Consider planet mass of 4.98 E 21 kilograms with a satellite orbitting it at distance of 3.11 E 6 meters from planet's center. Satellite has mass of 369 kilograms.
How long (inseconds) does it take the satellite to make one complete orbit around planet.
I just can't seem to get the correct answer - Please show me step by step.
Thanks
You need to set the gravitational force equal to the centripetal force. The period will be independent of the satellite's mass (m when it is small compared to the planet mass M.
GMm/R^2 = mV^2/R
m cancels out, so
GM/R = V^2
The period P is related to the Orbit radius R and velocity V by
V P = 2 pi R
Therefore
(2 pi R)^2/P^2 = GM/R
R^3 * (4 pi^2/GM)= P^2
G is the universal constant of gravity,
6.67*10^-11 m^3 kg^-1 s^-2
To calculate the time it takes for a satellite to make one complete orbit around a planet, you can use Kepler's third law of planetary motion. According to Kepler's third law:
T^2 = (4π^2 * R^3) / (G * M)
where T is the period of the orbit (time it takes for one complete orbit), R is the distance between the satellite and the planet's center, G is the gravitational constant, and M is the mass of the planet.
Let's break down the steps to calculate the time in seconds:
Step 1: Convert the given values to standard notation.
Mass of the planet (M) = 4.98 × 10^21 kilograms
Distance from the planet's center (R) = 3.11 × 10^6 meters
Mass of the satellite = 369 kilograms
Step 2: Convert the values to SI units.
The standard units used in physics are kilograms (kg), meters (m), and seconds (s). In this case, the given values are already in the correct units.
Step 3: Calculate the gravitational constant (G).
The gravitational constant (G) is a fundamental constant of nature and its value is approximately 6.67430 × 10^-11 m^3⋅kg^(-1)⋅s^(-2).
Step 4: Plug the values into the formula and solve for T^2.
T^2 = (4π^2 * R^3) / (G * M)
Step 5: Take the square root of T^2 to find T.
Now, let's calculate the time it takes for the satellite to make one complete orbit around the planet.
Step 1: Convert the given values to standard notation.
M = 4.98 × 10^21 kg
R = 3.11 × 10^6 m
m = 369 kg
Step 2: Calculate the gravitational constant (G).
G = 6.67430 × 10^-11 m^3⋅kg^(-1)⋅s^(-2)
Step 3: Plug the values into the formula and solve for T^2.
T^2 = (4π^2 * R^3) / (G * M)
T^2 = (4 * (3.14)^2 * (3.11 × 10^6)^3) / ((6.67430 × 10^-11) * (4.98 × 10^21))
Step 4: Calculate T.
T = sqrt(T^2)
By following these steps and plugging in the given values, you should be able to calculate the time it takes for the satellite to make one complete orbit around the planet.