Mechanical energy of satellite

In summary, the conversation discusses the calculation of the gravitational force exerted on a body by the Earth and the expression of the speed and mechanical energy of an artificial satellite in uniform circular motion at a specific height from the Earth's surface in terms of g, R, and m. The solution for the mechanical energy includes a negative sign due to the assumption that potential energy is zero at infinite distance.
  • #1
songoku
2,266
319

Homework Statement


the gravitational force exerted on a body of mass m by the Earth is GmM / r2

1. Express the speed of an artificial satellite which carries out uniform circular motion at height R from the surface of the Earth in terms of g and R

2. express the mechanical energy of the artificial satellite of (1) in terms of g, m, and R, where m is the mass of the artificial satellite and the potential energy is assumed to be zero when the distance r is infinite.

Homework Equations


Em = Ep + Ek


The Attempt at a Solution


1. done [ans : v =√(gR/2) ]

2.
Em = Ep + Ek = 0 + 1/2 mv2 = 1/2 m (gR/2) = mgR / 4

but the answer is - mgR / 4. Why is there negative sign?


thanks
 
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  • #2
It says, "assuming the potential energy is 0 when R is infinite". You assumed the potential energy was zero at the distance R. So there is a potential energy term you forgot to add in. As to why it is negative, the satellite has more potential energy when it is further away. This is clear if you think about an object at rest at infinity and falling inward. As it gains kinetic energy, it needs to lose potential energy so that total energy is conserved. So if the total energy is zero at infinity (since PE=0, and KE=0 since v=0), as it accelerates inward and KE increases positively, PE must increase negatively.
 
  • #3
phyzguy said:
It says, "assuming the potential energy is 0 when R is infinite". You assumed the potential energy was zero at the distance R. So there is a potential energy term you forgot to add in. As to why it is negative, the satellite has more potential energy when it is further away. This is clear if you think about an object at rest at infinity and falling inward. As it gains kinetic energy, it needs to lose potential energy so that total energy is conserved. So if the total energy is zero at infinity (since PE=0, and KE=0 since v=0), as it accelerates inward and KE increases positively, PE must increase negatively.

Oh, you're right. Got it now. Thanks a lot :smile:
 

1. What is mechanical energy?

Mechanical energy is the energy possessed by an object due to its motion or position. It can be in the form of kinetic energy (energy of motion) or potential energy (energy of position).

2. How is mechanical energy related to satellites?

Satellites have mechanical energy because they are in motion and have a specific position in orbit. The amount of mechanical energy a satellite has is determined by its mass, velocity, and position in relation to the Earth's gravitational pull.

3. What factors affect the mechanical energy of a satellite?

The mechanical energy of a satellite is affected by its mass, velocity, and distance from Earth. A satellite with a greater mass and higher velocity will have more mechanical energy, while a satellite farther from Earth will have less.

4. How is mechanical energy of a satellite calculated?

The mechanical energy of a satellite is calculated using the formula E=1/2mv^2-GMm/r, where E is the total mechanical energy, m is the mass of the satellite, v is its velocity, G is the gravitational constant, M is the mass of the Earth, and r is the distance between the satellite and the center of the Earth.

5. Can the mechanical energy of a satellite change over time?

Yes, the mechanical energy of a satellite can change over time. This can happen if the satellite's mass, velocity, or distance from Earth changes. Additionally, external forces such as atmospheric drag can also affect the mechanical energy of a satellite.

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