How to calculate work

Question

What is the work done by the force of gravity on a particle of mass $m$ as it moves radially from $7500 ~\text{km}$ to $9400 ~\text{km}$ from the center of the earth?

Answer 1

By the conservation of energy, the amount of work done by working against gravity is the difference in potential energy. That is, the difference in $$ -\dfrac{GMm}{r}\tag{1} $$ between the two points. If we know that $$ \frac{GM}{r^2}=9.80665 \text{ m/sec}^2\quad\text{at}\quad r=6.371\times10^6\text{ m}\tag{2} $$ we get that $GM=3.980\times10^{14}\text{ m}^3\text{/sec}^2$. Now just plug $m$, $r$, and $GM$ into $(1)$.

Answer 2

Hint: $$W = \int F \cdot dx = \int -\frac{GMm}{r^2} \cdot dr = -GMm\int\frac{1}{r^2} \ dr$$

where $M$ denotes the mass of the Earth and $m$ denotes the mass of the object. Integrate and plug in your bounds for the position.