A chord of length $64 m$ is used to connect a $100 k g$ astronaut to a spaceship whose mass is much larger than that of the astronaut. Estimate the value of the tension in the cord. Assume that the spaceship is orbiting near earth surface. Also assume that the spaceship and the astronaut fall on a straight line from the earth centre. The radius of the earth is $6400 k m$ .

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Solution

The tension in the string $T$ is given as, $T + F_{g r} = m a_{r}$
$\Rightarrow T + \frac{G M m}{(R + l)^{2}} = m (R + l) ω^{2}$
$\Rightarrow T = m (R + l) ω^{2} - \frac{G M m}{R^{2}} {(\frac{R}{R + l})}^{2}$
$= m (R + l) \frac{g}{R} - m g {[1 + \frac{l}{R}]}^{- 2} = \frac{3 m g l}{R}$
$= \frac{3 \times 10 \times 10 \times 64}{6400 \times 10^{3}} = 3 \times 10^{- 2} N$ .

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