Callisto orbits Jupiter at an average distance of 1.88 106 km with an orbital period of 0.0457 yr. Pluto, which is one of the satellites of the Sun, orbits its parent at an average distance of 5.90 109 km with an orbital period of 248.3 yr.
(a)
Use the above information to find the orbital speeds of Callisto around Jupiter and of Pluto around the Sun.
vCallisto
=8.2 m/s??
vPluto
=4.7m/s
(b)
What is the expression for the mass M of the parent in terms of the orbital speed v of the satellite, the orbital radius R of the satellite and the gravitational constant G? (Do not substitute numerical values; use variables only.)
M = (v^2r)/g
so for A I tried the equation 2pir/T = velocity and I converted the time from years to seconds for that. The number seem a bit low... I also wasn't sure how to do B. I set v^2/r equal to GM/r^2 because I figured in this scenario centripetal acceleration would be provided by g because that's the only thing impacting it.
I'm doing it again and my answers are definitely wrong. I'm so confused somebody help :(
spent 3 hours working on this problem and gave up. thanks so much <3
(a) To find the orbital speed of Callisto around Jupiter and Pluto around the Sun, we can use the formula:
v = 2πr / T,
where v is the orbital speed, r is the orbital radius, and T is the orbital period.
For Callisto:
v_Callisto = 2π(1.88 x 10^6 km) / 0.0457 yr
First, we need to convert the average distance from kilometers to meters and the orbital period from years to seconds:
v_Callisto = 2π(1.88 x 10^6 km * 10^3 m/km) / (0.0457 yr * 365.25 days/yr * 24 hours/day * 3600 seconds/hour)
v_Callisto ≈ 3.392 x 10^3 m/s
For Pluto:
v_Pluto = 2π(5.90 x 10^9 km) / 248.3 yr
Converting the average distance and the orbital period to meters and seconds:
v_Pluto = 2π(5.90 x 10^9 km * 10^3 m/km) / (248.3 yr * 365.25 days/yr * 24 hours/day * 3600 seconds/hour)
v_Pluto ≈ 4.712 m/s
Therefore, the orbital speeds are:
v_Callisto ≈ 3.39 x 10^3 m/s
v_Pluto ≈ 4.71 m/s
(b) To find the expression for the mass M of the parent object (Jupiter or the Sun), we can use the centripetal force equation:
F = m * v^2 / r = G * M * m / r^2,
where F is the centripetal force, m is the mass of the satellite, v is the orbital speed, r is the orbital radius, G is the gravitational constant, and M is the mass of the parent object.
Rearranging the equation:
v^2 = G * M / r,
Multiplying both sides by r:
v^2 * r = G * M.
Therefore, the expression for the mass M of the parent object is:
M = v^2 * r / G.
To find the orbital speed of Callisto and Pluto, we can use the formula for the orbital speed of a satellite:
v = 2πr / T
where v is the orbital speed, r is the orbital radius, and T is the orbital period.
(a) For Callisto around Jupiter:
Given that the average distance of Callisto from Jupiter is 1.88 * 10^6 km and the orbital period is 0.0457 yr, we can plug these values into the formula.
Convert the distance from km to meters:
r = 1.88 * 10^6 km * 1000 m/km = 1.88 * 10^9 m
Convert the period from years to seconds:
T = 0.0457 yr * 365.25 days/yr * 24 hours/day * 60 minutes/hour * 60 seconds/minute ≈ 1.4429 * 10^6 s
Substituting these values into the formula:
vCallisto = 2π * 1.88 * 10^9 m / 1.4429 * 10^6 s ≈ 8.2 m/s
For Pluto around the Sun:
Given that the average distance of Pluto from the Sun is 5.90 * 10^9 km and the orbital period is 248.3 yr, we can follow a similar process.
Convert the distance from km to meters:
r = 5.90 * 10^9 km * 1000 m/km = 5.90 * 10^12 m
Convert the period from years to seconds:
T = 248.3 yr * 365.25 days/yr * 24 hours/day * 60 minutes/hour * 60 seconds/minute ≈ 7.83 * 10^9 s
Substituting these values into the formula:
vPluto = 2π * 5.90 * 10^12 m / 7.83 * 10^9 s ≈ 4.7 m/s
Therefore, the orbital speeds are approximately vCallisto = 8.2 m/s and vPluto = 4.7 m/s.
(b) The expression for the mass M of the parent in terms of the orbital speed v of the satellite, the orbital radius R of the satellite, and the gravitational constant G is:
M = (v^2 * R) / G
This formula is derived from equating the centripetal force (mv^2 / R) to the gravitational force (GMm / R^2) and then canceling out the mass (m) of the satellite.
Where v is the orbital speed, R is the orbital radius, and G is the universal gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2).
Therefore, the expression for the mass of the parent is M = (v^2 * R) / G.