5:16 1 Roller Coasters... Roller Coaster Loops Today's coasters provide riders with weightless sensations and abnormally large G forces. Weightless sensations occur when the seat pushes on riders with a force that is less than their weight. Large G forces occur when the seat pushes with a force much greater than their weight. The number of Gs (Gz) is the ratio of this seat force to the rider's weight. The loop-the-loop provides much of this thrill. The earliest designs included circular loops as shown in Figure 1. Table 1 depicts the estimated speed (v) and the Gz for a rider starting from a 60-meter high drop. Table 1: Circular Loops Figure 1 hjoop (m) 15 VA * (m/s) 28.2 VB * (m/s) 32.5 Gz Gz at A at B 9.8 15.4 20 26.6 6.2 32.5 11.8 25 24.8 4.0 32.5 9.6 harop hloop 30 23.0 2.6 32.5 8.2 32.5 32.5 35 21.0 1.6 7.2 45 16.3 0.2 5.8 50 13.3 0.0 32.5 5.3 Figure 2 Because of the inherent dangers associated with circular loops and of the advent of tubular steel coasters, designers began to use non- circular loops in their designs. The teardrop-shaped loop is often called a clothoid loop. The shape is that of a cornu spiral, with a constantly changing radius of curvature as shown in Figure 2. Table 2 shows the effect of the radii (R) at the top and bottom of a loop upon the rider experience. Riop Rottam Rhottom Table 2: Clothoid Loops hloop RA VA * Gz RB VB Gz (m) (m/s) (m/s) at B (m) 10 at A (m) 30 23.0 4.4 20 32.5 6.4 36 12 20.6 2.6 24 32.5 5.5 40 13 18.8 1.8 26 32.5 5.2 40 13 18.8 1.8 30 32.5 4.6 1.8 1.8 1.4 40 13 18.8 34 32.5 4.2 13 18.8 18.8 40 38 32.5 3.8 40 15 26 32.5 5.2 40 17 18.8 1.1 26 32.5 5.2 45 15 16.3 0.8 30 32.5 4.6 45 15 16.3 0.8 34 32.5 4.2 45 15 16.3 0.8 38 32.5 3.8 45 15 16.3 0.8 42 32.5 3.6 45 18 16.3 0.5 34 32.5 4.2 45 21 16.3 0.3 34 32.5 4.2 * Calculations based on a 10% energy loss due to air resistance, track vibration, etc. O 1996-2012 The Physics Classroom, All rights reserved. c. At the bottom of a 30-m high loop. d. At the bottom of a 45-m high loop. 5. According to Table 2, what effect does an increase in the radius of the top of a loop have upon the speed of the riders at the top of the loop? a. Increasing the radius of the top of the loop increases the speed at this location. 4 of 4 b. Increasing the radius of the top of the loop decreases the speed at this location. c. Increasing the radius of the top of the loop has no effect upon the speed at this locauon. d. Increasing the radius of the top of the loop seems to affect the speed; the manner in which it does is not predictable. 6. According to Table 2, at which of the following locations on a clothoid-shaped loop will a rider experience the most Gz? a. At the top of a 30-m high loop having a radius of 10 m. b. At the top of a 40-m high loop having a radius of 15 m. c. At the bottom of a 36-m high loop having a radius of 24 m. d. At the bottom of a 40-m high loop having a radius of 26 m. © 1996-2012 The Physics Classroom, All rights reserved. 7. The number of Gz that can be considered safe depends upon a variety of factors - the age and health of the rider, the duration of those Gz and how abruptly they are experienced. Generally, the coaster industry seeks to keep the number of Gz at 5.0 or below. Given this concern, which one of the following clothoid loops would not be used in a 60-m high design? a. hjoop=40 m; RA = 13 m; Rg = 30 m b. hloop = 40 m; RA = 13 m; Rg 38 m c. hjoop= 40 m; RA=17 m; Rg = 26 m d. hjoop = 45 m; RA=15 m; Rg 34 m A coaster engineer wishes to include a clothoid loop in a 60-m high design that provides the riders the following experience: 8. | Top of Loop Bottom of Loop Between 1.0 and 1.5 Gz Between 4.0 and 4.5 Gz Use Table 2 to determine which of the following loop parameters would be most suitable for such a coaster? a. hjoop = 40 m; RA = 15 m; Rg = 30 m b. hjoop = 40 m; RA =17 m; RB = 26 m c. hloop = 45 m; RA=15 m; Rg = 34 m d. hjoop = 45 m; RA = 15 m; Rg = 42 m %3D As the number of Gz at the top of an inverted (upside down) loop begins to approach zero, designers must regard the possibility of riders falling out of their