Find the energy necessary to put 5kg , initially at rest on Earth's surface, into geosynchronous orbit.

Do I use the formula U=GMm/r

G=6.67*10^-11
M=2.5*10^24
m=5
r=41000

Is this sort of right, any help would be greatly appreciated.

The energy of the mass m in such an orbit is

E = P.E. + K.E.
...= GM m / (r - R) + m v^2 /2 ........(1)
v is the linear velocity of the mass m in gs orbit of radius r
This given by ,
GM m / r^2 = m v^2 / r
v^2 = GM / r ..........................(2)
Put this value in (1)
E = GM m / (r - R) + GM m / 2 r = ........(3)

For a gs orbit the angular velocity of the mass m must be same as that of the earth's rotation on its axis.
ω = 2π / T = 2 π / 24 x 3600 = 7.272 x10^-5 rad/s

From (2),
GM / r = r^2 ω^2
r^3 = GM / ω^2
G = 6.67x10^-11
M = 6x10^24 kg
That gives r , the radius of the gs orbit from the center of the earth, as
r = 4.23x10^7 m

R = 6.4 x10^6 m (radius of the earth)
r - R = (42.3 - 6.4)x10^6 = 35.9 x10^6 m

From (3) you can obtain the energy of the mass m = 5 kg as ,
E =

Yes, you are on the right track! To calculate the energy necessary to put an object into geosynchronous orbit, you can use the formula for gravitational potential energy:

U = GMm/r

where:
- U is the gravitational potential energy
- G is the gravitational constant (approximately 6.67 x 10^-11 Nm^2/kg^2)
- M is the mass of the Earth (approximately 5.98 x 10^24 kg)
- m is the mass of the object (5 kg)
- r is the distance between the center of the Earth and the object (41000 km or 4.1 x 10^7 m).

So, substituting the values into the formula:

U = (6.67 x 10^-11 Nm^2/kg^2) * (5.98 x 10^24 kg) * (5 kg) / (4.1 x 10^7 m)

Calculating this expression will give you the gravitational potential energy required to put the 5 kg object into geosynchronous orbit.

Yes, you are on the right track. To find the energy necessary to put an object into geosynchronous orbit, you can use the formula for gravitational potential energy.

The formula you mentioned, U = GMm/r, is indeed the correct formula for gravitational potential energy. However, please note that this formula calculates the potential energy between two objects separated by a distance r, where G is the gravitational constant, M is the mass of one of the objects, and m is the mass of the other object.

In this case, you are trying to find the energy necessary to put a 5kg object into geosynchronous orbit around the Earth. Geosynchronous orbit is generally about 35,786 kilometers above the Earth's surface, which is approximately 41,000 kilometers from the center of the Earth.

To calculate the gravitational potential energy, you need to substitute the correct values into the formula:

G = 6.67 * 10^-11 (gravitational constant)
M = 5.972 * 10^24 (mass of the Earth)
m = 5 (mass of the object in kilograms)
r = 41000 (distance from the center of the Earth to geosynchronous orbit)

Plugging these values into the formula, you get:

U = (6.67 * 10^-11) * (5.972 * 10^24) * 5 / 41000

Now, you can calculate the value of U:

U = 2.44 * 10^9 Joules

Therefore, the energy necessary to put a 5kg object into geosynchronous orbit around the Earth is approximately 2.44 * 10^9 Joules.