Mechanical Energy- Orbit

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  • #1
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Homework Statement


What is the total mechanical energy of a 380kg satellite in a circular orbit 3.0 Earth radii above the surface?


Homework Equations


W= E2- E1

E= 1/2 mv2- [itex]\frac{GmM}{r}[/itex]


The Attempt at a Solution



I'm not sure if the equations above are suitable to solve this or I just don't understand how to start the question.
 

Answers and Replies

  • #2
19
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Mechanical Energy is the sum of the potential energy, and the kinetic energy.

Emechanical = Ekinetic + EPotential

Since you're not close to the surface of the Earth, equating the potential energy to (mgh) is not applicable.

Epotential = (-GMm)/r

- r is the distance from the two objects centers,
- G is the gravitational constant of Earth
- M is the mass of Earth
- m is the mass of the satellite

Em = (1/2)mv2 - (GMm)/r

You have all the variables for the potential energy, and for the kinetic energy, you have the mass. So you need to solve for the velocity.

Since the mass of the satellite is just about negligible in relation to the earth, you can use the equation,

v = √(GM/r)

With that, you should be able to solve for the total mechanical energy.
 
  • #3
ehild
Homework Helper
15,543
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Homework Statement


What is the total mechanical energy of a 380kg satellite in a circular orbit 3.0 Earth radii above the surface?


Homework Equations


[STRIKE]W= E2- E1[/STRIKE]

E= 1/2 mv2- [itex]\frac{GmM}{r}[/itex]


The Attempt at a Solution



I'm not sure if the equations above are suitable to solve this or I just don't understand how to start the question.

The equation is good, you need to find the speed of the satellite.


The satellite travels along a circle with speed v. What is the radius of the orbit? What is its centripetal acceleration? What force provides the centripetal force?

ehild
 
  • #4
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The equation is good, you need to find the speed of the satellite.


The satellite travels along a circle with speed v. What is the radius of the orbit? What is its centripetal acceleration? What force provides the centripetal force?

ehild

so,

Fc= mac
(GmM)/RE2=(mv2)/RE
v2=(GM)/RE

where RE is the radius of the earth


then,

E= K+ UG where K=1/2(mv2) and UG=-(GmM)/r2 and r= 2.55x107m (equal to 4 earth radii)

so,

E= 1/2(mv2) - (GmM)/r2
E= 1/2(GM)/RE) - (GM)/r2
E= 15633739.23 J
 
  • #5
ehild
Homework Helper
15,543
1,913
so,

Fc= mac
(GmM)/RE2=(mv2)/RE
v2=(GM)/RE

where RE is the radius of the earth

Why do you calculate the speed of the satellite with the radius of Earth?? You know that the radius of the the circular orbit is 4 Earth-radius.

ehild
 
  • #6
27
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Why do you calculate the speed of the satellite with the radius of Earth?? You know that the radius of the the circular orbit is 4 Earth-radius.

ehild

so it would just be

E= 1/2(GM/r) - (GM)/r
E= -7819322.353 J
 
  • #7
gneill
Mentor
20,925
2,867
so it would just be

E= 1/2(GM/r) - (GM)/r
E= -7819322.353 J

You haven't included the mass of the satellite. So as it stands so far, what you've calculated is the Specific Mechanical Energy (energy per kg).
 
  • #8
27
0
You haven't included the mass of the satellite. So as it stands so far, what you've calculated is the Specific Mechanical Energy (energy per kg).

would it be,

E= K + UG
E= 1/2 (mv2) - (GmM)/r
E= 1/2 m(GM/r) - (GmM)/r
E= 2.377074x1010 J
 
  • #9
gneill
Mentor
20,925
2,867
would it be,

E= K + UG
E= 1/2 (mv2) - (GmM)/r
E= 1/2 m(GM/r) - (GmM)/r
E= 2.377074x1010 J

No. The formula's okay, but something went wrong in the execution.

Your previous value for the specific energy was good. Just multiply that by the mass of the satellite!
 

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