Height at which pilot must pull out of dive before hitting 5g

[negation]

In post #26:
I feel misread! Please read it again:
While going straight down at constant speed, the pilot experiences 1g downward. This is a vector.
As soon as he pulls the wheel (or what do they call it -- A stick, I realized since then) towards him, a uniform circular motion sets in that requires 4g of acceleration towards the centre. His seat provides the force required for that acceleration. Also a vector, initially pointing to the right. So initially he feels an acceleration of 1 g down, 4g to the right. √17 magnitude total. Gradually going towards 1 g down and 4 g down as well just before point q. Magnitude now 5 g (same direction). He can stand that as per given in the OP. If he's smart he pulls up a little longer, haha. The moment he is going straight again, he experiences 1 g towards the center of the earth.

In post#29:

The 1 g acceleration down does not lead to a free fall. His seat pushes him up to oppose the gravitational force and it also pushes him up with 4g to keep him in a circular motion. These seat forces are in the same direction, hence the 5g.

I'm glad rcgldr helps too. Lok is amused, which is nice. But I resent him saying you are being ignored. I try to help you patiently as best I can to deal with the exercise. Where a circular motion is not exactly a given, but already complicated enough for us, right?

Without a detailed sequential diagram, it can be hard to conceive of a mental modelling. Why is it 4g down? Acceleration is always in the direction towards the center, isn't it? Let's just say that when the pilot is at the point (0,-r), he experiences a 4g acceleration towards the center. At the same time, there is a 1g acceleration in the direction towards the ground. But, why, as you'e put it, would the 4g acceleration be in the direction towards the ground? How did this 4g acceleration, initially towards the center changes vector suddenly. It doesn't make sense.

View attachment 66192

 

[lewando]

Mental models are important to establish. Maybe think about it this way:

Scenario 1: You sit in a chair attached securely to the Earth. Let this be a zero-velocity reference frame. The chair is applying a force on you in the upward direction. Earth's gravity, and the fact that you have mass, counters this force exactly, so you don't accelerate.

Scenario 2: Now take away the Earth and attach a rocket under your chair that applies the same equivalent "upward" force as in Scenario 1. You are now accelerating "upwards" at 1g. The amount of blood in your brain remains the same as before and if you close your eyes, it feels exactly like Scenario 1.

Scenario 3: You are sitting in the chair now resting on the curved floor of a rotating centrifuge of an interstellar spacecraft . The centrifuge is a giant "hamster-wheel" that spins with a certain angular velocity that can creates artificial Earthlike gravity. When you close you eyes, it feels exactly like you are sitting in the chair under conditions of either Scenarios 1 or 2. The force of the chair upon your body feels just like it did when you were on the Earth-- "upwards", which is actually towards the center of the centrifuge. As a result of this force, you are accelerating in the direction of "upwards" from your perspective or towards the center of the centrifuge, in reality--"upwards" is relative to you (the direction above your head).

The "g force" (the calculated one) at the start of the turn can be 5g, as the gravitational 1g is 90 deg from the centrifugal force. So the two do not mix in the down direction which is actually the horizontal at the beginning of the turn. And the story is not finished yet. The 5g should end up being a 4g at the end of the turn where the gravitational and centrifugal mix at 0 deg one from the other so add up.

If this problem was in an aeronautical setting the 5g varies to 4g during the turn and turn height is bigger. If only a entry level physics problem do ignore the variation. So the ~2300 should be close enough.

Interesting point, Lok (variable centripetal acceleration). But I think the minimum height at which you begin your turn would still be the same as it would if the turn trajectory was circular (constant centripetal acceleration)

 

[Lok]

Without a detailed sequential diagram, it can be hard to conceive of a mental modelling. Why is it 4g down? Acceleration is always in the direction towards the center, isn't it? Let's just say that when the pilot is at the point (0,-r), he experiences a 4g acceleration towards the center. At the same time, there is a 1g acceleration in the direction towards the ground. But, why, as you'e put it, would the 4g acceleration be in the direction towards the ground? How did this 4g acceleration, initially towards the center changes vector suddenly. It doesn't make sense.
View attachment 66192

I've made a small picture for this.
cg=centrifugal g
gg=gravitational g
As a g force for a human only the downward g(from a human orientation reference) is significant as it is the one that increases or decreases the pressure in the head.
All I am saying is that you would have to have a varying radius in order to keep 5g's. Otherwise you would get more at the end of the turn.

