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

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

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Answer 1 · 2018-04-04T15:08:31

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

Explanation:

The work done when displacing these two masses, $W$ $W$ , is the amount that the gravitational potential changes. Note that

$W = \int_{48}^{18} F (r) d r$ $W=int_48^18F(r)dr$

Where $F (r)$ $F(r)$ is the gravitational force between the two objects as a function of the distance between the objects. From Newton's Law of Gravity

$F = \frac{G m_{1} m_{2}}{r^{2}}$ $F=(Gm_1m_2)/r^2$

where

$G =$ $G=$ the gravitational constant $\approx 6.674 \times 10^{- 11} m^{3} k g^{- 1} s^{- 2}$ $~~6.674xx10^-11m^3kg^-1s^-2$

$r =$ $r=$ the distance between the objects,

$m_{1} =$ $m_1=$ the mass of the first object = 45,000 kg, and

$m_{2} =$ $m_2=$ the mass of the second object = 36,000 kg.

Our integral now looks like

$W = \int_{48}^{18} \frac{G m_{1} m_{2}}{r^{2}} d r = G m_{1} m_{2} \int_{48}^{18} \frac{d r}{r^{2}}$ $W=int_48^18(Gm_1m_2)/r^2dr=Gm_1m_2int_48^18(dr)/r^2$

$W = - (G m_{1} m_{2}) \frac{1}{r}$ $W=-(Gm_1m_2)1/r$ evaluated from 48 to 18.

$W = - 6.674 \times 10^{- 11} \cdot 45000 \cdot 36000 (\frac{1}{18} - \frac{1}{48})$ $W=-6.674xx10^-11*45000*36000(1/18-1/48)$

$W \approx - 0.00375$ $W~~-0.00375$ 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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Two objects have masses of $45 M G$ $45 MG$ and $36 M G$ $36 MG$ . How much does the gravitational potential energy between the objects change if the distance between them changes from $48 m$ $48 m$ to $18 m$ $18 m$ ?

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Explanation:

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Two objects have masses of 45 MG45MG45 MG and 36 MG36MG36 MG. How much does the gravitational potential energy between the objects change if the distance between them changes from 48 m48m48 m to 18 m18m18 m?

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