Question

You are an astronaut, and you observe a small planet to be spherical. After landing on the planet, you set off, walk straight ahead, and find yourself returning to your spacecraft from the opposite side after completing a lap of 24.0 km. You hold a hammer and a falcon feather at a height of 1.40 m, release them, and observe them to fall together to the surface in 28.7 s. Determine the mass of the planet.

Answer 1

Answer:
`7.219(10)^(14) kg`

Step-by-step explanation:

According to the Law of Universal Gravitation:

The force
`F` exerted between two bodies of masses
`m` and
`M` and separated by a distance
`d` is equal to the product of their masses and inversely proportional to the square of the distance.

Written in a mathematicall form is:

`F=G(mM)/(d^(2))` (1)

Where:

`G=6.674(10)^(-11)(m^(3))/(kgs^(2))`is the gravitational constant

`m` is the mass of the object

`M` is the mass of the planet

`d` is the distance from the center of the planet and the object

In addition, this gravitational force is also equal to:

`F=mg` (2)

Where
`g` is the acceleration due gravity in this planet

On the other hand, we are told the astronaut walks straight ahead, and find itself returning to its spacecraft from the opposite side after completing a lap of
`24 km=24000 m`. This means the measure of the circumference
`C` of this planet is:

`C=24000 m=2 \pi r` (3)

Where
`r` is the radius of the planet.

Finding
`r`:

`r=(2400 m)/(2 \pi)=3819.71 m` (4)

At this point we know the distance between the center of the planet and its surface. If we want to know the distance between the center and any of the mentioned objects we will have to add its height
`h=1.4 m`:

`d=r+h=3819.71 m+1.4 m=3821.11 m` (5)

Now, we need to find the acceleration due gravity in this planet, which can be found by the following equation of an object falling to ground from rest:

`h=(1)/(2)gt^(2)` (6)

Where
`t=28.7 s`

Isolating
`g`:

`g=(2h)/(t^(2))` (7)

`g=(2(1.4 m))/((28.7 s)^(2))` (8)

`g=0.0033 m/s^(2)` (9)

Now that we have all the data, we can find the mass by making (1)=(2):

`G(mM)/(d^(2))=mg` (10)

Isolating
`M`:

`M=(gd^(2))/(G)` (11)

`M=((0.0033 m/s^(2))(3821.11 m)^(2))/(6.674(10)^(-11)(m^(3))/(kgs^(2)))` (12)

Finally:

`M=7.219(10)^(14) kg` This is the mass of the planet