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#?

Answer 1

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.

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Answer 2

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.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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