Horizons: Exploring the Universe (MindTap Course List)
Horizons: Exploring the Universe (MindTap Course List)
14th Edition
ISBN: 9781305960961
Author: Michael A. Seeds, Dana Backman
Publisher: Cengage Learning
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Chapter 17, Problem 5P

Phobos obits Mars at a distance of 9376 km from the center of the planet and has a period of 0.3 189 days. Assume Phobos’s orbit is circular. Calculate the mass of Mars. (Hints: Use the circular orbit velocity formula in Reasoning with Numbers 4-1; make sure to convert relevant quantities to units of meters, kilograms. and seconds.)

Expert Solution & Answer
Check Mark
To determine

To calculate:

The mass of Mars based on the distance and orbit angle of Phobos

Answer to Problem 5P

The mass of Mars = 6.45 x 1023 Kg.

Explanation of Solution

Given Information:

Radius of orbit = 9376 Km

Period of Orbit = 0.3189 Days

Formula Used:

Circular orbit velocity = VS=G.Mr

Also, VS=distance/time

Calculation:

Assume Phobos (Moons) orbit to be circular.

First convert the quantities into proper units

  1)     Radius of orbit = 9376 Km =9376 x 1000 m=9.376 x  106m

  2)     Period of orbit=0.3189 Days                                    = 0.3189 x ( 24 hr ) x ( 60 min ) x ( 60 sec )= 27552.96 Sec

Now,

Circular Velocity = VS=G.Mr -------------------------------- (Assume moon is circular)

Where, G = Gravitational Constant = 6.67 x 1011m3/s2Kg

r = Radius of Planet. (m)

m= Mass of Planet (Kg)

Rewriting in terms of m

∴ m = Vs^2.rG

For finding circular velocity that is distance covered per unit time.

∴ VS =Distance/time

Distance between but circumference of orbit.

circumference C = 2πr                             = 2 x π x 9.376 x  106                                        = 58911145.44                            = 5.9 x  107m

∴ Moon must cover 5.9 x 107 m distance in 27552.96 sec

∴ its circular velocity must be,

  VS=5.9x10^727552.9=2141.33 m/s

∴ put all the Values in mass of Mars formula,

  m=Vs^2.rG

= 2141.33X2141.33X9.376X10^66.67X10^11

= 6.45 x 1023Kg

Conclusion:

Mass of Mars is 6.45 x 1023 kg, which is around 10.7 % of Earth’s mass.

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