I'm glad rcgldr helps too. Lok is amused, which is nice. But I resent him saying you are being ignored. I try to help you patiently as best I can to deal with the exercise. Where a circular motion is not exactly a given, but already complicated enough for us, right?

@BvU, I did not mean it in a bad way. If you can imagine.

 

[negation]

Mental models are important to establish. Maybe think about it this way:

Scenario 1: You sit in a chair attached securely to the Earth. Let this be a zero-velocity reference frame. The chair is applying a force on you in the upward direction. Earth's gravity, and the fact that you have mass, counters this force exactly, so you don't accelerate.

Scenario 2: Now take away the Earth and attach a rocket under your chair that applies the same equivalent "upward" force as in Scenario 1. You are now accelerating "upwards" at 1g. The amount of blood in your brain remains the same as before and if you close your eyes, it feels exactly like Scenario 1.


I haven't got to the next chapter on forces-will soon- but I shall try to build a vague understanding for the bigger picture to come. I love thinking the big-picture way.

And I suppose, in scenario 1, this occurs arises from Newton's third law: succinctly, that every action has an equal and opposing reaction?
An entity sitting on a chair applies a down ward force on the chair equivalent to the product of the mass and acceleration from the gravity-this translate mathematically to weight since w = mg.
In turn, the chair counters this with an opposing force.

In scenario 2, the net acceleration is 1g. So this implies that the upward acceleration must be 1g greater than the down ward acceleration from gravity. Am I right?

Scenario 3: You are sitting in the chair now resting on the curved floor of a rotating centrifuge of an interstellar spacecraft . The centrifuge is a giant "hamster-wheel" that spins with a certain angular velocity that can creates artificial Earthlike gravity. When you close you eyes, it feels exactly like you are sitting in the chair under conditions of either Scenarios 1 or 2. The force of the chair upon your body feels just like it did when you were on the Earth-- "upwards", which is actually towards the center of the centrifuge. As a result of this force, you are accelerating in the direction of "upwards" from your perspective or towards the center of the centrifuge, in reality--"upwards" is relative to you (the direction above your head).

 

[negation]

I've made a small picture for this.
cg=centrifugal g
gg=gravitational g
As a g force for a human only the downward g(from a human orientation reference) is significant as it is the one that increases or decreases the pressure in the head.
All I am saying is that you would have to have a varying radius in order to keep 5g's. Otherwise you would get more at the end of the turn.


@BvU, I did not mean it in a bad way. If you can imagine.

My modelling is this. Is there any parts I ought to correct?

 

[negation]

I've made a small picture for this.
cg=centrifugal g
gg=gravitational g
As a g force for a human only the downward g(from a human orientation reference) is significant as it is the one that increases or decreases the pressure in the head.
All I am saying is that you would have to have a varying radius in order to keep 5g's. Otherwise you would get more at the end of the turn.


@BvU, I did not mean it in a bad way. If you can imagine.

View attachment 66204

My modelling is this. Is there any parts I ought to correct?
By the way, it occurred to me that in moving around a predictable circular path, the entire phenomenon takes on a SHM modelling.

 

[lewando]

And I suppose, in scenario 1, this occurs arises from Newton's third law: succinctly, that every action has an equal and opposing reaction?

Sure, the 3rd law applies. So does the 2nd law: ∑(all forces) = ma. Since a is zero, ∑(all forces) must be zero, which is the case when adding equal magnitude and opposite direction forces.

In scenario 2, the net acceleration is 1g. So this implies that the upward acceleration must be 1g greater than the down ward acceleration from gravity. Am I right?

In scenario 2, there is no gravity because we disappeared the Earth. Acceleration is a reuslt of the force applied by the rocket: a = F_rocket/m

 

[negation]

Sure, the 3rd law applies. So does the 2nd law: ∑(all forces) = ma. Since a is zero, ∑(all forces) must be zero, which is the case when adding equal magnitude and opposite direction forces.



In scenario 2, there is no gravity because we disappeared the Earth. Acceleration is a reuslt of the force applied by the rocket: a = F_rocket/m

That explains. It's much clearer.

