A rotating cylinder 7.95 kilometers in radius is designed to be used as a space colony. With what period must it rotate so the residents on it feel as heavy as they would on Earth?
Set the centripetal acceleration to the acceleration of gravity
v^2/r = g
To determine the required period of rotation for the space colony, we need to consider the centrifugal force acting on the residents.
The centrifugal force is the force experienced by an object moving in a circular path, directed outward from the center of rotation. It depends on the radius of rotation and the angular velocity. The force experienced should be equal to the force of gravity experienced on Earth to make the residents feel "as heavy" as they would on Earth.
The formula for centrifugal force is given by:
F_cent = m * a_c = m * (r * ω^2)
Where:
F_cent is the centrifugal force,
m is the mass of a resident,
a_c is the centripetal acceleration,
r is the radius of rotation,
ω (omega) is the angular velocity.
On Earth, the gravitational force (F_grav) experienced by a resident is given by:
F_grav = m * g
Where:
g is the acceleration due to gravity (9.8 m/s^2 on Earth).
To make the residents feel "as heavy" as on Earth, the centrifugal force (F_cent) should be equal to the gravitational force (F_grav):
F_cent = F_grav
m * (r * ω^2) = m * g
m cancels out from both sides, leaving:
r * ω^2 = g
Rearranging the equation to solve for ω (angular velocity):
ω = √(g/r)
Substituting the given values:
r = 7.95 kilometers = 7950 meters (converted to meters)
g = 9.8 m/s^2
ω = √(9.8/7950)
Calculating the value:
ω ≈ √0.001231445 = 0.03509 rad/s
The angular velocity (ω) represents the radians covered per unit of time. To convert the angular velocity to period (T), which represents the time taken to complete one full rotation, we use the formula:
T = 2π/ω
Substituting the value of ω:
T = 2π/0.03509
Calculating the value:
T ≈ 179.53 seconds
Therefore, the rotating cylinder must rotate with a period of approximately 179.53 seconds for the residents to feel as heavy as they would on Earth.