13. A roller coaster at the Six FlagsQIC Great America amusement parkin Gurnee, Illinois, incorporatessome clever design technologyand some basic physics. Each ver-tical loop, instead of being cir-cular, is shaped like a teardrop(Fig. P6.13). The cars ride onthe inside of the loop at the top,and the speeds are fast enoughto ensure the cars remain on thetrack. The biggest loop is 40.0 mhigh. Suppose the speed at thetop of the loop is 13.0 m/s andthe corresponding centripetalacceleration of the riders is 2g. (a) What is the radius of thearc of the teardrop at the top? (b) If the total mass of a carplus the riders is M, what force does the rail exert on the carat the top? (c) Suppose the roller coaster had a circular loopof radius 20.0 m. If the cars have the same speed, 13.0 m/sat the top, what is the centripetal acceleration of the ridersat the top? (d) Comment on the normal force at the top inthe situation described in part (c) and on the advantages ofhaving teardrop-shaped loopsFigure P6.13Frank Cezus/Photographer's Choice/Getty Images

Question
Asked Oct 25, 2019
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13. A roller coaster at the Six Flags
QIC Great America amusement park
in Gurnee, Illinois, incorporates
some clever design technology
and some basic physics. Each ver-
tical loop, instead of being cir-
cular, is shaped like a teardrop
(Fig. P6.13). The cars ride on
the inside of the loop at the top,
and the speeds are fast enough
to ensure the cars remain on the
track. The biggest loop is 40.0 m
high. Suppose the speed at the
top of the loop is 13.0 m/s and
the corresponding centripetal
acceleration of the riders is 2g. (a) What is the radius of the
arc of the teardrop at the top? (b) If the total mass of a car
plus the riders is M, what force does the rail exert on the car
at the top? (c) Suppose the roller coaster had a circular loop
of radius 20.0 m. If the cars have the same speed, 13.0 m/s
at the top, what is the centripetal acceleration of the riders
at the top? (d) Comment on the normal force at the top in
the situation described in part (c) and on the advantages of
having teardrop-shaped loops
Figure P6.13
Frank Cezus/Photographer's Choice/Getty Images
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13. A roller coaster at the Six Flags QIC Great America amusement park in Gurnee, Illinois, incorporates some clever design technology and some basic physics. Each ver- tical loop, instead of being cir- cular, is shaped like a teardrop (Fig. P6.13). The cars ride on the inside of the loop at the top, and the speeds are fast enough to ensure the cars remain on the track. The biggest loop is 40.0 m high. Suppose the speed at the top of the loop is 13.0 m/s and the corresponding centripetal acceleration of the riders is 2g. (a) What is the radius of the arc of the teardrop at the top? (b) If the total mass of a car plus the riders is M, what force does the rail exert on the car at the top? (c) Suppose the roller coaster had a circular loop of radius 20.0 m. If the cars have the same speed, 13.0 m/s at the top, what is the centripetal acceleration of the riders at the top? (d) Comment on the normal force at the top in the situation described in part (c) and on the advantages of having teardrop-shaped loops Figure P6.13 Frank Cezus/Photographer's Choice/Getty Images

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Expert Answer

Step 1

Let v be the velocity of the roller coaster at the top of the teardrop, ac be the centripetal acceleration at the top, and r be the arc radius.

Write the expression for the centripetal acceleration at the top, and substitute the known values.

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2g 2g (13 m/s) 2(9.8 m/s2 8.622 m

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Step 2

Let N be the force applied by the rails on the roller coaster car.

Write the expression for newton’s second law of motion.

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N Mg N M (a g —М (2g - s) = Mg

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Step 3

Let the radius of the circular loop is r1.

Write the expression for cent...

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(13 m/s) 20 m 8.45 m/s

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