two spacecraft

Recently, two spacecraft successfully landed on two small planets labeled α and β. Both ships were equipped with sensitive sensors that measured the basic parameters of the asteroids. The sensors found that the day α took six times longer on planet α than on planet β, and they also found that the radius α of the planet α was four times that of the planet β.

After a while, however, both sensors broke due to excessive centrifugal force acting on them. Find out which of the sensors was subjected to the greater centrifugal force if you know that the sensor measuring on planet α weighed mα = 9 kg, while the sensor on the other planet weighed mβ = 1 kg. The relationship applies to the centrifugal force on the surface of any planet.

Fo = mv² / r,

where m is the mass of the considered sensor, v is the circumferential velocity of the planet given by its rotation, and r is the radius of the given planet.

Result

α

β

Step-by-step explanation:

$$\begin{gathered} \alpha :T=6d,R=4r,m=9kg \\ \beta :T=d,R=r,m=1kg \\ F=m{v}^2/R=m(2\pi R/T{)}^2/R=4{\pi }^{2}mR/T^2 \\ F(\alpha )=4{\pi }^2\cdot 9\cdot 4r/(6d{)}^2 \\ F(\beta )=4{\pi }^2\cdot 1\cdot r/d^2 \\ F(\alpha )/F(\beta )=(9\cdot 4/6^2)/1=36/36=1 \\ F(\alpha )=F(\beta ) \end{gathered}$$

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