 

[BvU]

Post #26:

1.Without a detailed sequential diagram, it can be hard to conceive of a mental modelling.
2.Why is it 4g down?
3.Acceleration is always in the direction towards the center, isn't it?
4.Let's just say that when the pilot is at the point (0,-r), he experiences a 4g acceleration towards the center.
5.At the same time, there is a 1g acceleration in the direction towards the ground.
6.But, why, as you'e put it, would the 4g acceleration be in the direction towards the ground? How did this 4g acceleration, initially towards the center changes vector suddenly. It doesn't make sense.

1. I agree. Here's another picture, a continuation from the post#22 picture, with the origin shifted to where you have it too.

A. Coming down along a straight line with constant speed. 1mg downward force from the gravitational field of the Earth is counteracted by the pilot's restraining belts. The pilot experiences 1g downwards (for him: forward, towards the nose of the plane, towards the earth) but doesn't accelerate because the forces executed by the belts hold him at the speed of the plane. This is what we call the normal force in static cases: eg. pilot's weight on chair in office: force mg down, equal but opposite normal force up: zero sum, no motion. What you feel in such situation is called 1g. In the picture: 'felt' 1g down, 'felt' acceleration: red Arrow, size g, up. This is present constantly throughout the entire flight of the plane.

B. The moment he pulls the stick and the plane executes the command, he starts to describe a circle. The OP wording and our state of knowledge about kinematics kind of dictate that the plane's trajectory is a quarter part of a circle. Required centripetal force: 4 mg (value to be justified later) towards center, so at point B that is in positive x direction. Required acceleration: 4g in positive x direction. In addition to that, the 1g from before still applies.(What is an SHM model?). Acceleration 4g in positive x direction plus 1g in positive y direction add up to the purple vector sum. Dotted lines: this is how you add vectors, with a parallellogram. Magnitude of the actual acceleration: √17 g. The pilot experiences this √ 17 g in exactly the opposite direction: he was already leaning in his belts (1g), now he's also pushed into his chair (4g).

C. Centripetal force has turned a bit and aligns more with the normal force. Less strain in the belts, more push from the chair.

D. even less belts, more chair

E. Lowest point, just before pilot pushes back the stick to go to level flight. Acceleration to describe circle still 4g, straight up. Acceleration to counteract gravity and NOT fall towards the Earth any further 1g, straight up. Two independent effects, combined to add up to the pilot experiencing 5g pushing him down in his seat. Just what he can stand while remaining conscious. Hence the 4g at B.

Still E. Plane is now horizontal and pilot pushes back the stick to go to level flight. Plane obeys instantly. No more circle. Acceleration to counteract gravity and NOT fall towards the Earth 1g, straight up.

F. Level flight. Force to counteract gravity 1 mg, delivered by the seat. But the pilot feels 1g down.
​

2. It is not 4g down. I used

So initially he feels an acceleration of 1 g down, 4g to the right

Which was wrong and confusing. My apologies.(*) He feels 1 down, 4 to the left. He is accelerated by an acceleration 1g up (belts) plus 1 g down (gravity) and 4g to the right. That way the net force is towards the center of the circle .

This centripetal force rotates $+\pi/2$ until it is 4mg up. Gravity is still there and needs to be counteracted by another 1mg up. Pilot feels 5g pushing him down in his chair.

(*)I realize I repeated this error by copying to post #35 without checking. Sorry once more. Lot has predictive gifts!​

3. For uniform circular motion in the absence of other forces: yes. In this exercise, there is another (constant) force present: Earth gravity.

4. Yes.

5. There is a 1mg force down. But we were told the plane moves at 1200 km/h, a constant speed. Not unreasonable: the air resistance at such speeds is hefty. I would vote for counteracting this 1mg so the trajectory becomes a circle and we can apply our ^v<sup>2/r

6. See the apology under 2. I confused you and didn't realize I had done so. Then I made it worse by repeating.

So much on the quote. I'm running behind. What a dilemma: too quick and you confuse people, too slow and a thread becomes a plate of spaghetti. Fortunately we are having good fun.

Lewando throws in the scenarios, very instructive.

Lok wants a more complicated trajectory by using the full 5g as much as he can. I wonder if he would stay conscious when braking at 5g while simultaneously turning up at 5g. My pilot son tells me the braking is worse because you have the sensation of falling over forward. Nauseating for sure; don't know about influence on consciousness. Will interrogate him some more on this. Any aeronaut listening in?
Lot also turns to the left. What is it with these guys ;-) Good thing none of us means it in a bad way ;-) (I should brush up my smiley knowledge).

