ethys, one of Saturn's moons, travels in a circular orbit at a speed of 1.1x104 m/s. The
mass of Saturn is 5.67x1026 kg. Calculate
a) the orbital radius in kilometres.( i know how to do this part, the answer is 3.1x10^8 m)
b) the orbital period in Earth days.
you know the radius and the speed, so
time = distance/speed = (2π * 3.1*10^8)/(1.1*10^4)
that gives the time in seconds. I figure you can convert that to earth days ...
To calculate the orbital period of ethys, we can use the formula:
T = (2πr) / v
Where:
T = orbital period (in seconds)
π = Pi (approximately 3.14159)
r = orbital radius (in meters)
v = orbital speed (in meters per second)
Since we have already calculated the orbital radius as 3.1x10^8 m, and the orbital speed is given as 1.1x10^4 m/s, we can substitute these values into the formula:
T = (2π * 3.1x10^8) / (1.1x10^4)
T ≈ (2 * 3.14159 * 3.1x10^8) / (1.1x10^4)
T ≈ 6.18313x10^8 / 1.1x10^4
T ≈ 56119363.6 seconds
Now, to convert this to Earth days, we need to divide by the number of seconds in a day.
There are 86400 seconds in a day, so:
T (in days) ≈ 56119363.6 / 86400
T (in days) ≈ 649.465
Therefore, the orbital period of ethys is approximately 649.465 Earth days.
To calculate the orbital radius and orbital period of ethys, one of Saturn's moons, we can use the formulas related to circular motion and gravity.
a) Orbital Radius:
The formula to calculate the orbital radius (r) is:
r = v^2 / (G * M)
where:
- v is the speed of ethys in meters per second (1.1x10^4 m/s in this case).
- G is the gravitational constant (approximated as 6.67430x10^-11 N(m/kg)^2).
- M is the mass of Saturn (5.67x10^26 kg in this case).
Plugging in the values:
r = (1.1x10^4 m/s)^2 / (6.67430x10^-11 N(m/kg)^2 * 5.67x10^26 kg)
= 1.21x10^8 m^2/s^2 / (3.7782571x10^15 N(m/kg)^2)
≈ 3.208x10^-8 m^3/(N*s^2) * 1 km/1000 m
≈ 3.208x10^-11 km^3/(N*s^2)
Therefore, the orbital radius is approximately 3.208x10^-11 km^3/(N*s^2) or simply 3.1x10^8 km. (which matches your answer)
b) Orbital Period:
The formula to calculate the orbital period (T) is:
T = 2πr / v
Using the value of r from part (a) and plugging it in:
T = 2π * 3.1x10^8 km / (1.1x10^4 m/s)
= 2 * 3.14159 * 3.1x10^8 * 1000 km / 1.1x10^4 m
≈ 5.67722x10^8 km*m / m
= 5.67722x10^8 km
Since there are 24 hours in a day, we can convert the result to Earth days by dividing it by 24:
T ≈ 5.67722x10^8 km / 24
≈ 2.3655x10^7 km
Therefore, the orbital period of ethys is approximately 2.3655x10^7 Earth days.