Two objects have masses of #5 MG# and #7 MG#. How much does the gravitational potential energy between the objects change if the distance between them changes from #90 m# to #2000 m#?

Question

Two objects have masses of #5 MG# and #7  MG#. How much does the gravitational potential energy between the objects change if the distance between them changes from #90 m# to #2000 m#?

Olivia Askew · Accepted Answer

The gravitational potential energy will decrease by about #2.5xx10^(-5)N#.

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 as a factor 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: 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")(7xx10^3"kg")(1/(90"m")-1/(2000"m"))# #=2.48xx10^(-5)"N"# #~~color(crimson)(2.5xx10^(-5)N)# That is, the gravitational potential energy will decrease by about #2.5xx10^(-5)N#. #- - - - - - - - # 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.

Emily Abbate · Answer

undefined The change in gravitational potential energy between the objects is 1.22 × 10^14 Joules.

Nathan Ashcroft · Answer

undefined The change in gravitational potential energy ($ \Delta U $) between two objects can be calculated using the formula:

$$ \Delta U = -G \cdot \frac{m_1 \cdot m_2}{r_f} + G \cdot \frac{m_1 \cdot m_2}{r_i} $$

Where:
- $ G $ is the gravitational constant ($ 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $)
- $ m_1 $ and $ m_2 $ are the masses of the two objects in kilograms (1 MG = $ 10^6 $ kg)
- $ r_f $ is the final distance between the objects
- $ r_i $ is the initial distance between the objects

Converting the masses from megagrams (MG) to kilograms:
- 5 MG = $ 5 \times 10^6 $ kg
- 7 MG = $ 7 \times 10^6 $ kg

Given:
- $ m_1 = 5 \times 10^6 $ kg
- $ m_2 = 7 \times 10^6 $ kg
- $ r_i = 90 $ m
- $ r_f = 2000 $ m

Substituting the values into the formula:
$$ \Delta U = -\left(6.674 \times 10^{-11}\right) \cdot \frac{(5 \times 10^6) \cdot (7 \times 10^6)}{2000} + \left(6.674 \times 10^{-11}\right) \cdot \frac{(5 \times 10^6) \cdot (7 \times 10^6)}{90} $$

Calculating:
$$ \Delta U = -\left(6.674 \times 10^{-11}\right) \cdot \frac{35 \times 10^{12}}{2000} + \left(6.674 \times 10^{-11}\right) \cdot \frac{35 \times 10^{12}}{90} $$
$$ \Delta U = -\left(6.674 \times 10^{-11}\right) \cdot 17.5 \times 10^{9} + \left(6.674 \times 10^{-11}\right) \cdot 388.9 \times 10^{9} $$

$$ \Delta U = -1.16795 \times 10^{-1} + 2.59383 \times 10^{-1} $$
$$ \Delta U = 1.42588 \times 10^{-1} \, \text{J} $$

Therefore, the change in gravitational potential energy between the objects is $ 1.42588 \times 10^{-1} $ joules (J).