Time for me to turn in. Look forward to negation's next exercise! Good thing he likes big-picture thinking (rockets, space stations, no eart nearby and all that)

This surely has become some kind of knotted cable instead of a didactically responsible thread!
Anyway, the ensuing complications are highly undesirable for negation.</sup>

 

[negation]

Post #26:


1. I agree. Here's another picture, a continuation from the post#22 picture, with the origin shifted to where you have it too.

A. Coming down along a straight line with constant speed. 1mg downward force from the gravitational field of the Earth is counteracted by the pilot's restraining belts. The pilot experiences 1g downwards (for him: forward, towards the nose of the plane, towards the earth) but doesn't accelerate because the forces executed by the belts hold him at the speed of the plane. This is what we call the normal force in static cases: eg. pilot's weight on chair in office: force mg down, equal but opposite normal force up: zero sum, no motion. What you feel in such situation is called 1g. In the picture: 'felt' 1g down, 'felt' acceleration: red Arrow, size g, up. This is present constantly throughout the entire flight of the plane.

B. The moment he pulls the stick and the plane executes the command, he starts to describe a circle. The OP wording and our state of knowledge about kinematics kind of dictate that the plane's trajectory is a quarter part of a circle. Required centripetal force: 4 mg (value to be justified later) towards center, so at point B that is in positive x direction. Required acceleration: 4g in positive x direction. In addition to that, the 1g from before still applies.(What is an SHM model?). Acceleration 4g in positive x direction plus 1g in positive y direction add up to the purple vector sum. Dotted lines: this is how you add vectors, with a parallellogram. Magnitude of the actual acceleration: √17 g. The pilot experiences this √ 17 g in exactly the opposite direction: he was already leaning in his belts (1g), now he's also pushed into his chair (4g).

C. Centripetal force has turned a bit and aligns more with the normal force. Less strain in the belts, more push from the chair.

D. even less belts, more chair

E. Lowest point, just before pilot pushes back the stick to go to level flight. Acceleration to describe circle still 4g, straight up. Acceleration to counteract gravity and NOT fall towards the Earth any further 1g, straight up. Two independent effects, combined to add up to the pilot experiencing 5g pushing him down in his seat. Just what he can stand while remaining conscious. Hence the 4g at B.

Still E. Plane is now horizontal and pilot pushes back the stick to go to level flight. Plane obeys instantly. No more circle. Acceleration to counteract gravity and NOT fall towards the Earth 1g, straight up.

F. Level flight. Force to counteract gravity 1 mg, delivered by the seat. But the pilot feels 1g down.
​

2. It is not 4g down. I used Which was wrong and confusing. My apologies.(*) He feels 1 down, 4 to the left. He is accelerated by an acceleration 1g up (belts) plus 1 g down (gravity) and 4g to the right. That way the net force is towards the center of the circle .

This centripetal force rotates $+\pi/2$ until it is 4mg up. Gravity is still there and needs to be counteracted by another 1mg up. Pilot feels 5g pushing him down in his chair.

(*)I realize I repeated this error by copying to post #35 without checking. Sorry once more. Lot has predictive gifts!​

3. For uniform circular motion in the absence of other forces: yes. In this exercise, there is another (constant) force present: Earth gravity.

4. Yes.

5. There is a 1mg force down. But we were told the plane moves at 1200 km/h, a constant speed. Not unreasonable: the air resistance at such speeds is hefty. I would vote for counteracting this 1mg so the trajectory becomes a circle and we can apply our ^v<sup>2/r

6. See the apology under 2. I confused you and didn't realize I had done so. Then I made it worse by repeating.

So much on the quote. I'm running behind. What a dilemma: too quick and you confuse people, too slow and a thread becomes a plate of spaghetti. Fortunately we are having good fun.

Lewando throws in the scenarios, very instructive.

Lok wants a more complicated trajectory by using the full 5g as much as he can. I wonder if he would stay conscious when braking at 5g while simultaneously turning up at 5g. My pilot son tells me the braking is worse because you have the sensation of falling over forward. Nauseating for sure; don't know about influence on consciousness. Will interrogate him some more on this. Any aeronaut listening in?
Lot also turns to the left. What is it with these guys ;-) Good thing none of us means it in a bad way ;-) (I should brush up my smiley knowledge).

