A new geosynchronous satellite needs an orbit of 35.7 km . The mass of the Earth is 5.97x 10^24 Kg and the radius of Earth 6.4 x 10^3 km. The mass of the satellite is 200 Kg. What velocity will it need to achieve that orbit?
A string 1.0 m long breaks when its tension is 100 N. What is the greatest speed at which it can be used to whirl a 1.0 kg stone? (Neglect the gravitational pull of the earth on the stone.)
To determine the velocity needed for the satellite to achieve the desired orbit, we can use the formula for the centripetal force acting on an object in circular motion:
F = (mv^2)/r
Where:
F is the gravitational force between the satellite and the Earth,
m is the mass of the satellite,
v is the velocity of the satellite, and
r is the radius of the orbit.
In this case, the gravitational force is given by the equation:
F = (G * m1 * m2) / r^2
Where:
G is the gravitational constant (6.67 x 10^-11 N*m^2/kg^2),
m1 is the mass of Earth, and
m2 is the mass of the satellite.
Setting these two equations equal to each other, we have:
(G * m1 * m2) / r^2 = (m * v^2) / r
Rearranging the equation to solve for v:
v = sqrt((G * m1 * m2) / r)
Now we can substitute the given values:
m1 = 5.97 x 10^24 kg
m2 = 200 kg
r = 35.7 km = 35.7 x 10^3 m
G = 6.67 x 10^-11 N*m^2/kg^2
Plugging these values into the equation, we can calculate the velocity needed for the satellite to achieve the desired orbit.