Two objects have masses of #4 MG# and #7 MG#. How much does the gravitational potential energy between the objects change if the distance between them changes from #320 m# to #450 m#?

Question

Two objects have masses of #4 MG# and #7  MG#. How much does the gravitational potential energy between the objects change if the distance between them changes from #320 m# to #450 m#?

Amelia Adams · Accepted Answer

Change in gravitational potential energy
#=1.69xx10^-6J# rounded to two decimal places

Gravitational potential energy #PE_g# between two objects of masses #m_1 and m_2# is given by the relation #PE_g = − (Gm_1m_2)/r#, where #r# is the separation between their centres and #G# is the universal gravitational constant (#6.67 xx 10^(−11) m^3 kg^-1 s^-1#) As the distance between their centres changes from #r_I=320m# to #r_F=450m#, the change in gravitational potential energy can be found from #Delta PE_g=− (Gm_1m_2)/r_F-(− (Gm_1m_2)/r_I)# #=− (Gm_1m_2)[1/r_F-1/r_I]# Inserting the given values, mass #1Mg=10^3kg# #=-(6.67 xx 10^(−11)xx4xx10^3xx7xx10^3)[1/450-1/320]# or #=-(186.76 xx 10^(−5))xx(-9.02 dot7xx10^-4)# #=1.69xx10^-6J# rounded to two decimal places

Charles Aeschliman · Answer

undefined To calculate the change in gravitational potential energy between the two objects, we can use the formula for gravitational potential energy:

ΔU = G * (m1 * m2) * (1/r2 - 1/r1)

Where:
ΔU is the change in gravitational potential energy,
G is the gravitational constant (6.674 × 10^-11 N m^2/kg^2),
m1 and m2 are the masses of the two objects (in kilograms),
r1 and r2 are the initial and final distances between the objects (in meters).

Using the given masses and distances:
m1 = 4 MG = 4 × 10^6 kg,
m2 = 7 MG = 7 × 10^6 kg,
r1 = 320 m,
r2 = 450 m.

Plugging these values into the formula:

ΔU = (6.674 × 10^-11 N m^2/kg^2) * (4 × 10^6 kg * 7 × 10^6 kg) * (1/450 m - 1/320 m)

Calculating the values inside the parentheses first:

1/450 m ≈ 0.0022222 m^-1,
1/320 m ≈ 0.003125 m^-1.

Now, plug these values back into the equation:

ΔU = (6.674 × 10^-11 N m^2/kg^2) * (4 × 10^6 kg * 7 × 10^6 kg) * (0.0022222 m^-1 - 0.003125 m^-1)

Now, calculate the difference in the reciprocal distances:

0.0022222 m^-1 - 0.003125 m^-1 ≈ -0.0009028 m^-1.

Now, plug this value back into the equation:

ΔU = (6.674 × 10^-11 N m^2/kg^2) * (4 × 10^6 kg * 7 × 10^6 kg) * (-0.0009028 m^-1)

Calculate the final result:

ΔU ≈ -1.758 × 10^9 J.

Therefore, the change in gravitational potential energy between the objects is approximately -1.758 × 10^9 joules.