The planet Mars has a satellite, Deimos, which travels in an orbit of radius 2.346×107 m with a period of 1.26 days. Calculate the mass of Mars from this information.
Force on Deimos= centripetal force
GMm*Ms/r^2=Ms v^2/r
but v=2PIr/T and you know r and T
Solve for Mmars
To calculate the mass of Mars using the given information, we can apply Kepler's Third Law of Planetary Motion.
Kepler's Third Law states that the square of the period of revolution of a satellite is directly proportional to the cube of the average distance from the satellite to the planet it orbits.
Mathematically, it can be expressed as:
T^2 = (4π^2 / GM) * r^3
Where T is the period of revolution of the satellite, r is the average distance from the satellite to the planet, G is the gravitational constant, and M is the mass of the planet.
First, let's convert the given period of 1.26 days to seconds:
T = 1.26 days = 1.26 * 24 * 60 * 60 seconds = 108,864 seconds.
Next, let's substitute the given values into the formula:
108,864^2 = (4π^2 / G) * (2.346×10^7)^3
Now, we can rearrange the equation to solve for the mass of Mars (M):
M = (4π^2 / G) * (2.346×10^7)^3 / 108,864^2
The value of the gravitational constant (G) is approximately 6.67430 × 10^-11 m^3/(kg·s^2).
Plugging in the values and calculating, we can find the mass of Mars.