A satellite used in a cellular telephone network has a mass of 2010kg and is in a circular orbit at a height of 770km above the surface of the earthTake the gravitational constant to be G = 6.67×10−11N⋅m2/kg2 , the mass of the earth to be me = 5.97×1024kg , and the radius of the Earth to be re = 6.38×106m .
What fraction is this of the satellite's weight at the surface of the earth?
Take the free-fall acceleration at the surface of the earth to be g = 9.80m/s2
To find the fraction of the satellite's weight at the surface of the earth, we need to compare the gravitational force acting on the satellite in its orbit to the weight of the satellite at the surface of the earth.
First, let's find the weight of the satellite at the surface of the earth:
Weight = mass x acceleration due to gravity
Weight = 2010 kg x 9.80 m/s^2
Weight = 19698 N
We can calculate the gravitational force acting on the satellite in its orbit using the formula:
Gravitational Force = (G x Mass of Earth x Mass of Satellite) / (Radius of Earth + Height)^2
Gravitational Force = (6.67x10^-11 N⋅m^2/kg^2) x (5.97x10^24 kg) x (2010 kg) / (6.38x10^6 m + 770000 m)^2
Gravitational Force = 1.9386x10^9 N
To find the fraction of the satellite's weight, we divide the gravitational force by the weight:
Fraction of weight = Gravitational Force / Weight
Fraction of weight = (1.9386x10^9 N) / (19698 N)
Fraction of weight = 98390
Therefore, the weight of the satellite in its orbit is approximately 98,390 times less than its weight at the surface of the Earth.
To find the fraction of the satellite's weight at the surface of the Earth, we first need to calculate the weight of the satellite at that height above the Earth's surface.
The weight of an object is given by the formula:
Weight = Mass × Acceleration due to gravity
At the surface of the Earth, the weight of an object can be calculated using the following formula:
Weight_surface = Mass × g
where g is the acceleration due to gravity at the surface of the Earth, which is given as 9.80 m/s^2.
Using the given mass of the satellite (2010 kg), we can calculate its weight at the surface of the Earth:
Weight_surface = 2010 kg × 9.80 m/s^2
Now, let's calculate the weight of the satellite at its orbital height.
The acceleration due to gravity at an altitude is given by the formula:
g_altitude = (G × me) / (r + altitude)^2
where G is the gravitational constant, me is the mass of the Earth, r is the radius of the Earth, and altitude is the height above the Earth's surface. In this case, altitude is given as 770 km.
Plugging in the given values:
g_altitude = (6.67×10^(-11) N⋅m^2/kg^2 × 5.97×10^24 kg) / (6.38×10^6 m + 770×10^3 m)^2
Now we can calculate the weight of the satellite at its orbital height:
Weight_altitude = Mass × g_altitude
Weight_altitude = 2010 kg × g_altitude
Now, to find the fraction of the satellite's weight at the surface of the Earth, we divide the weight at the satellite's orbital height by the weight at the surface:
Fraction = Weight_altitude / Weight_surface
By plugging in the calculated values, you will arrive at the final answer.
Tehe
(Gm1m2)/r^2
so
(6.67x10^-11)(2010)(5.97x10^24)/[(6.38x10^6)+770000]^2
note: 770km = 770000m