Two objects have masses of #45 MG# and #36 MG#. How much does the gravitational potential energy between the objects change if the distance between them changes from #48 m# to #18 m#?

Question

Two objects have masses of #45   MG# and #36    MG#. How much does the gravitational potential energy between the objects change if the distance between them changes from #48  m# to #18  m#?

Lily Adams · Accepted Answer

The objects have approximately 0.00375 FEWER Joules of gravitational potential energy after they are allowed to come closer together.

The work done when displacing these two masses, #W#, is the amount that the gravitational potential changes. Note that #W=int_48^18F(r)dr# Where #F(r)# is the gravitational force between the two objects as a function of the distance between the objects. From Newton's Law of Gravity #F=(Gm_1m_2)/r^2# where #G=# the gravitational constant #~~6.674xx10^-11m^3kg^-1s^-2# #r=# the distance between the objects, #m_1=# the mass of the first object = 45,000 kg, and #m_2=# the mass of the second object = 36,000 kg. Our integral now looks like #W=int_48^18(Gm_1m_2)/r^2dr=Gm_1m_2int_48^18(dr)/r^2# #W=-(Gm_1m_2)1/r# evaluated from 48 to 18. #W=-6.674xx10^-11*45000*36000(1/18-1/48)# #W~~-0.00375# Joules Because this is negative, the objects have 0.00375 FEWER Joules of gravitational potential energy after they are allowed to come closer together.

Xavier Adame · Answer

undefined To calculate the change in gravitational potential energy, you can use the formula:

ΔU = -G * (m1 * m2) * (1/r2 - 1/r1)

Where:
ΔU = change in gravitational potential energy
G = gravitational constant (6.674 × 10^-11 N m^2/kg^2)
m1 and m2 = masses of the objects (in this case, 45 MG and 36 MG)
r1 and r2 = initial and final distances between the objects (in this case, 48 m and 18 m)

Plugging in the values:

ΔU = -6.674 × 10^-11 * (45 * 36) * (1/18^2 - 1/48^2)
ΔU ≈ -6.674 × 10^-11 * (45 * 36) * (1/324 - 1/2304)
ΔU ≈ -6.674 × 10^-11 * (45 * 36) * (0.00308642 - 0.00043403)
ΔU ≈ -6.674 × 10^-11 * (45 * 36) * 0.00265239
ΔU ≈ -6.674 × 10^-11 * 1620 * 0.00265239
ΔU ≈ -0.04419789 J

Therefore, the change in gravitational potential energy between the objects is approximately -0.0442 joules.