seats as being the top safety hazard. If a design includes a clothoid loop with Gz less than 1.0 at the top, what change could be made to improve the safety at this location? a. Switch to a circular loop. c. Increase the radius at the loop top. 9. b. Increase the height of the loop. d. Decrease the radius at the loop top. 10. A coaster engineer has included a clothoid loop in his coaster design. As currently designed, riders run the risk of falling out of their seats at the top of the loop. Which one of the following design changes would NOT fix this safety issue? a. Decrease the height of the loop. b. Increase the height of the first drop. c. Decrease the radius of the top of the loop. d. Increase the height of the loop and/or increase the radius of the top of the loop. 11. For the same loop heights, how are the values of the clothoid-shaped loops of Table 2 different than the values for the circular loops of Table 1? a. The speed at the top of the loops is less for clothoid-shaped loops. b. The Gz at the top of the loops are greater for clothoid-shaped loops. c. The Gz at both the top and the bottom of the loops are less for clothoid-shaped loops. d. The speed at both the top and the bottom of the loops are less for clothoid-shaped loops. O 1996-2012 The Physics Classroom, All rights reserved.
5:16 1 Roller Coasters... Roller Coaster Loops Today's coasters provide riders with weightless sensations and abnormally large G forces. Weightless sensations occur when the seat pushes on riders with a force that is less than their weight. Large G forces occur when the seat pushes with a force much greater than their weight. The number of Gs (Gz) is the ratio of this seat force to the rider's weight. The loop-the-loop provides much of this thrill. The earliest designs included circular loops as shown in Figure 1. Table 1 depicts the estimated speed (v) and the Gz for a rider starting from a 60-meter high drop. Table 1: Circular Loops Figure 1 hjoop (m) 15 VA * (m/s) 28.2 VB * (m/s) 32.5 Gz Gz at A at B 9.8 15.4 20 26.6 6.2 32.5 11.8 25 24.8 4.0 32.5 9.6 harop hloop 30 23.0 2.6 32.5 8.2 32.5 32.5 35 21.0 1.6 7.2 45 16.3 0.2 5.8 50 13.3 0.0 32.5 5.3 Figure 2 Because of the inherent dangers associated with circular loops and of the advent of tubular steel coasters, designers began to use non- circular loops in their designs. The teardrop-shaped loop is often called a clothoid loop. The shape is that of a cornu spiral, with a constantly changing radius of curvature as shown in Figure 2. Table 2 shows the effect of the radii (R) at the top and bottom of a loop upon the rider experience. Riop Rottam Rhottom Table 2: Clothoid Loops hloop RA VA * Gz RB VB Gz (m) (m/s) (m/s) at B (m) 10 at A (m) 30 23.0 4.4 20 32.5 6.4 36 12 20.6 2.6 24 32.5 5.5 40 13 18.8 1.8 26 32.5 5.2 40 13 18.8 1.8 30 32.5 4.6 1.8 1.8 1.4 40 13 18.8 34 32.5 4.2 13 18.8 18.8 40 38 32.5 3.8 40 15 26 32.5 5.2 40 17 18.8 1.1 26 32.5 5.2 45 15 16.3 0.8 30 32.5 4.6 45 15 16.3 0.8 34 32.5 4.2 45 15 16.3 0.8 38 32.5 3.8 45 15 16.3 0.8 42 32.5 3.6 45 18 16.3 0.5 34 32.5 4.2 45 21 16.3 0.3 34 32.5 4.2 * Calculations based on a 10% energy loss due to air resistance, track vibration, etc. O 1996-2012 The Physics Classroom, All rights reserved. c. At the bottom of a 30-m high loop. d. At the bottom of a 45-m high loop. 5. According to Table 2, what effect does an increase in the radius of the top of a loop have upon the speed of the riders at the top of the loop? a. Increasing the radius of the top of the loop increases the speed at this location. 4 of 4 b. Increasing the radius of the top of the loop decreases the speed at this location. c. Increasing the radius of the top of the loop has no effect upon the speed at this locauon. d. Increasing the radius of the top of the loop seems to affect the speed; the manner in which it does is not predictable. 6. According to Table 2, at which of the following locations on a clothoid-shaped loop will a rider experience the most Gz? a. At the top of a 30-m high loop having a radius of 10 m. b. At the top of a 40-m high loop having a radius of 15 m. c. At the bottom of a 36-m high loop having a radius of 24 m. d. At the bottom of a 40-m high loop having a radius of 26 m. © 1996-2012 The Physics Classroom, All rights reserved. 