Time for me to turn in. Look forward to negation's next exercise! Good thing he likes big-picture thinking (rockets, space stations, no eart nearby and all that)

This surely has become some kind of knotted cable instead of a didactically responsible thread!
Anyway, the ensuing complications are highly undesirable for negation.</sup>

<sup>And so much more clearer. I was rather confused at the outset with the whole "pilot feels 1g acceleration down ward but has a 1g acceleration upward"-rather contradictory initially- but it seems the arrow in the diagram indicates the corollary of what the pilot is subject to-that is, it indicates an opposing acceleration.
And so, the diagram now makes sense. It's so much easier to work with a real-time visual model in my head.

To end the problem, all I've to do is apply the g = v^2/r with g = SQRT(17g^2), yes?</sup>

 

[lewando]

A couple comments on your notations: "g = v^2/r" is more meaningful and better written as a_centripetal = v²/r

(use the [Go Advanced] button and use the X² and X₂ tools. Use the [Preview Post] button you see if what you did looks like what you expected)

"g = SQRT(17g^2)" is based on what, again?

The point made clear in BvU's the last post was that at every point in the turning phase of the flight, there is a vector sum of the acceleration due to gravity, a_g, and the acceleration due to the circular motion, a_centripetal. The magnitude of this sum cannot exceed 5m/s².

Can you tell, from inspection, which of the points in BvU's diagram (B, C, D, or E) represents the largest magnitude of the vector sum?

From that point, you get what you are looking for: the maximum a_centriptetal.

From a_centripetal = v²/r, the maximum a_centripetal gives the smallest r, which is the height that you are looking for.

Edit: I meant "5g m/s²"

 

[negation]

A couple comments on your notations: "g = v^2/r" is more meaningful and better written as a_centripetal = v²/r

(use the [Go Advanced] button and use the X² and X₂ tools. Use the [Preview Post] button you see if what you did looks like what you expected)

"g = SQRT(17g^2)" is based on what, again?

The point made clear in BvM's the last post was that at every point in the turning phase of the flight, there is a vector sum of the acceleration due to gravity, a_g, and the acceleration due to the circular motion, a_centripetal. The magnitude of this sum cannot exceed 5m/s².

Can you tell, from inspection, which of the points in BvM's diagram (B, C, D, or E) represents the largest magnitude of the vector sum?

From that point, you get what you are looking for: the maximum a_centriptetal.

From a_centripetal = v²/r, the maximum a_centripetal gives the smallest r, which is the height that you are looking for.

E is the greatest. It has an acceleration of magnitude 5g.

 

[lewando]

Alright, so... wrap it up, man! :tongue2:

 

[negation]

Alright, so... wrap it up, man! :tongue2:

r =( 333ms^-1)^2 / 5g

Thanks!

 

[BvU]

Oops. I still would use a 4 here instead of a 5. The difference is reserved to deal with the omnipresent small red arrow pointing straightupwards. In particular at point E in the last moment before letting go of the stick. To explain this, I took quite some time to compose #45. Might have made mistakes (again -- if so, please point them out), but it's better than post #25.

The 5 is applicable in Iewando's scenario number 3. But the Earth is damn nearby.

I sure wish I could come up with an animated version of the picture in #44. Things like this make me want to learn a new software tool to do so efficiently. Any tips?

 

[negation]

Oops. I still would use a 4 here instead of a 5. The difference is reserved to deal with the omnipresent small red arrow pointing straightupwards. In particular at point E in the last moment before letting go of the stick. To explain this, I took quite some time to compose #45. Might have made mistakes (again -- if so, please point them out), but it's better than post #25.

The 5 is applicable in Iewando's scenario number 3. But the Earth is damn nearby.

I sure wish I could come up with an animated version of the picture in #44. Things like this make me want to learn a new software tool to do so efficiently. Any tips?

Regardless the 4g or 5g, how did you arrived at the 4g to the right? I would really appreciate the explanation

 

[BvU]

To the right of what ? I thought we had at last had agreed upon a coordinate system. This by now is a long thread.

Let me make a wild but educated guess. (btw It's the guessing what the other guy means exactly, that makes this whole thing so time consuming, almost exasperating. For all parties involved.).

