whenever two Apollo astronauts were on the surface of the moon, a third astronaut orbited the moon, assume the orbit to be circular and 100 km above the surface of the moon, where the acceleration due to gravity is 1.52 m/s^2. the radius of the moon is 1.70* 10^6 m.(a) determine the astronaut's orbital speed, and (b) the period of the orbit
To determine the astronaut's orbital speed, you can use the formulas for circular motion. The centripetal force required to keep an object in circular motion is provided by gravity in this case.
(a) To find the astronaut's orbital speed, we can equate the gravitational force to the centripetal force:
F_gravity = F_centripetal
The gravitational force is given by the equation:
F_gravity = (G * m_Moon * m_astronaut) / r^2
Where:
G is the gravitational constant (6.67 * 10^-11 N*m^2/kg^2)
m_Moon is the mass of the moon (5.97 * 10^24 kg)
m_astronaut is the mass of the astronaut (assume 80 kg)
r is the distance between the center of the moon and the astronaut's orbit (radius of the moon + altitude).
The centripetal force is given by the equation:
F_centripetal = (m_astronaut * v^2) / r
Where:
v is the orbital speed of the astronaut.
By equating the two forces, we can solve for v:
(G * m_Moon * m_astronaut) / r^2 = (m_astronaut * v^2) / r
Now, plug in the given values:
G = 6.67 * 10^-11 N*m^2/kg^2
m_Moon = 5.97 * 10^24 kg
m_astronaut = 80 kg
r = (1.70 * 10^6 m) + (100,000 m) = 1.80 * 10^6 m
Simplifying the equation:
(G * m_Moon * m_astronaut) = (m_astronaut * v^2) * r
Dividing both sides by m_astronaut:
(G * m_Moon) = v^2 * r
Now, solve for v:
v^2 = (G * m_Moon) / r
v = sqrt((G * m_Moon) / r)
Substitute the given values and calculate:
v = sqrt((6.67 * 10^-11 N*m^2/kg^2 * 5.97 * 10^24 kg) / (1.80 * 10^6 m))
Calculating the square root will give you the astronaut's orbital speed.
(b) To find the period of the orbit, you can use the formula:
T = (2 * pi * r) / v
Where:
T is the period of the orbit (time taken for one complete revolution)
Substitute the given values and the orbital speed calculated in part (a) to find the period of the orbit.