The angular speed of the lander at the end of the rotating arm is 4.98 rpm.
It is given to us that -
The radius of the rotating arm = 4.25 m
Mass of the Europa = [tex]4.80 * 10^{22}[/tex] kg
Diameter of the Europa = 3120 km
=> Radius of the Europa = 3120/2 = 1560 km = 1,560,000 m
We have to find out the angular speed (in rpm) at which the rotating arm should spin so that the acceleration of the lander is the same as the acceleration due to gravity at the surface of Europa.
Firstly, we have to find out the acceleration due to gravity of the Europa. We know that the force on the Europa can be given as -
[tex]F = Mg\\= > Mg = \frac{G*m*M}{R^{2} } \\= > g = \frac{G*m}{R^{2}}\\= > g = \frac{(6.67 * 10^{-11} )*(4.80*10^{22} )}{1560000^{2} } \\= > g = 1.19 m/s^{2}[/tex] ----- (1)
It is given to us that -
Acceleration of the lander = Acceleration due to gravity at the Europa
[tex]= > a_{l} = g\\= > a_{l} = 1.19 m/s^{2}[/tex] ----- (2) [From equation (1)]
Now, to calculate the angular speed of the lander, we can use the formula as -
[tex]= > a_{l} = w^{2} r\\= > w^{2} = \frac{a_{l} }{r} \\[/tex] ----- (3)
where,
[tex]a_{l}[/tex] = Acceleration of the lander
[tex]w[/tex] = angular speed of the lander
r = radius of the rotating arm
Substituting the values of [tex]a_{l}[/tex] from equation (2) and r in equation (3), we have -
[tex]w^{2} = \frac{a_{l} }{r}\\= > w^{2} = \frac{1.19 }{4.25}\\= > w^{2} = 0.28\\= > w = 0.529 rad/s[/tex]
In order to convert the angular speed in rpm, we have to use -
[tex]w = 0.529 rad/s\\= > w = 0.529 * \frac{1 rev * 60 s}{2\pi rad*1 min}\\ = > w = 4.98 rpm[/tex]
Therefore, the angular speed of the lander at the end of the rotating arm is 4.98 rpm.
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