I thought we had at last agreed upon a coordinate system. I use it. I posted a bunch of pictures with letters and arrows. I refer to picture in post 44 and my educated guess is you are referring to point B, point (0,0). There is an eight line explanation with a cliffhanger: to be justified later. Under point E explanation the veil is lifted: the happy end is "hence the 4g at B".

So the way I think about this thing is:
1. Don't complicate it more than necessary: assume constant speed.
2. You suffer 1g behind your desktop. If you pass out at 5 g, you can stand 4g extra from acrobatics like looping and such -- in a worst case where Earth and acrobatics pull in the same direction.
3. If all you want is to get out of a nosedive, a quarter circle is what you need and have sufficient knowledge to do some calculations on. Iewando's complications are for SMH (what on Earth or in heaven is that?) or post-BS.
4. Combine 2 and 3: a quarter circle with at max 4g normal force in the vertical upward direction to deliver the centripetal force needed to describe a circular motion is what the exercise wants us to work on.
5. We know a constant acceleration is needed for circular motion. Acceleration is a vector. It points to the center of the circle. Acceleration from Earth gravity is also a vector. It points downward (or rather: towards the center of gravity of the Earth - don't ask). From a simple sketch it becomes clear that the worst case (i.e. Earth and acrobatics pulling in the same direction, so that the vector addition comes down to a simple magnitude addition is at point E, at the very end of the quarter circle.
6. Combine 1 and 5: 4g is available for the quarter circle.

So in response to your question (as I guess it was meant): 4g is the max available. It the trajectory is to be a quarter circle and the speed constant, the centripetal acceleration has to be constant in magnitude and has to point to the center of that circle. So initially, at point B, after a jerk on the stick, it has to point in the positive x-direction in the coordinate system we agreed upon. And its magnitude is 4g. Big red arrow.

Now, what makes this exercise so difficult for you: There is a difference between what the pilot feels and the net acceleration. Behind a desktop (on earth, please), you feel 1g. But you don't fall because the chair holds you in place by exercising exactly 1g upwards. The normal force in school books. In this exercise the pilot feels an acceleration that is equal to the opposite of the purple arrows (purple is unique so far, so I don't specify that it's in the picture of post #45). The net acceleration is ZERO until point B (hence a straight line, constant speed. Newton!), then from B to E the net acceleration is a vector, magnitude 4g and pointing to (r,0), the centre of the circle (hence the circle, and hence a radius v²/4g), and after point E the net acceleration is ZERO again, hence a straight line, grazing the surface of the Earth at constant speed.

Unhealthy, but we were asked to calculate the minimum height, regardless of tree heights ;-)



(Why does my W7 IE11 all of a sudden wants to capitalize each occurrence of the word Arrow, but not the word arrows? Some copyright or trademark that is suddenly a part of the english language ?)

 

[negation]

To the right of what ? I thought we had at last had agreed upon a coordinate system. This by now is a long thread.

Let me make a wild but educated guess. (btw It's the guessing what the other guy means exactly, that makes this whole thing so time consuming, almost exasperating. For all parties involved.).

I thought we had at last agreed upon a coordinate system. I use it. I posted a bunch of pictures with letters and arrows. I refer to picture in post 44 and my educated guess is you are referring to point B, point (0,0). There is an eight line explanation with a cliffhanger: to be justified later. Under point E explanation the veil is lifted: the happy end is "hence the 4g at B".

So the way I think about this thing is:
1. Don't complicate it more than necessary: assume constant speed.
2. You suffer 1g behind your desktop. If you pass out at 5 g, you can stand 4g extra from acrobatics like looping and such -- in a worst case where Earth and acrobatics pull in the same direction.
3. If all you want is to get out of a nosedive, a quarter circle is what you need and have sufficient knowledge to do some calculations on. Iewando's complications are for SMH (what on Earth or in heaven is that?) or post-BS.
4. Combine 2 and 3: a quarter circle with at max 4g normal force in the vertical upward direction to deliver the centripetal force needed to describe a circular motion is what the exercise wants us to work on.
5. We know a constant acceleration is needed for circular motion. Acceleration is a vector. It points to the center of the circle. Acceleration from Earth gravity is also a vector. It points downward (or rather: towards the center of gravity of the Earth - don't ask). From a simple sketch it becomes clear that the worst case (i.e. Earth and acrobatics pulling in the same direction, so that the vector addition comes down to a simple magnitude addition is at point E, at the very end of the quarter circle.
6. Combine 1 and 5: 4g is available for the quarter circle.

