MadSci Network: Physics
Query:

Re: Would a pilot feel less G's if the cockpit were in the center of the ship?

Date: Sun Mar 25 13:06:38 2001
Posted By: Bradley Kelley, Grad student, Mechanical Engineering, Colorado State University
Area of science: Physics
ID: 984664270.Ph
Message:

Lynton,

The quick answer is 'no' because the g-forces that a fighter pilot tends to 
feel are not from rotations about the center-of-gravity (CG), but rotation 
about a point in space.

Here is the long answer.  An airplane (or a car) can't change direction 
instantaneously, it follows an arc.  At some point in the curve, the arc 
you are following will have a minimum radius.  This is the steepest part of 
the turn.  So lets say you are driving along in you car on a country road 
and you come upon a 90° turn.  Do you have to slow down for it?  It 
depends, doesn't it, on how sharp the corner is.  If it is a big, sweeping 
turn with a bank (inward slope) then you may not have to slow down.  If it 
a sharp corner you may need to slow down a lot.  The difference (ignoring 
the bank) is the radius of the curve.  The big sweeping one has a large 
radius and the sharp one has a small one.  You should know from experience 
that the faster you go around a sharp curve, the greater force you feel 
inside, frequently called centrifugal force.  This is the sensation of 
being pulled outward away from the curve, but in reality the car is pushing 
you toward the center of the curve.  This is the direction of acceleration 
assuming you are moving forward at a constant speed.  (Think of it this 
way.  When you take off in you car from a stop, as you accelerate forward, 
you feel like someone is pulling you into the back seat.  The same is true 
in a corner, you feel like you are being pulled outward from the corner, so 
the direction of acceleration is in toward the center!)  The following 
website has a decent animation that shows the velocity and acceleration 
vectors in a circle. http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/circmot/ucm.html

The magnitude of this acceleration is Ar=V^2/r
Where:  Ar=Radial Acceleration
V=Velocity 
r=Radius of curve

This is the case when V is constant, and Ar always points toward the center 
of the arc.  So lets do a quick example for your car (we will do one for 
the jet later).  Say you are traveling at 90 km/hr (about 56 mph) and are 
going to go around a curve at that speed.  What radius does it need to be 
(with no bank or slope) so that your car doesn't slide off into the ditch. 
 We'll give you a Corvette to drive so you can be cool and because the 
'Vette has a radial acceleration limit of 1.0 g's.  (Most cars are .7 to 
.8).  One g=9.81 m/s^2 and 90 km/hr = 25 m/s, so our equation looks like:
9.81 m/s^2 = (25 m/s)^2 / r
Solving for r we get r = 63.7 meters or 209 feet!

Now this is treating the car like a point with mass all at its center of 
gravity.  If we were to treat it as a rigid body, each portion of the car 
will see slightly different acceleration forces because each point will 
have a slightly different radial distance to the center of the arc than the 
CG.  However, this distance is usually insignificant compared to the large 
distance of the arc radius.  For example, say you were sitting a foot on 
the outside of the CG of the car (like in the drivers seat).  Using a 
radius now of 210 feet (64 meters) and 25m/s we get and acceleration of 
9.77 m/s^2, a difference of .04, which would hardly be noticeable.  You may 
have experience with this, no matter which seat in the car you are in, the 
forces you feel in the corner are pretty much the same.

In the case of the jet, the same principle holds true, although your 
acceleration limits are more dependent on what the human body can 
withstand.  The general acceleration limit for fighter pilots is around 9 
g's, or 88.3 m/s^2.  For a fighter jet traveling at a constant speed of 900 
km/hr (250 m/s or 560 mph) what is the minimum arc radius the jet can 
travel to keep the pilot under 9 g's?  Using our equation again, 88.3 m/s^2 
= (250 m/s)^2 / r .  Solving for r we get:
r = 708 m (2322 ft, or almost ½ a mile!)

So you can see, g-force that pilots are concerned with is from rotation 
about an arc in space, not about the Center-of-Gravity.  Fighters do rotate 
about the long axis during barrel rolls (turning upside down and upright 
again) but the pilot sits very close to that axis of rotation and will see 
little force do to that rotation.  If the plane is flipping end-over-end or 
around in circles, then the pilot would certainly be rotating about the CG 
and will see forces from that, however, if these motions are occurring 
those g-forces are likely the least of his concerns!  I hope that this 
helps and wasn't too technical.  Best of luck.
BK

Equations were verified in my trusty physics book, "Physics for Scientists 
and Engineers, 2nd edition" by Douglas Giancoli.



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