Circular Motion## Newton's Second Law

## applied to a

## Vertical Circle

What forces come into play as an aerobatic pilot does a barrel roll?

Or, for that matter, what forces come into play as you or I ride on one of today's roller coasters with hills and valleys and loops?

Please do

notattempt to memorize -- and keep straight -- the force equations that we will derive for the different situations.

Rather,make a cleardiagramand then applyF_{net}ac whereac is thecentripetal accelerationso it -- and thenet forceor thecentripetal force-- points toward the center of the circle. That's all there is!>

Alwaysdraw good, detailed "free body diagrams"! That isalwaysimportant and it is especially important now.Neverstart by plugging numbers into equations and calculating an answer.Alwaysstart with a diagram!

On top of a hillThe only forces acting on the rider are the upward normal force

nexerted by the car and the downward force of gravityw, the rider's weight. These add together, as vectors, to provide the net forceF_{net}whichisthe centripetal forceF_{c}, directedtoward the centerof the circle. Thenormal forcemay also be called the rider's "apparent weight" for this is the force of the seatonthe rider and also describes what the rider "feels" (in addition to terror!).F _{net}= w - n = mg - nF

_{net}= F_{c}= m v^{2}/ rmg - n = m v

^{2}/ rn = mg - m v

^{2}/ rWhat does all this mean? The normal force or the rider's apparent weight is

lessthan the rider's real weight. The seat can not exert a negative force on the rider. If we approach a situation where the apparent weight might become negative, there should be a good safetyrestraintsystem -- seat belts, lap bars, shoulder restraints, or something of that sort.We might ask how fast the coaster can go until the rider just (barely) looses contact with the seat. That means the normal force between seat and rider is

zero. That occurs forn = mg - m v ^{2}/ r = 0m v

^{2}/ r = mgv

^{2}/ r = gv

^{2}= g rWe have described this with a diagram showing a guest on the top of a hill of a roller coaster. The same ideas would apply, of course, to a pilot right-side-up on the top of a "roll" as shown here.

Bottom of a Valley

The only forces acting on the rider are the upward normal force

nexerted by the car and the downward force of gravityw, the rider's weight. These add together, as vectors, to provide the net forceF_{net}whichisthe centripetal forceF_{c}, directedtoward the centerof the circle. Notice, of course, that the center of the circle is nowupfrom the rider. As always, thenormal forcemay also be called the rider's "apparent weight" for this is the force of the seatonthe rider and also describes what the rider "feels" (in addition to terror!).F _{net}= n - w = n - mgF

_{net}= F_{c}= m v^{2}/ rn - mg = m v

^{2}/ rn = mg + m v

^{2}/ rWhat does all this mean? The normal force or the rider's apparent weight is now

morethan the rider's real weight. The rider feels pressed down into the seat. And the feeling is more general. The rider's hands and arms are hard to move. The rider's blood is even hard to move. Airplane pilots are in this situation as they pull out of a dive. The ratio of this "apparent weight" to real weight may be described as the apparent effect of gravity or may be described as "g-forces". "g-forces" of six or seven -- meaning an apparent weight of six or seven times one's real weight -- can mean that enough blood will not be circulated to the brain and a pilot -- or other passenger -- may pass out.We have described this with a diagram showing a guest on the bottom of a valley of a roller coaster. The same ideas would apply, of course, to a pilot on the bottom of a "dive" as shown here.

Upside-down at the Top of a LoopNow both the normal force

nand the weightwpoint in the same direction so the net force is thesumof these two forces,F _{net}= n + w = n + mgF

_{net}= F_{c}= m v^{2}/ rn + mg = m v

^{2}/ rn = ( m v

^{2}/ r ) - mgWhat does all this mean? If the speed is

too low, this equation says the normal force will benegative. What does a negative normal force mean? Since the seat can not reach out and pull you back, you will fall out of the car. And, since the track can not reach out and pull back on the roller coaster car, the car would fall off the track! To prevent this, roller coasters have wheels onboth sides of the track! We might think of designing a ride so the normal force just vanishes and the victims do, indeed, feel weightless at the top. Or we might design a ride so the normal force is equal to the real weight at the top.We have described this with a diagram showing a guest on the top of a loop of a roller coaster. The same ideas would apply, of course, to a pilot upside-down on the top of a roll as shown here.

Of course, there are other situations that provide vertical circles besides just a roller coaster. Aerobatic pilots or military pilots in dives and loops experience these same things. As you drive a hilly road you experience two of these three situations.

Table of Contents(c) Doug Davis, 2001; all rights reserved