A satellite orbits a planet at a distance of 3.80 108 m. Assume that this distance is between the centers of the planet and the satellite and that the mass of the planet is 3.90 1024 kg. Find the period for the moon's motion around the earth. Express the answers in days.
1) find the circumference (c=2(3.14)r)
2(pi)(3.8E8)= 2387610417m = 1 orbit/rev
2) find velocity v=√(G*M)/(r)
v=√(6.67E-11*3.9E24)/(3.8E8)=827.38m/s
3)convert velocity into m/days; so if there is 86400 seconds in a day...
827.38*86400=69996090.7m/days
4)divide the orbit/revolution by m/day
2387610417m/69996090.7= 34.111 days :)
To find the period for the satellite's motion around the planet, we can use Kepler's third law of planetary motion, which states that the square of the period of revolution (T) is proportional to the cube of the semi-major axis of the orbit (r).
1. First, convert the given distance of 3.80 x 10^8 m to kilometers for consistency. There are 1000 meters in a kilometer, so the distance is 3.80 x 10^5 km.
2. Next, we need to convert the period from seconds to days. There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute. Therefore, there are 24 x 60 x 60 = 86400 seconds in a day.
3. Now, let's use the formula for Kepler's third law:
T^2 = k * r^3
where T is the period, r is the distance between the centers of the planet and the satellite, and k is a constant.
4. Rearrange the formula to solve for the period (T):
T^2 = k * r^3
T = sqrt(k * r^3)
5. Substitute the values into the formula:
T = sqrt(k * (3.80 x 10^5)^3)
6. To find the constant k, we need to convert the mass of the planet to solar masses. The mass of the Sun is approximately 1.989 × 10^30 kg.
k = (4π^2)/(G * M)
where G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2) and M is the mass of the planet.
7. Substitute the values into the formula:
k = (4π^2) / (6.67430 × 10^-11 * 3.90 x 10^24)
8. Calculate the value of k using a calculator and substitute it back into the period formula.
Finally, calculate the value of T in seconds and then convert it to days by dividing by 86400.
Note: The exact numerical value of the constant k will depend on the units used and any specific values provided in the problem.