Question

A sky laboratory of mass $2\times {10}^{3}kg$ has to be lifted from one circular orbit of radius $2R$ into another circular orbit of radius $3R$.The minimum energy required, if the radius of earth $R=6.37\times {10}^{6}m$ and $g=9.8m{s}^{-2}$. is

• A. $1.04\times {10}^{10}J$
• B. $2.04\times {10}^{10}J$
• C. $1.04\times {10}^{5}J$
• D. $2.04\times {10}^{5}J$

Answer 1

The correct option is A $1.04\times {10}^{10}J$

For an orbitting satellite, the gravitational force provides the centripetal force:

$\frac{m{v}^2}{R}=\frac{GMm}{R^2}\Rightarrow \frac{1}{2}m{v}^2=\frac{GmM}{2R}$.

Now, by energy conservation, we have:

$\frac{1}{2}m{v}_1^2-\frac{GmM}{R_1}+W_1=\frac{1}{2}m{v}_2^2-\frac{GmM}{R_2}\Rightarrow -\frac{GMm}{2R_1}+W_1=-\frac{GMm}{2R_2}\Rightarrow W_1=GMm(\frac{1}{{2R}_1}-\frac{1}{{2R}_2})R_1=2R,R_2=3R\Rightarrow W_1=\frac{GMm}{12R}=\frac{mgR}{12}=1.04\times {10}^{10}J$

where $W_1$ is the work done.