations focus on the trajectory of a planet's satellite, more specifically the distance between the planet and its satellite. Combining the results with Newton's law of gravitation and Newton's laws of motion helps determine the planet's acceleration due to gravity.
The planet's mass can then be calculated if the acceleration due to gravity is determined, provided that the planet's radius is known from independent astronomical observations. The method relies on making multiple observations because there are no direct means to measure the mass of distant planets.
This text is adapted from Openstax, University Physics Volume 1, Section 13.2: Gravitation Near Earth's Surface.