In this problem, be especially wary of round-off; keep 3 sigfigs in each answer

Question

In this problem, be especially wary of round-off; keep 3 sigfigs in each answer and throughout your calculations.

Based on the following data about planet X (which orbits around the Sun):

Planet X's distance from Sun = 3.4*1012 m
Planet X's radius = 1.8*106 m
Planet X's mass = 9*1022 kg

a.) Find gx, the size of the acceleration due to gravity on the surface of Planet X.  m/s2

b.) What is the weight of a 10 kg mass on the surface of Planet X?  N
(How does this compare to its weight on Earth?)

c.) How long would it take for a ball dropped from a height of 10 m to hit the ground?  s
(How does this compare to the time it would take on Earth?)

d.) At 1 of Planet X's radii above the planet's surface, what is gx?  m/s2

e.) For Planet X's orbit around the Sun, please find:
its orbital speed:  m/s
its centripetal acceleration:  m/s2

f.) How long is a year on Planet X? Express your answers in both seconds and Earth years:
s
Earth years

I solved part but not all and need help making sure they are write and with the rest.

a) acc to g is 1.856

b) 18.6N

c)ball to fall is 3.28s

d)ac due to g is .464

i cant do the rest. please help!

thanks

Explanation / Answer

mass of Planet X   M = 9 x 10^22 kg

                Radious R = 1.8 x 10^6 m

when a mass m is on the surface of the planet gx is gravitational acceleration on the planet

Gravitaional force Fg = GMm/R2

foorce due to gravitational acceleration gx   = mgx

mgx   = GMm/R2

gx = GM/R2    = 6.67 x 10^-11 x 9 x 10^22 / (1.8 x 10^6)2

                         = 1.853 m/s/s

b) weight = mgx = 10 x 1.853 =18.53 N

c) t = SQRT(2h/g) = SQRT(2x10/1.853) = 3.285 s

d) if the hieght is eqaul to R the distance will be 2R and the gx will be 1/4th

      new gx = 1.853/4 = 0.463

e) for this you need mass of Sun

Gravitational force between the SUn and planet must be equal to the centrifugal force due to orbiting

Let Ms be the mass of Sun then we have = 2 x 10^30 kg

GMsMx/D2 = MxV2/D, D idstnace from Sun

V2 = GMs/D = 6.67 x 10^-11 x 2 x 10^30 / 3.4 x 10^12 = 39.2 x 10^6 m/s

V = 6.26 x 10^3 m/s

centripetal acceleration = V2/R = 39.2x 10^6/3.4x10^12 = 11.53 x 10^-6 m/s/s

angular velocity w = V/R = 6.26x10^3/3.4x10^12 = 1.84 x 10^-9 rad/s

two complete one orbit(2pi radians) t =  2x3.14/(1.84x10^-9)

                                                             = 3.41 x 10^9 secs i.e one year

                                        = 3.41x10^9/(360x24x3600) = 109.6 earth years

Earth year = 360x24x3600 secs

orbital

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