Roller coasters are all about acceleration Roller coasters are all about acceleration experienced within loops, hills, dips, etc. experienced within loops, hills, dips, etc.

LoopLoop

HillHill Dip

•Centripetal acceleration occurs within Centripetal acceleration occurs within Clothoid LoopsClothoid Loops

• ClothoidClothoid refers to the teardrop shape of the loop (radius is constantly refers to the teardrop shape of the loop (radius is constantly changing)changing)

• Radius is much larger at the bottom than at the topRadius is much larger at the bottom than at the top

• As a rider travels through a clothoid loop, he/she experiences an As a rider travels through a clothoid loop, he/she experiences an acceleration due to change in speed and change in directionacceleration due to change in speed and change in direction

Clothoid LoopsClothoid Loops

SAMPLE PROBLEM 1:SAMPLE PROBLEM 1:

Wario is riding the Wario is riding the Toadster CoasterToadster Coaster and goes through a loop with radius and goes through a loop with radius 4.0 m at the top and radius 17.3 m at 4.0 m at the top and radius 17.3 m at the bottom. He travels at a speed of the bottom. He travels at a speed of 8.2 m/s at the top and a speed of 8.2 m/s at the top and a speed of 20.1 m/s at the bottom. What is the 20.1 m/s at the bottom. What is the normal force acting on him (at the top normal force acting on him (at the top and bottom of the loop) if his mass is 90 kg?and bottom of the loop) if his mass is 90 kg?

Givens:Givens:

TOPTOP BOTTOMBOTTOMv v toptop = 8.2 m/s = 8.2 m/s v v bottombottom = 20.1 m/s = 20.1 m/sr r toptop = 4.0 m = 4.0 m r r bottombottom = 17.3 m = 17.3 m

m = 90 kgm = 90 kg

FREE BODY DIAGRAMS:FREE BODY DIAGRAMS:

TopTop BottomBottom

FFggFFNN FFgg

FNFNyyyy

CALCULATIONS:CALCULATIONS:

TopTop

a = va = vtoptop22 / r / r

= (8.2 m/s) = (8.2 m/s)22 / 4.0 m / 4.0 m = 16.81 m/s = 16.81 m/s22

FFRR = F = Fgg + F + FNN

ma = mg + Fma = mg + FNN

FFNN = (90)(16.81) – (90)(9.81) = (90)(16.81) – (90)(9.81)FFNN = 630.0 N [down] = 630.0 N [down]

Therefore, the Therefore, the normal force normal force acting on Wario acting on Wario is 630.0 N [up] at is 630.0 N [up] at the top of the the top of the loop. loop.

CALCULATIONS:CALCULATIONS:

BottomBottom

a = va = vbottombottom22 / r / r

= (20.1 m/s) = (20.1 m/s)22 / 17.3 m / 17.3 m = 23.35 m/s = 23.35 m/s22

FFRR = F = Fgg + F + FNN

ma = mg + Fma = mg + FNN

FFNN = (90)(23.35) – (90) = (90)(23.35) – (90)(9.81)(9.81)FFNN = 1219 N [up] = 1219 N [up]

Therefore, the Therefore, the normal force normal force acting on acting on Wario is 1219 Wario is 1219 N [up] at the N [up] at the bottom of the bottom of the loop. loop.

SAMPLE PROBLEM 2:SAMPLE PROBLEM 2:

Aneri and Saba are riding Aneri and Saba are riding Moors’ Monster Moors’ Monster Mayhem Mayhem on Fizzix Island. Their speed is 19.2 on Fizzix Island. Their speed is 19.2 m/s while at the top of the hill that has a radius m/s while at the top of the hill that has a radius of 13.1 m. What is the applied force of the of 13.1 m. What is the applied force of the track pulling down on their car, which has a track pulling down on their car, which has a mass of 636 kg.mass of 636 kg.

Givens:Givens: FBDFBD

v = 19.2 m/sv = 19.2 m/sr = 13.1 mr = 13.1 mm = 636 kgm = 636 kg FFgg

FFaa

CALCULATIONS:CALCULATIONS:

a = va = v22 / r / r a = (19.2 m/s)a = (19.2 m/s)22 / 13.1 m / 13.1 ma = 28.14 m/sa = 28.14 m/s22

FFRR = F = Fgg + F + Faa

ma = mg + Fma = mg + Faa

FFaa = (636 kg)(28.14 m/s = (636 kg)(28.14 m/s22) – (636 kg)(9.81 m/s) – (636 kg)(9.81 m/s22))FFaa = 1.166 x 10 = 1.166 x 1044 N [down] N [down]

Therefore, the applied force is 1.166 x 10Therefore, the applied force is 1.166 x 1044 N N [down][down]