The answers on the model of the human centrifuge ready for the launch to each question of the statement are listed below:
a) A force of 2479.210 newtons is acting on the astronaut's back.
b) A <em>net centripetal</em> force of 2479.210 newtons is acting on the astronaut.
c) The <em>centripetal</em> acceleration of the astronaut is 30.990 meters per square second.
d) The astronaut has a <em>linear</em> speed of approximately 19.284 meters per second.
e) The <em>angular</em> speed of the astronaut is 1.607 radians per second (15.346 revolutions per minute).
<h3>How to apply Newton's laws to analyze a process in a human centrifuge training</h3>The human centrifuge experiments a <em>centripetal</em> acceleration when it reaches a <em>peak</em> angular speed. In this question we must apply Newton's laws of motion and concepts of <em>centripete</em> and <em>centrifugal</em> forces to answer the questions. Now we proceed to answer the questions:
<h3>How much force is acting on the astronaut's back?</h3>By the third Newton's law the astronaut experiments a <em>rection</em> force (<em>F</em>), in newtons, which has the same magnitude to <em>centrifugal</em> force but opposed to that force. The magnitude of the force acting on the back of the astronaut is equal to:
A force of 2479.210 newtons is acting on the astronaut's back.
By the second and third Newton's laws we know that the <em>net centripetal</em> force on the astronaut is equal to the magnitude of the force found in the previous question. Thus, a <em>net centripetal</em> force of 2479.210 newtons is acting on the astronaut.
The centripetal acceleration of the astronaut (<em>a</em>), in meters per square second, is found by dividing the result of the previous question by the mass of the astronaut (<em>m</em>), in kilograms:
(1)
If we know that F = 2479.210 newtons and m = 80 kilograms, then the centripetal acceleration of the astronaut is:
The <em>centripetal</em> acceleration of the astronaut is 30.990 meters per square second.
By definition of <em>uniform circular</em> motion, we have the following formula for the <em>linear</em> velocity of the astronaut (<em>v</em>):
(1)
Where <em>r</em> is the radius of the human centrifuge, in meters.
If we know that and
, then linear velocity of the astronaut is:
<em>v ≈ 19.284 m/s</em>
The astronaut has a <em>linear</em> speed of approximately 19.284 meters per second.
The <em>angular</em> speed of the astronaut (ω), in radians per second, is found by the following <em>kinematic</em> relationship:
(1)
If we know that <em>v ≈ 19.284 m/s</em> and <em>R = 12 m</em>, then the angular speed is:
<em>ω = 1.607 rad/s (15.346 rev/m)</em>
The <em>angular</em> speed of the astronaut is 1.607 radians per second (15.346 revolutions per minute).
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