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 12.5g. (b) What is the difference between the acceleration of his head and feet if the astronaut is 2.00 m tall?

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Hey everybody. So today we're dealing with some problems regarding centripetal acceleration and force. So we're told that a centrifuge conveniently creates any desired value of acceleration using rotation. Now, a specific human being measuring 1.65 m tall is strapped into a 7.85 long centrifuge arm with their head at the outermost end with this, we're being asked to calculate the difference in acceleration experienced by the head and the feet. It's a difference in acceleration between the head and feet of the person they are subjected to and the maximum acceleration of 9GS. So this is a bit of a loaded question. But let's take it one step at a time. We're being asked to determine the difference between the acceleration, centripetal acceleration at the head and this centripetal acceleration at the feet. This is what we are looking for. Now, before moving any further, I'd like to go ahead and take a look at the or at the conceptual representation of this straw circle. Oh, nice. Okay, so we have our circle and let's just say that's approximately the center and we have our centrifuge arm. Now, if we have a person on this arm, right? And let's just say there is indeed a person, mother's arm, Our 1.65 m person. Right? So we'll say that the radius here, excuse me, we can say that the radius here is 7.85 m, right, because that is the length of the arm. Maybe you drive this a little bit better to scale this look makes it look a little confusing. This is our center. This is the central future arm. The arm represents a radius of 7.85 m. Right? And we have a person on here Who measures 1. m. So the person himself Is 1.65 m. Cool. The acceleration at the two parts the head and the feet are dependent upon the radius, so they cannot be the same. So a of the head cannot equal a of defeat. There are different points along the radius and will experience different acceleration As a result, however, we are told That the maximum acceleration is at 9G, which means a C. Head. We write that a little better. A sea of the head, centripetal acceleration of the head is nine G. So with that in mind we can go ahead and solve for uh the period. And why would we want to find the period? Well, it'll help us later on to find exactly how long it takes for the. Excuse me. It'll give us the time necessary, right? Will give us the time necessary to figure out the acceleration of the feet as well. Since we already know the acceleration of the head. But working our way down there, we can recall that the formula for centripetal acceleration is four pi four pi squared, multiplied by the radius into f squared. Now recall F is equal to one over T. So this can be rewritten as four pi r squared or four pi squared R over T squirt plugging in. The values that we have for this after rewriting for or rearranging for T. T squared is equal to four pi squared R over a C. Which can then be substituted. Right? We can have four pi squared into 7.85 m .85 m divided by nine G. Which is just nine and 2 9.81 9. m per second squared, Which equals 3.51 seconds squared. So then T. Is therefore equal to the square root of 3. Second squared, which is simply 1.87 seconds. So the period experienced by the head. Excuse me, the period experienced by the head, right? And therefore by the rest of the body as well should be 1.87 seconds. And let me just scroll down so I have a little more space now. So if this is at the head and for the entire body as well to plug in what this is going to be. Excuse me too too Assume what the acceleration at the feet is going to be. We need to change the radius by 1.65 because that is the height of the person after all. So using blue as a color, we need to determine or we need to plug this value in of the period to determine for a C. A C. Feet. No, that is horribly messy. A sea of defeat. When are the radius is 7.85 -1. Because the center future arms 7.85 m and the person is 1.65 m which means their feet will be 1. m away from the edge of the centrifuge arm. So that equals 6.2 m and plugging that back in. You see the feet now, oops, come on. There we go. For the feet now, which will be equal to four pi squared r over T squirt. Right? So substituting in our values, we get four pi squared into 6.20 m Divided by 1.87 seconds squared. or to give us an answer of 70 meters per second squared. So the difference in accelerations therefore Will be the acceleration of the head, which is 9G. It'll be. And let me just write this as hoops. Let me write this as oh boy. The change or difference in the centripetal acceleration from head to feed is equal to nine g minus 70 m per second squared, 70 m per second squared, which is simply nine into 9.8, m/s squared minus 70 m per second squared, which equals 18.2 m per second squared. So therefore the answer choice a will uh, describe the difference between the centripetal acceleration experience by the head and the feet. I hope this helps. And I look forward to seeing you all in the next one.

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Key Concepts

Centripetal Acceleration

Gravitational Acceleration (g)

Height and Acceleration Difference