The National Aeronautics and Space Administration (NASA) studies the physiological effects of large accelerations on astronauts. Some of these studies use a machine known as a centrifuge. This machine consists of a long arm, to one end of which is attached a chamber in which the astronaut sits. The other end of the arm is connected to an axis about which the arm and chamber can be rotated. The astronaut moves on a circular path, much like a model airplane flying in a circle on a guideline. The chamber is located 14.0 m from the center of the circle. At what speed must the chamber move so that an astronaut is subjected to 3.75 times the acceleration due to gravity?
To determine the speed at which the chamber must move, we can use the equation for centripetal acceleration:
a = (v^2) / r
where:
a is the centripetal acceleration,
v is the speed of the chamber,
r is the radius of the circular path.
In this case, we want the astronaut to experience 3.75 times the acceleration due to gravity (denoted as "g"). So, the centripetal acceleration will be 3.75g.
We also know that the chamber is located 14.0 m from the center of the circle (radius, r).
Using the equation, we can rearrange it to solve for the speed (v):
v = sqrt(a * r)
Substituting the given values:
a = 3.75g
r = 14.0 m
v = sqrt((3.75g) * (14.0 m))
Next, we need to determine the value of g, which is the acceleration due to gravity. The standard value is approximately 9.8 m/s^2.
Substituting g = 9.8 m/s^2:
v = sqrt((3.75 * 9.8 m/s^2) * (14.0 m))
Now, we can calculate the speed:
v = sqrt((36.75) * (14.0)) m/s
v = sqrt(514.5) m/s
v ≈ 22.68 m/s
Therefore, the chamber must move at a speed of approximately 22.68 m/s for the astronaut to experience 3.75 times the acceleration due to gravity.