Two objects have masses of 5 MG and 12 MG. How much does the gravitational potential energy between the objects change if the distance between them changes from 90 m to 270 m?

1 Answer
Aug 9, 2017

DeltaU_G=3.0xx10^(-5)N

Explanation:

The gravitational potential energy of two masses m_1 and m_2 is given by:

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

where G is the gravitation constant, m_1 and m_2 are the masses of the objects, and r 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)_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=5xx10^3"kg"
  • |->m_2=12xx10^3"kg"
  • |->r_i=90"m"
  • |->r_f=270"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)(5xx10^3"kg")(12xx10^3"kg")(1/(90"m")-1/(270"m"))

=2.96xx10^(-5)"N"

~~color(crimson)(3.0xx10^(-5)N)

That is, the gravitational potential energy will decrease by about 3.0xx10^(-5)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.