Question

At its Ames Research Center, NASA uses its large "20-G" centrifuge to test the effects of very large accelerations ("hypergravity") on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and an astronaut is strapped in at the other end. Suppose that he is aligned along the centrifuge’s arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this device is typically 12.5 g. (a) How fast must the astronaut’s head be moving to experience this maximum acceleration? (b) What is the difference between the acceleration of his head and feet if the astronaut is 2.00 m tall? (c) How fast in rpm (rev/min) is the arm turning to produce the maximum sustained acceleration?

Answer 1

Answer:

32.9242311983 m/s

1.40506567727g

35.3405770759 rpm

Explanation:

v = Linear Velocity of the capsule

$a_c$ = Centripetal acceleration = $12.5g=12.5\times 9.81$

r = Radius of the centrifuge = 8.84 m

l = Person's height = 2 m

Centripetal acceleration is given by

$a_c=\frac{v^2}{r}\\\Rightarrow v=a_cr\\\Rightarrow v=\sqrt{12.5\times 9.81\times 8.84}$

The linear speed of the capsule is 32.9242311983 m/s

The radius would be

$r=\sqrt{r^2+\dfrac{l^2}{4}}\\\Rightarrow r=\sqrt{8.84^2+\dfrac{2^2}{4}}\\\Rightarrow r=8.89638128679\ m$

The centripetal acceleration

$a_{c2}=\dfrac{32.9242311983^2}{8.89638128679\times 9.81}g\\\Rightarrow a_{c2}=12.4207805891g$

Change in acceleration from Pythagoras law

$a=\sqrt{a_{ch}^2-a_{c2}^2}\\\Rightarrow a=\sqrt{12.5^2g^2-12.4207805891^2g^2}\\\Rightarrow a=1.40506567727g$

The difference is 1.40506567727g

Velocity

$v=\omega r\\\Rightarrow v=2\pi Nr\\\Rightarrow N=\dfrac{v}{2\pi r}\\\Rightarrow N=\dfrac{32.9242311983}{2\pi 8.89638128679}\\\Rightarrow N=0.589009617932\ rev/s$

$N=0.589009617932\times 60=35.3405770759\ rpm$

The speed is 35.3405770759 rpm