A 1 048-kg satellite orbits the Earth at a constant altitude of 110-km.
A.How much energy must be added to the system to move the satellite into a circular orbit with altitude 210 km?
B.What is the change in the system's kinetic energy?
C.What is the change in the system's potential energy?
A.How much energy must be added to the system to move the satellite into a circular orbit with altitude 210 km?
B.What is the change in the system's kinetic energy?
C.What is the change in the system's potential energy?
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A. There are no nonconservative forces acting on the satellite, so the mechanical energy is conserved going from orbit 1 into orbit 2 except for the added energy. Note that E1 = PE1 + KE1 and E2 = PE2 + KE2 where E is the mechanical energy, PE is the gravitational potential energy, and KE is the kinetic energy.
We have that PE = mg(R + h) where R is the radius of the Earth (= 6.38 x 10^6 m), h is the height of the orbit and m is the mass of the satellite. We can assume g does not change by going from orbit 1 to orbit 2 since the height difference is very small. Also we have KE = 1/2 m v^2 ignoring any rotational effects. It is given that h1 = 110 km and h2 = 210 km. But what are v1 and v2?
The velocity v can be calculated using Newton's Law of Universal Gravitation. Since F = ma by the 2nd law of dynamics, then F = G m M / (R + h)^2 = m v^2 / (R + h) = m a, where G is a universal constant, m is satellite's mass, M is the Earth's mass, and v^2 / (R + h) is the centripetal acceleration for the circular orbit.
If you do a little algebra, you can find that v = sqrt(GM/(R + h)). So to put it all together, we have E1 + added energy = E2 which gives mg(R + h1) + 1/2 m GM/(R + h1) + added energy = mg(R + h2) + 1/2 m GM/(R + h2). Put in the values and calculate the added energy.
B. This is simply 1/2 m (v2)^2 - 1/2 m (v1)^2
C. This is simply mg(R + h2) - mg(R + h1)
I will leave the numerical calculations to yourself.
We have that PE = mg(R + h) where R is the radius of the Earth (= 6.38 x 10^6 m), h is the height of the orbit and m is the mass of the satellite. We can assume g does not change by going from orbit 1 to orbit 2 since the height difference is very small. Also we have KE = 1/2 m v^2 ignoring any rotational effects. It is given that h1 = 110 km and h2 = 210 km. But what are v1 and v2?
The velocity v can be calculated using Newton's Law of Universal Gravitation. Since F = ma by the 2nd law of dynamics, then F = G m M / (R + h)^2 = m v^2 / (R + h) = m a, where G is a universal constant, m is satellite's mass, M is the Earth's mass, and v^2 / (R + h) is the centripetal acceleration for the circular orbit.
If you do a little algebra, you can find that v = sqrt(GM/(R + h)). So to put it all together, we have E1 + added energy = E2 which gives mg(R + h1) + 1/2 m GM/(R + h1) + added energy = mg(R + h2) + 1/2 m GM/(R + h2). Put in the values and calculate the added energy.
B. This is simply 1/2 m (v2)^2 - 1/2 m (v1)^2
C. This is simply mg(R + h2) - mg(R + h1)
I will leave the numerical calculations to yourself.
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I'm so dumbfounded, I don't even know what to say. If you figure it out, I'm going to start praying to you.