Ganymede is a moon of jupiter mass=1.90 x 10^27 kg. it orbits jupiter in a circle of radius 1.07 x 10^9 m. what is ganymede's period in hours?
To find Ganymede's period in hours, we can use the formula for the period of a circular orbit:
T = 2π * √(r³ / G * M)
Where:
T is the period (in seconds)
π is a mathematical constant approximately equal to 3.14159
r is the distance between the center of Jupiter and Ganymede (in meters)
G is the gravitational constant (approximately 6.67430 x 10^-11 m³ kg^-1 s^-2)
M is the mass of Jupiter (in kilograms)
First, let's convert the radius to meters:
r = 1.07 x 10^9 m
Next, let's convert the mass of Jupiter to kilograms:
M = 1.90 x 10^27 kg
Now we can substitute these values into the formula to calculate the period in seconds:
T = 2π * √((1.07 x 10^9)^3 / (6.67430 x 10^-11) * (1.90 x 10^27))
After substituting the values and performing the calculation, we should be able to determine the period in seconds. Finally, we can convert the period from seconds to hours.
Let me calculate that for you.
G m M / R^2 = m v^2/R
G M/R = v^2
so
v = sqrt (G M/R)
and
T = 2 pi R/v