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 net centripetal force of 2479.210 newtons is acting on the astronaut.
c) The centripetal acceleration of the astronaut is 30.990 meters per square second.
d) The astronaut has a linear speed of approximately 19.284 meters per second.
e) The angular speed of the astronaut is 1.607 radians per second (15.346 revolutions per minute).
How to apply Newton's laws to analyze a process in a human centrifuge training
The human centrifuge experiments a centripetal acceleration when it reaches a peak angular speed. In this question we must apply Newton's laws of motion and concepts of centripete and centrifugal forces to answer the questions. Now we proceed to answer the questions:
How much force is acting on the astronaut's back?
By the third Newton's law the astronaut experiments a rection force (F), in newtons, which has the same magnitude to centrifugal force but opposed to that force. The magnitude of the force acting on the back of the astronaut is equal to:
[tex]F = 3.16\cdot (80\,kg)\cdot \left(9.807\,\frac{m}{s^{2}} \right)[/tex]
[tex]F = 2479.210\,N[/tex]
A force of 2479.210 newtons is acting on the astronaut's back. [tex]\blacksquare[/tex]
What is the net centripetal force on the astronaut?
By the second and third Newton's laws we know that the net centripetal force on the astronaut is equal to the magnitude of the force found in the previous question. Thus, a net centripetal force of 2479.210 newtons is acting on the astronaut. [tex]\blacksquare[/tex]
What is the astronaut's centripetal acceleration?
The centripetal acceleration of the astronaut (a), in meters per square second, is found by dividing the result of the previous question by the mass of the astronaut (m), in kilograms:
[tex]a = \frac{F}{m}[/tex] (1)
If we know that F = 2479.210 newtons and m = 80 kilograms, then the centripetal acceleration of the astronaut is:
[tex]a = \frac{2479.210\,N}{80\,kg}[/tex]
[tex]a = 30.990\,\frac{m}{s^{2}}[/tex]
The centripetal acceleration of the astronaut is 30.990 meters per square second. [tex]\blacksquare[/tex]
What is the astronaut's linear speed?
By definition of uniform circular motion, we have the following formula for the linear velocity of the astronaut (v):
[tex]v = \sqrt{a\cdot r}[/tex] (1)
Where r is the radius of the human centrifuge, in meters.
If we know that [tex]a = 30.990\,\frac{m}{s^{2}}[/tex] and [tex]r = 12\,m[/tex], then linear velocity of the astronaut is:
[tex]v = \sqrt{\left(30.990\,\frac{m}{s^{2}} \right)\cdot (12\,m)}[/tex]
v ≈ 19.284 m/s
The astronaut has a linear speed of approximately 19.284 meters per second. [tex]\blacksquare[/tex]
What is the astronaut's angular speed?
The angular speed of the astronaut (ω), in radians per second, is found by the following kinematic relationship:
[tex]\omega = \frac{v}{R}[/tex] (1)
If we know that v ≈ 19.284 m/s and R = 12 m, then the angular speed is:
[tex]\omega = \frac{19.284\,\frac{m}{s} }{12\,m}[/tex]
ω = 1.607 rad/s (15.346 rev/m)
The angular speed of the astronaut is 1.607 radians per second (15.346 revolutions per minute). [tex]\blacksquare[/tex]
To learn more on centripetal forces, we kindly invite to check this verified question: https://brainly.com/question/11324711
