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Derive an expression for the gravitational potential energy U(r) of the object-earth system as a function of the object's distance from the center of the earth. Take the potential energy to be zero when the object is at the center of the earth.

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Answer:

[tex]U(r)=-\frac{Gm_Emr^2}{2R^3_E}[/tex]

Explanation:

We are given that

Gravitational force=[tex]F_g=\frac{Gm_Emr}{R^3_E}[/tex]

r=0,U(0)=0

We know that

Gravitational potential energy=[tex]-\int F_gdr[/tex]

[tex]U(r)=-\int\frac{Gm_Emr}{R^3_E}dr[/tex]

[tex]U(r)=-\frac{Gm_Em}{R^3_E}\times \frac{r^2}{2}+C[/tex]

Substitute r=0 ,U(0)=0

[tex]0=0+C[/tex]

[tex]C=0[/tex]

Substitute the value

[tex]U(r)=-\frac{Gm_Emr^2}{2R^3_E}[/tex]

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