Question

Six artificial satellites complete one circular orbit around a space station in the same amount of time. Each satellite has mass m and radius of orbit L. The satellites fire rockets that provide the force needed to maintain a circular orbit around the space station. The gravitational force is negligible.Rank the net force acting on each satellite from their rockets.Rank from largest to smallest. To rank items as equivalent, overlap themA) m=200 kg and L= 5000mB) m=400 kg and L=2500mC)m=100kg and L=2500mD)m=100kg and L=10000mE)m=800kg and L=5000mF)m=300kg and L=7500m

Answer 1

Answer: The rank of the items is E,B,F,A,D,C..

Explanation:      

In the given problem, The satellites fire rockets that provide the force needed to maintain a circular orbit around the space station. The centripetal force is acting to keep up the continuous circular motion.

The expression for the centripetal force is as follows;

$F=\frac{m\times v^{2}}{r}$

Here, m is the mass of the object, v is the velocity and r is the radius.

In the given problem, the orbital period is constant.

$r\propto v$

Thus,

$F\propto m\times r$

In the given problem,  each satellite has mass m and radius of orbit L.

$r=L+R$

Calculate the net force acting on each satellite from their rockets for the part A.

$mr=m(L+R)$

Put m= 200 kg, R=6378 km and L=5000 m.

$mr=200(5000+6378)$

$mr=1.28\times 10^{9}kg\times m$

Calculate the net force acting on each satellite from their rockets for the part B.

$mr=m(L+R)$

Put m= 400 kg, R=6378 km and L=2500 m.

$mr=400(2500+6378)$

$mr=2.55\times 10^{9}kg\times m$  

Calculate the net force acting on each satellite from their rockets for the part C.

$mr=m(L+R)$

Put m= 100 kg, R=6378 km and L=2500 m.

$mr=100(2500+6378)$

$mr=6.38\times 10^{8}kg\times m$

Calculate the net force acting on each satellite from their rockets for the part D.

$mr=m(L+R)$

Put m= 100 kg, R=6378 km and L=10000 m.

$mr=100(10000+6378)$

$mr=6.39\times 10^{8}kg\times m$

Calculate the net force acting on each satellite from their rockets for the part E.

$mr=m(L+R)$

Put m= 800 kg, R=6378 km and L=5000 m.

$mr=800(5000+6378)$

$mr=5.11\times 10^{9}kg\times m$

Calculate the net force acting on each satellite from their rockets for the part F.

$mr=m(L+R)$

Put m= 300 kg, R=6378 km and L=7500 m.

$mr=300(7500+6378)$

$mr=1.92\times 10^{9}kg\times m$

Therefore, the rank of the items is E,B,F,A,D,C.