Answer:
The centripetal acceleration of Deimos is [tex]0.077m/s^{2}[/tex].
Explanation:
The centripetal acceleration is defined as:
[tex]a = \frac{v^{2}}{r}[/tex] (1)
Where v is the velocity of Deimos and r is the orbital distance.
Notice that is necessary to determine the velocity first.
The speed of the Deimos can be found by means of the Universal law of gravity:
[tex]F = G\frac{M \cdot m}{r^{2}}[/tex] (2)
Then, replacing Newton's second law in equation 2 it is gotten:
[tex]m\cdot a = G\frac{M \cdot m}{r^{2}}[/tex] (3)
However, a is the centripetal acceleration since Deimos almost describes a circular motion around Mars:
[tex]a = \frac{v^{2}}{r}[/tex] (4)
Replacing equation 4 in equation 3 it is gotten:
[tex]m\frac{v^{2}}{r} = G\frac{M \cdot m}{r^{2}}[/tex]
[tex]m \cdot v^{2} = G \frac{M \cdot m}{r^{2}}r[/tex]
[tex]m \cdot v^{2} = G \frac{M \cdot m}{r}[/tex]
[tex]v^{2} = G \frac{M \cdot m}{rm}[/tex]
[tex]v^{2} = G \frac{M}{r}[/tex]
[tex]v = \sqrt{\frac{G M}{r}}[/tex] (5)
Where v is the orbital speed, G is the gravitational constant, M is the mass of Mars, and r is the orbital radius.
[tex]v = \sqrt{\frac{(6.67x10^{-11}N.m^{2}/kg^{2})(6.39x10^{23}kg)}{2.35x10^{7}m}}[/tex]
[tex]v = 1346m/s[/tex]
Finally, equation 4 can be used:
[tex]a = \frac{(1346m/s)^{2}}{2.35x10^{7}m}[/tex]
[tex]a = 0.077m/s^{2}[/tex]
Hence, the centripetal acceleration of Deimos is [tex]0.077m/s^{2}[/tex].
