Two objects have masses of 27 MG27MG and 13 MG13MG. How much does the gravitational potential energy between the objects change if the distance between them changes from 48 m48m to 90 m90m?

1 Answer
Aug 10, 2017

The gravitational potential energy will decrease by about 2.3xx10^(-4)N2.3×104N.

Explanation:

This seems to be a popular question.

The gravitational potential energy of two masses m_1m1 and m_2m2 is given by:

color(crimson)(U_g=-(Gm_1m_2)/r)Ug=Gm1m2r

where GG is the gravitation constant, m_1m1 and m_2m2 are the masses of the objects, and rr is the distance between them

The change in gravitational potential energy will then be:

DeltaU_g=(-(Gm_1m_2)/r)_f-(-(Gm_1m_2)/r)_iΔUg=(Gm1m2r)f(Gm1m2r)i

color(darkblue)(=>Gm_1m_2(1/r_i-1/r_f))

There are two ways I can see to interpret MG at the end of the mass values. The first case is that G is the gravitational constant and M is a variable, leaving the final answer with a variable.

The second is that MG is meant to be the unit "megagram" (notated Mg), which is 10^3"kg". I will interpret the question this way.

We have the following information:

  • |->m_1=27xx10^3"kg"
  • |->m_2=13xx10^3"kg"
  • |->r_i=48"m"
  • |->r_f=90"m"
  • |->"G"=6.67xx10^(-11)"Nm"//"kg"^2

Substituting in these values into the above equation:

Gm_1m_2(1/r_i-1/r_f)

=>(6.67xx10^(-11)"Nm"//"kg"^2)(27xx10^3"kg")(13xx10^3"kg")(1/(48"m")-1/(90"m"))

=2.276xx10^(-4)"N"

~~color(crimson)(2.3xx10^(-4)N)

That is, the gravitational potential energy will decrease by about 2.3xx10^(-4)N.

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Note: It may seem odd that we would state a potential energy as being negative, but this is due to the way this particular form of potential energy it is defined—it was chosen that U=0 when r=oo (when two objects are infinitely far apart), and so all the negative sign means is that the potential energy of the two masses at separation r is less than their potential energy at infinite separation. It is only the change in potential energy that has physical significance, which will be the same no matter where we place the zero of potential energy.