Io, one of the moons of Jupiter, circles Jupiter once every 1.77 days. The radius of its orbit is 4.22 X 108 m. If this orbit is circular, what is the mass of Jupiter?

To find the mass of Jupiter, we can use the following formula:

F = (G * m1 * m2) / r^2

Where:
F is the gravitational force between Jupiter and Io,
G is the gravitational constant (6.67 × 10^-11 N m^2/kg^2),
m1 is the mass of Jupiter,
m2 is the mass of Io, and
r is the radius of the orbit.

We can rearrange the formula to solve for m1:

m1 = (F * r^2) / (G * m2)

First, let's calculate the value of F using the centripetal force equation:

F = (m2 * v^2) / r

Where:
v is the orbital velocity of Io.

The orbital velocity can be calculated using the formula:

v = (2 * π * r) / T

Where:
T is the period of Io's orbit.

Given that T = 1.77 days, and a day has 24 hours, we convert T to seconds:

T = 1.77 * 24 * 60 * 60 = 152,928 seconds

Now, we can calculate v:

v = (2 * π * 4.22 * 10^8) / 152,928

Next, we can substitute the value of v into the formula for F:

F = (m2 * ((2 * π * 4.22 * 10^8) / 152,928)^2) / (4.22 * 10^8)

Let's calculate the value of F:

F = (m2 * 4 * π^2 * (4.22 * 10^8)^2) / (152,928)^2

Now, substituting the known values for G, m2, r, and F into the equation for mass:

m1 = (F * r^2) / (G * m2)

Let's substitute the values and calculate the mass of Jupiter.

To find the mass of Jupiter, we can use the formula that relates the gravitational force between two objects with their masses and the distance between them. The formula is:

F = (G * m1 * m2) / r^2

Where:
F is the gravitational force
G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^2)
m1 and m2 are the masses of the two objects
r is the distance between the centers of the two objects

For the given scenario, we know:
- The radius of Io's orbit around Jupiter is 4.22 X 10^8 m.
- Io takes 1.77 days to complete one orbit.

First, we need to convert the time from days to seconds since the units of the constants are in SI (International System of Units).

1 day = 24 hours
1 hour = 60 minutes
1 minute = 60 seconds

So, 1.77 days can be converted to seconds as follows:

1.77 days * 24 hours * 60 minutes * 60 seconds ≈ 152,928 seconds

Now we can use this time along with the known radius of Io's orbit to find the mass of Jupiter.

The formula for the velocity of a satellite in circular orbit is given by:

v = (2 * π * r) / T

Where:
v is the velocity of the satellite
r is the radius of the orbit
T is the time period of the orbit

By rearranging the equation, we can solve for the velocity:

v = (2 * π * r) / T

Substituting the given values:

v = (2 * 3.14159 * (4.22 X 10^8)) / 152,928

Now that we have the velocity, we can calculate the mass of Jupiter using the following formula:

v = √(G * M / r)

Where:
v is the velocity of the satellite
G is the gravitational constant
M is the mass of Jupiter
r is the radius of the orbit

Rearranging the equation to solve for M:

M = (v^2 * r) / G

Now, substituting the known values:

M = ((2 * 3.14159 * (4.22 X 10^8)) / 152,928)^2 * (4.22 X 10^8) / (6.67430 × 10^-11)

Evaluating this expression will give us the mass of Jupiter.

m=mass of satellite, M=mass of Jupiter

R=4.22•10^8 m, T= 1.74 days=1.74•24•3600 seconds,
G =6.67•10^-11 N•m²/kg²,

F= G•M•m /R²
Centripetal acceleration
a=v²/R =(2•π•R/T)² /R=4• π²•R/T². (1)
Newton’s 2Law
F=m•a = G•M•m /R².
a= G•M /R² (2)
Equate (1) and (2)
4• π²•R/T² = G•M /R²
M= 4• π²•R³/T²•G = ...