A communcations satellite of mass 200kg is in circular orbit of radis 4x10^4 km measured from the center of the earth. What is the gravitational force on the satellite and what fraction is this of its weight on the surface of the earth?
On the surface of the earth, the weight weight would be
W = M*g = 200*9.8 = 1960 Newtons
The Earth's radius is 6370 km. Your orbit radius is 40,000 km. The weight is proportional to 1/R^2, so at 40,000 km the weight is
1960*(6370/40,000)^2 = 49.7 Newtons
To calculate the gravitational force on the satellite and the fraction of its weight on the surface of the Earth, you will need to use the equation for gravitational force:
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force,
G is the gravitational constant (6.67430 × 10^-11 N m^2/kg^2),
m1 and m2 are the masses of the two objects (in this case, the satellite and the Earth),
r is the distance between the centers of the two objects.
To find the gravitational force, we need to calculate the mass of the Earth. The mass of the Earth is approximately 5.972 × 10^24 kg.
Now we can calculate the gravitational force on the satellite:
F = (G * m_satellite * m_earth) / r^2
Given:
m_satellite = 200 kg
m_earth = 5.972 × 10^24 kg
r = 4 × 10^7 m (converted from km to m)
Plugging in the values:
F = (6.67430 × 10^-11 N m^2/kg^2 * 200 kg * 5.972 × 10^24 kg) / (4 × 10^7 m)^2
Now we can solve for F.
After calculating, the gravitational force on the satellite is approximately 261.68 N.
To calculate the fraction of the satellite's weight, we need to compare it to the weight on the surface of the Earth.
The weight on the surface of the Earth is given by:
W = m * g
Where:
W is the weight,
m is the mass of the satellite,
g is the acceleration due to gravity on the surface of the Earth (approximately 9.8 m/s^2).
Plugging in the values:
W = 200 kg * 9.8 m/s^2
Simplifying, we find that the weight on the surface of the Earth is 1960 N.
To find the fraction of the satellite's weight, we divide the gravitational force by the weight on the surface of the Earth:
Fraction = F / W
Plugging in the values:
Fraction = 261.68 N / 1960 N
Simplifying, we find that the fraction of the satellite's weight on the surface of the Earth is approximately 0.1337, or about 13.37%.