7. The number of Gz that can be considered safe depends upon a variety of factors - the age and health of the rider, the duration of those Gz and how abruptly they are experienced. Generally, the coaster industry seeks to keep the number of Gz at 5.0 or below. Given this concern, which one of the following clothoid loops would not be used in a 60-m high design? a. hjoop=40 m; RA = 13 m; Rg = 30 m b. hloop = 40 m; RA = 13 m; Rg 38 m c. hjoop= 40 m; RA=17 m; Rg = 26 m d. hjoop = 45 m; RA=15 m; Rg 34 m A coaster engineer wishes to include a clothoid loop in a 60-m high design that provides the riders the following experience: 8. | Top of Loop Bottom of Loop Between 1.0 and 1.5 Gz Between 4.0 and 4.5 Gz Use Table 2 to determine which of the following loop parameters would be most suitable for such a coaster? a. hjoop = 40 m; RA = 15 m; Rg = 30 m b. hjoop = 40 m; RA =17 m; RB = 26 m c. hloop = 45 m; RA=15 m; Rg = 34 m d. hjoop = 45 m; RA = 15 m; Rg = 42 m %3D As the number of Gz at the top of an inverted (upside down) loop begins to approach zero, designers must regard the possibility of riders falling out of their seats as being the top safety hazard. If a design includes a clothoid loop with Gz less than 1.0 at the top, what change could be made to improve the safety at this location? a. Switch to a circular loop. c. Increase the radius at the loop top. 9. b. Increase the height of the loop. d. Decrease the radius at the loop top. 10. A coaster engineer has included a clothoid loop in his coaster design. As currently designed, riders run the risk of falling out of their seats at the top of the loop. Which one of the following design changes would NOT fix this safety issue? a. Decrease the height of the loop. b. Increase the height of the first drop. c. Decrease the radius of the top of the loop. d. Increase the height of the loop and/or increase the radius of the top of the loop. 11. For the same loop heights, how are the values of the clothoid-shaped loops of Table 2 different than the values for the circular loops of Table 1? a. The speed at the top of the loops is less for clothoid-shaped loops. b. The Gz at the top of the loops are greater for clothoid-shaped loops. c. The Gz at both the top and the bottom of the loops are less for clothoid-shaped loops. d. The speed at both the top and the bottom of the loops are less for clothoid-shaped loops. O 1996-2012 The Physics Classroom, All rights reserved.
College Physics
11th Edition
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Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Transcribed Image Text:5:16 1
Roller Coasters...
Roller Coaster Loops
Today's coasters provide riders with weightless sensations and abnormally large G forces.
Weightless sensations occur when the seat pushes on riders with a force that is less than their
weight. Large G forces occur when the seat pushes with a force much greater than their weight.
The number of Gs (Gz) is the ratio of this seat force to the rider's weight. The loop-the-loop
provides much of this thrill. The earliest designs included circular loops as shown in Figure 1.
Table 1 depicts the estimated speed (v) and the Gz for a rider starting from a 60-meter high drop.
Table 1: Circular Loops
Figure 1
hjoop
(m)
15
VA *
(m/s)
28.2
VB *
(m/s)
32.5
Gz
Gz
at A
at B
9.8
15.4
20
26.6
6.2
32.5
11.8
25
24.8
4.0
32.5
9.6
harop
hloop
30
23.0
2.6
32.5
8.2
32.5
32.5
35
21.0
1.6
7.2
45
16.3
0.2
5.8
50
13.3
0.0
32.5
5.3
Figure 2
Because of the inherent dangers associated with circular loops and
of the advent of tubular steel coasters, designers began to use non-
circular loops in their designs. The teardrop-shaped loop is often
called a clothoid loop. The shape is that of a cornu spiral, with a
constantly changing radius of curvature as shown in Figure 2.
Table 2 shows the effect of the radii (R) at the top and bottom of a
loop upon the rider experience.