So in response to your question (as I guess it was meant): 4g is the max available. It the trajectory is to be a quarter circle and the speed constant, the centripetal acceleration has to be constant in magnitude and has to point to the center of that circle. So initially, at point B, after a jerk on the stick, it has to point in the positive x-direction in the coordinate system we agreed upon. And its magnitude is 4g. Big red arrow.

Sorry mate, I should have been more specific. I was only curious about how the value of 4g (towards the right, in direction of the positive x-axis) was arrived at.
You meant SHM? I think if the entity kept to a constant predictable loop, indefinitely, a simple harmonic motion model could be drawn out-a sine phase.

Now, what makes this exercise so difficult for you: There is a difference between what the pilot feels and the net acceleration. Behind a desktop (on earth, please), you feel 1g. But you don't fall because the chair holds you in place by exercising exactly 1g upwards. The normal force in school books. In this exercise the pilot feels an acceleration that is equal to the opposite of the purple arrows (purple is unique so far, so I don't specify that it's in the picture of post #45). The net acceleration is ZERO until point B (hence a straight line, constant speed. Newton!), then from B to E the net acceleration is a vector, magnitude 4g and pointing to (r,0), the centre of the circle (hence the circle, and hence a radius v²/4g), and after point E the net acceleration is ZERO again, hence a straight line, grazing the surface of the Earth at constant speed.

Unhealthy, but we were asked to calculate the minimum height, regardless of tree heights ;-)

Yes, there was an initial confusion arising from the conflation of what the pilot experience and the opposing force indicated schematically. But, in time, I realize only the opposing force to what the pilot experience matters.

 

[BvU]

Ah, I guessed sophisticated human modelling, so there you see what damage an abbrev can cause!

You Australian, mate?

And no, there is nothing pointing to SHM here. I can imagine a sine pops up if you start decomposing, but a sine is not Always going to SHM. (Always is capitalized automatically too!?)
SHM tell-tales: force opposite wrt motion and proportional to deviation from equilibrium, periodicity, etc. Proportional being a nice word for linear, which is often a first approximation. You'll get plenty exercise with SHM, don't worry. All the way through to QFT (an abbreviation!)

This confusing g scale pilots and entourage use is a boon for teachers looking for exercises. Same thing about pressure measurements in Barg (or worse, psi without mentioning the g) instead of N/m². So: beware.

But real world physics is also physics, so we have to deal with the jargon

If you study physics a bit longer, everything becomes relative. I go on and on about this because you need a good basis further on, no matter what you do or study.

 

[negation]

Ah, I guessed sophisticated human modelling, so there you see what damage an abbrev can cause!

You Australian, mate?

And no, there is nothing pointing to SHM here. I can imagine a sine pops up if you start decomposing, but a sine is not Always going to SHM. (Always is capitalized automatically too!?)
SHM tell-tales: force opposite wrt motion and proportional to deviation from equilibrium, periodicity, etc. Proportional being a nice word for linear, which is often a first approximation. You'll get plenty exercise with SHM, don't worry. All the way through to QFT (an abbreviation!)

This confusing g scale pilots and entourage use is a boon for teachers looking for exercises. Same thing about pressure measurements in Barg (or worse, psi without mentioning the g) instead of N/m². So: beware.

But real world physics is also physics, so we have to deal with the jargon

If you study physics a bit longer, everything becomes relative. I go on and on about this because you need a good basis further on, no matter what you do or study.

I'm not Aussie but been here for a while. Yes, I understand the importance of relativity. And yes I will be doing Physic. I'm going into my 2nd year next month- first year was foundation. I'm reading Astrophysics and Math as majors. Been doing lots of self-study during this break for preparation.
The reading materials aren't sufficient to give deep analysis and most lecturers, in my experience, don't pile a deep underlying foundation either.

 

[BvU]

Good. I'm in NL, so GMT+1. Use your time efficiently: go to the next exercise.

 

[homemade]

What if a pilot were submersed in a liquid container in flight, wouldn't that significantly reduce g force? The liquid you can breathe would be even better.