A cord of length h = 64m is used to connect an astronaut (m = 100 kg) to a spaceship (M > > m)which is orbiting near the earth's surface as shown in the figure. Assuming that the spaceship and the astronaut fall on a straight line from the earth's centre, then the tension in the cord is close to [R = 6400 km]

Answer

`3xx10^(-2)N`

Explanation

for satellite `(GMm_(1))/(R^(2))-T=m_(1)omega^(2)R` ..(i)
for astronaut `(GMm_(2))/((R+r)^(2))+T=m_(2)omega^(2)(R+r)` ..(ii)
Dividing `eq^(n)` (i) by (ii) `(gm_(1)-T)/((gm_(2)R^(2))/((R+r)^(2))+T)=(m_(1)Romega^(2))/(m_(2)(R+r)omega^(2))`
`impliesgm_(1)m_(2)[(R+r)-(R^(3))/((R+r)^(2))]=T[m_(1)R+m_(2)(R+r)]`
`impliesT=(m_(2)g)/(R)[(R+r)-(R^(3))/((R+r)^(2))](because(m_(2))/(m_(1))=0)`
`=m_(2)g[(1+(r)/(R))-(1-(2r)/(R))]=(3m_(2)gr)/(R)`
`=(3xx100xx10xx64)/(6400xx1000)=0.03N`