The Work Done Against Gravity in Propelling an Object with Mass

question 125

Essay

The work done against gravity in propelling an object with mass m kg to an altitude of h m above the surface of the earth is given by $W = \int _ { 637 \times 10 ^ { 6 } } ^ { 637 \times 10 ^ { 6 } + h } \frac { G M _ { E } m } { r ^ { 2 } } d r$ where 6.37 $\times$ 10⁶ m is the radius of the earth, $G \approx 6.667 \times 10 ^ { - 11 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { kg } ^ { 2 }$ and $M _ { E } \approx 5.90 \times 10 ^ { 24 } \mathrm {~kg}$ is the mass of the earth.(a) Find the work required to launch a 1000-kilogram satellite vertically to an altitude of 1000 km.(b) The formula shows that the work is dependent on h. Show that $\int _ { 637 \times 10 ^ { 6 } } ^ { \infty } \frac { G M _ { Z } m } { r ^ { 2 } } d r$ is convergent. What is the physical significance of the fact that this improper integral is finite?

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Definitions:

Negotiating

The process of discussing something with one or more people in order to reach an agreement or compromise.

Project Liaison

A project liaison acts as a communicator and coordinator between different groups or departments involved in a project, ensuring smooth operation and information flow.

Linking Pins

A concept describing individuals who connect different groups or levels within an organization, facilitating coordination and communication.

Interdependence

The mutual reliance between two or more groups, individuals, or systems where the actions of one affect the outcomes of the others.