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Deffense [45]
3 years ago
8

A satellite of Mars, called Phobos, has an orbital radius of 9.4 ✕ 106 m and a period of 2.8 ✕ 104 s. Assuming the orbit is circ

ular, determine the mass of Mars. (kg)
Physics
1 answer:
FromTheMoon [43]3 years ago
6 0

Answer:

6.27*10^{23}kg

Explanation:

assume

M= mass of Mars

m=mass of phobos

r=orbital radius

T=period

we can apply F=ma to this orbital motion (considering the cricular motion laws)

where,

F=\frac{GMm}{r^{2} }  and a=rω^2

where ω=\frac{2\pi }{T} and G is the universal gravitational constant.

G = 6.67 x 10-11 N m2 / kg2

F=ma\\\frac{GMm}{r^{2} }=mr(\frac{2\pi }{T} )^{2}\\  M=\frac{r^{3}}{G}  (\frac{2\pi }{T} )^{2}\\M=\frac{(9.4*10x^{6} )^{3}*(2\pi )^{2} }{(2.8*10^{4}) ^{2} *6.67*10^{-11} } \\M=6.27*10^{23}kg

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and an ideal spring of unknown force constant. The oscillator is found to have a period of 0.157 s and a maximum speed of 2 m/s
DENIUS [597]

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k = 1073.09 N/m

A = 0.05 m

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Given:

- Time period T = 0.147 s

- maximum speed V_max = 2 m/s

- mass of the block m = 0.67 kg

Find:

- The spring constant k

- The amplitude of the motion A.

Solution:

- A general simple harmonic motion is modeled by:

                                     x (t) = A*sin(w*t)

- The velocity of the above modeled SHM is:

                                     v = dx / dt

                                     v(t) = A*w*cos(w*t)

- Where A is the amplitude in meters, w is the angular speed rad/s and time t is in seconds.

- We can see that maximum velocity occurs when (cos(w*t)) maximizes i.e it is equal to 1 or -1. Hence,

-                                      V_max = A*w

- Where w is related to mass of the object and spring constant k as follows,

                                       w = sqrt ( k / m )

- The relationship between w angular speed and Time period T is:

                                       w = 2*pi / T

- Equating the above two equations we have,

                                        m*(2*pi / T)^2 = k

- Hence,                           k = 0.67*(2*pi / 0.157)^2

                                        k = 1073.09 N / m

- So, amplitude A is:

                                        A = V_max*sqrt ( m / k )

                                        A = 2*sqrt ( 0.67 / 1073.09 )

                                        A = 0.05 m      

6 0
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