Riop
Rottam
Rhottom
Table 2: Clothoid Loops
hloop
RA
VA *
Gz
RB
VB
Gz
(m)
(m/s)
(m/s)
at B
(m)
10
at A
(m)
30
23.0
4.4
20
32.5
6.4
36
12
20.6
2.6
24
32.5
5.5
40
13
18.8
1.8
26
32.5
5.2
40
13
18.8
1.8
30
32.5
4.6
1.8
1.8
1.4
40
13
18.8
34
32.5
4.2
13
18.8
18.8
40
38
32.5
3.8
40
15
26
32.5
5.2
40
17
18.8
1.1
26
32.5
5.2
45
15
16.3
0.8
30
32.5
4.6
45
15
16.3
0.8
34
32.5
4.2
45
15
16.3
0.8
38
32.5
3.8
45
15
16.3
0.8
42
32.5
3.6
45
18
16.3
0.5
34
32.5
4.2
45
21
16.3
0.3
34
32.5
4.2
* Calculations based on a 10% energy loss due to air resistance, track vibration, etc.
O 1996-2012 The Physics Classroom, All rights reserved.
Transcribed Image Text:c. At the bottom of a 30-m high loop.
d. At the bottom of a 45-m high loop.
5. According to Table 2, what effect does an increase in the radius of the top of a loop have
upon the speed of the riders at the top of the loop?
a. Increasing the radius of the top of the loop increases the speed at this location. 4 of 4
b. Increasing the radius of the top of the loop decreases the speed at this location.
c. Increasing the radius of the top of the loop has no effect upon the speed at this locauon.
d. Increasing the radius of the top of the loop seems to affect the speed; the manner in which
it does is not predictable.
6. According to Table 2, at which of the following locations on a clothoid-shaped loop will a
rider experience the most Gz?
a. At the top of a 30-m high loop having a radius of 10 m.
b. At the top of a 40-m high loop having a radius of 15 m.
c. At the bottom of a 36-m high loop having a radius of 24 m.
d. At the bottom of a 40-m high loop having a radius of 26 m.
© 1996-2012 The Physics Classroom, All rights reserved.
7. The number of Gz that can be considered safe depends upon a variety of factors - the age
and health of the rider, the duration of those Gz and how abruptly they are experienced.
Generally, the coaster industry seeks to keep the number of Gz at 5.0 or below. Given this
concern, which one of the following clothoid loops would not be used in a 60-m high
design?
a. hjoop=40 m; RA = 13 m; Rg = 30 m
b. hloop = 40 m; RA = 13 m; Rg 38 m
c. hjoop= 40 m; RA=17 m; Rg = 26 m
d. hjoop = 45 m; RA=15 m; Rg 34 m
A coaster engineer wishes to include a clothoid loop in a 60-m high design that provides the
riders the following experience:
8.
| Top of Loop
Bottom of Loop
Between 1.0 and 1.5 Gz
Between 4.0 and 4.5 Gz
Use Table 2 to determine which of the following loop parameters would be most suitable
for such a coaster?
a. hjoop = 40 m; RA = 15 m; Rg = 30 m
b. hjoop = 40 m; RA =17 m; RB = 26 m
c. hloop = 45 m; RA=15 m; Rg = 34 m
d. hjoop = 45 m; RA = 15 m; Rg = 42 m
%3D
As the number of Gz at the top of an inverted (upside down) loop begins to approach zero,
designers must regard the possibility of riders falling out of their seats as being the top
safety hazard. If a design includes a clothoid loop with Gz less than 1.0 at the top, what
change could be made to improve the safety at this location?
a. Switch to a circular loop.
c. Increase the radius at the loop top.
9.
b. Increase the height of the loop.
d. Decrease the radius at the loop top.
10. A coaster engineer has included a clothoid loop in his coaster design. As currently designed,
riders run the risk of falling out of their seats at the top of the loop. Which one of the
following design changes would NOT fix this safety issue?
a. Decrease the height of the loop.
b. Increase the height of the first drop.
c. Decrease the radius of the top of the loop.
d. Increase the height of the loop and/or increase the radius of the top of the loop.
11. For the same loop heights, how are the values of the clothoid-shaped loops of Table 2
different than the values for the circular loops of Table 1?
a. The speed at the top of the loops is less for clothoid-shaped loops.
b. The Gz at the top of the loops are greater for clothoid-shaped loops.
c. The Gz at both the top and the bottom of the loops are less for clothoid-shaped loops.
d. The speed at both the top and the bottom of the loops are less for clothoid-shaped loops.
O 1996-2012 The Physics Classroom, All rights reserved.
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