Unformatted text preview: circular motion, the mégninide of the acceleration is .vzlr; So the com. patients of Newto‘rt‘ 8 law become r?
C 11111
x: itsi'nti ="% ' y: itcosﬂ.=0
r . where the 0 on the righthand. side of the .y. equation reﬂects the fact that we don‘t want the .car to accelerate in the verticai direction Solv~
ing the )2 equation gives 11: mgicoso Then using this result in the _x_
equation gives mg sine/cos H: mvzfr, or g tanti — vzlr The mass can
celed, which is good news because it means our banked road will work
for a vehicle of any. maSs Now we can solve for the Banking angle: . merit 5.7: Circtliar Motioré. Looping the Loop ' 7 The “Great American Revolution" roller coaster at Valencia Califorw I nia, includes a loopthciOop section Whose radius 15 6. 3. m at the top (see the chapter opening photo). What is the minimum speed for a ' rollercoaSter car. atthc top of the loop if it’sto stay on the track?  INTERPRET Again, we have. circular motion described by Newton’ 5 sec—
ond law. We re asked about the minimum speed for the car to stay on
the track. What does it mean to stay on the track? It means there must
he a normal force: bettVeen car anti track; otherwise, the two aren’ t in contactso we canidentify tw0 forces acting on the car; gravity and ' the normal force from the track. Davao? Figure 5.15 shows the phySic‘al situation The situation is es—
pecially simple at the top of the track, Where both forces point in the
same direction. We show this in our freebody diagram, Fig; 5 16. Since that common direction' is downward, it makes sense to choose a ' coordinate system With the y axis downward. The applicable eqi'iation
is Newton 1} second law, and with the. hive forces we ve identiﬁed, that
becomes" 11 + F= ma EvALuan With both forces in the same direction, we need only the
y cemponen't cf NeWton’ s law. With the doanard direction gositive, =1: and F —mg At the' top of the loop, the? car is in circular
motion,lso its acceleration is toward the center—'doanard—and has At the top, bothferries point doiimwérd‘andthe I
'  caris‘moiiienieﬁlyin..___
uniform circular ‘ _ 1 
_ motion ' " ' Gravityis airways downward '
 __ . but at this pomtthe normal force is horizontal. The net
fume ism “Howard the center,
and the car isslowiiig as well
 as changing direction. Eliillﬁlimi ‘ Forces on the roller—coaster car. 5.3 Circuiar Motion 75 ‘ 10 % finjig) Z: %n_1( (9s $153313; 1.11).: 180 assess Make sense? At low speed 1} or large radius r, the car 5 motion
changes gently and it doesn 1 take a large force to keep it on its circu
lar. path But as v increases or. r decreases, the required force increases
and so does the banking angle That’s because the horizontal compo
nent of the dermal force 15 what keréps the car in circular motion, and /' I _ the steeper the angle. the greater that cemponeiit. ' I Elam5.]; Our freebody diagram at the tap of theioep. I magnitude v'izlr So. a =_21r,and the y componcht of Newtoii’ s. law
becomes _  mvz n+mg.——'
r Solving for the speed gives v2 V (iv/m) +gr 'NoW, the minimum possible speed for contact with the track occurs when 11 gets arbihfaii ly small right at the top of. the track, so we ﬁnd this minimum limit. by
setting 11:9. Then the rinswer is  '(9sm132)(63m}—79m/s _A'sse_ss Do you see what’ s happening here? With the minimum Speed I the normal force 'vaniizh‘es at the top of the LOOP, and gravity alone pro ' vides the force that iceLegs the object. in its circular path. Since the mo~ tion is circular, that force must have magnitude. owl/r. But the force of  gravity. alone is mg; and rm: 'ng sfollow's directly from equating those two quantities A car moving any slower than. 11mm would lose
contact with the track and go into the parabolic; najeetory of a proiee
tile. For a car moving faster, there wrinid be. a nonzero nonrial force
contributing to the downward acceleration .at the top of; the loop In the
“Great American Revolution, the acttial speed at the loop 5 top is
9.72111/5 to provide 211 margin of safety. As with many problems invalv
ing gravity, the mass cancels. That’ s'. .a good .thii'rg because it means the
safe spreéd doesn’t depend ori the number or mass of the riders I  ...
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 Spring '07
 KOPP
 Physics

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