given that the radius of the earth is 6.37×10^3km and the period of revolution of the moon round the earth is 27.3 days, calculate the average distance from the earth to the moon.
Well, you know the Gravitational Field Strength at Earth's surface, 9.8N/kg, and that this is reduced by distance squared, so at the moons location, it is...
field strength= 9.8N/kg * (re/d)^2 where re is the radius of Earth, and d is the distance from the EArth cneter to the Moon center.
This field strength is also equal to the acceleration falling toward EArth, so at the moon...
9.8(re/d)^2=V^2/d=(2PId/period)^2/d
9.8 re^2=4PI^2 d^3/period
Solve for d change period in days to seconds first, and re to meters
To calculate the average distance from the Earth to the Moon, we can use Kepler's third law. This law states that the square of the period of revolution of a celestial body around another is proportional to the cube of the semi-major axis of its elliptical orbit.
First, we need to convert the period of revolution from days to seconds:
27.3 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute = 2,360,320 seconds (approximately)
Now, we can use Kepler's third law to find the average distance (semi-major axis) from the Earth to the Moon. Let's denote the distance as 'd' (in km).
(T1^2 / T2^2) = (d1^3 / d2^3)
We know that T1 (period of revolution of the Moon around the Earth) is 2,360,320 seconds, and d2 (average distance from the Earth to the Moon) is what we want to find.
Let's assume T2 is the period of revolution of the Earth around the Sun, which is approximately 365.25 days (in seconds: 365.25 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute).
Therefore:
(2,360,320 seconds^2 / (365.25 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute)^2) = (6.37×10^3 km^3 / d2^3)
Solving for d2:
d2^3 = (6.37×10^3 km^3) * (365.25 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute)^2 / 2,360,320 seconds^2
Taking the cube root of both sides, we can find d2:
d2 = cube root [ (6.37×10^3 km^3) * (365.25 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute)^2 / 2,360,320 seconds^2 ]
Evaluating this expression will give us the average distance from the Earth to the Moon.