The gravitational force between two objects of mass m and M,...

The gravitational force between two objects of mass $m$ and $M,$ separated by a distance $r,$ has magnitude $G M m / r^{2},$ where $G=6.67 \times 10^{-11} \mathrm{m}^{3} \mathrm{kg}^{-1} \mathrm{s}^{-1}.$ \begin{equation} \begin{array}{l}{\text {Use the result of Exercise } 35 \text { to calculate the work required to place }} \\ {\text { a } 2000 \text { -kg satellite in an orbit } 1200 \mathrm{km} \text { above the surface of the earth. }} \\ {\text { Assume that the earth is a sphere of radius } R_{e}=6.37 \times 10^{6} \mathrm{m} \text { and }} \\ {\text { mass } M_{e}=5.98 \times 10^{24} \mathrm{kg} . \text { Treat the satellite as a point mass. }}\end{array} \end{equation}

The gravitational force between two objects of mass $m$ and $M,$ separated by a distance $r,$ has magnitude $G M m / r^{2},$ where $G=6.67 \times 10^{-11} \mathrm{m}^{3} \mathrm{kg}^{-1} \mathrm{s}^{-1}.$
\begin{equation}
\begin{array}{l}{\text {Use the result of Exercise } 35 \text { to calculate the work required to place }} \\ {\text { a } 2000 \text { -kg satellite in an orbit } 1200 \mathrm{km} \text { above the surface of the earth. }} \\ {\text { Assume that the earth is a sphere of radius } R_{e}=6.37 \times 10^{6} \mathrm{m} \text { and }} \\ {\text { mass } M_{e}=5.98 \times 10^{24} \mathrm{kg} . \text { Treat the satellite as a point mass. }}\end{array}
\end{equation}

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Key Concepts

Gravitational Force

This is the fundamental interaction attracting two masses together. Its magnitude is given by an inverse-square law, meaning it decreases with the square of the distance between the centers of the masses. Understanding this force is critical when analyzing the behavior of objects in a gravitational field, such as planets and satellites.

Gravitational Potential Energy

Gravitational potential energy represents the energy due to an object's position in a gravitational field. It depends inversely on the distance from the source of gravity, and changes in this energy correspond to the work required to move an object between different positions within the field.

Work Done Against Gravity

This concept involves calculating the energy needed to move an object in a gravitational field. Because the gravitational force changes with distance, the work required to move an object from one point to another is determined by the change in gravitational potential energy, often computed using integration.

Integration of Variable Forces

When dealing with forces that vary with position, such as gravity, integration is used to sum the incremental work contributions over a continuous path. This mathematical approach is essential for accurately finding the total work done in moving objects in non-uniform force fields.

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