In a large centrifuge used for training pilots and astronauts, a small chamber i

Question

In a large centrifuge used for training pilots and astronauts, a small chamber is fixed at the end of a rigid arm that rotates in a horizontal circle. A trainee riding in the chamber of a centrifuge rotating with a constant angular speed of 1.9 rad/s experiences a centripetal acceleration of 3.0 times the acceleration due to gravity. In a second training exercise, the centrifuge speeds up from rest with a constant angular acceleration. When the centrifuge reaches an angular speed of 1.9 rad/s, the trainee experiences a total acceleration equal to 4.4 times the acceleration due to gravity. (a) How long is the arm of the centrifuge? Incorrect: Your answer is incorrect. m (b) What is the angular acceleration (magnitude) of the centrifuge in the second training exercise? rad/s2

Explanation / Answer

centrip accel = w^2 r where w is the angular velocity and r is the distance from the center of rotation

in the first experiment, we are told w=1.9 rad/s, and accel = 3g=3*9.8m/s/s or a=29.4m/s/s, so we can solve for r:

29.4m/s/s = (1.9rad/s)^2 r => r= 8.14m

in the second experiment, we are told there are two sources of acceleration, the angular acceleration and the radial acceleration

we know the radial (centripetal) accleration = w^2 r = 29.4m/s/s

the total acceleration has magnitude sqrt[at^2 + ac^2] where at is the tangential acceleration and ac is the centripetal (radial acceleration)

if the total acceleration is 4.4g, it equals 4.4*9.8m/s/s = 43.12m/s/s

therefore we have 43.12^2=at^2+ac^2
43.12^2=at^2+29.4^2 or at=31.54m/s/s

now...be careful to realize this is the tangential acceleration, and you are asked for the angular acceleration, these are related by

alpha= angular acceleration = at/r = 31.54m/s/s / 8.14m=3.87rad/s/s

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