Upload
others
View
8
Download
0
Embed Size (px)
Citation preview
Auxiliary Tidal
Force Effects [AUTHOR(S) OMITTED FOR WEB POSTING]
h
Background Tides
Gravity
Celestial bodies
Constructive Gravitational Forces Tidal Phases
Outline Problem Scenario
Assumptions
Mathematical Development and Solution
Results and Conclusion
Extensions
Problem Scenario The Sun’s size is increasing.
What are the tidal effects?
Compare tidal shift to the RMS Titanic.
Assumptions 1. Considering only 3 bodies: The Earth, Sun, and Moon
2. Neap Tide Position
3. These bodies are stationary
4. Sun’s size increases, all other sizes remain constant
5. Each body possesses uniform density
6. As the Sun grows, it will continually have uniform density
7. Traveling momentum of the Titanic is negated
Mathematical Development 1: Modeling Tides
Newton’s Law of Universal Gravitation:
A New Formula:
center of mass (G)
center of buoyancy (B) (the center of the submerged
volume of the boat).
metacenter (M)
Mathematical Development 2: Capsizing Vessels and GZ Curves
Righting Arm (GZ length)
Mathematical Development 2: Capsizing Vessels and GZ Curves
Solution Assume time variable, and assume initial Lunar
displacement
Find a path from increasing Sun mass to Solar tidal
displacement by use of formulae:
Sun’s Mass(t) ⇒
Volume ⇒
Radius ⇒
Distance from Sun to Earth ⇒
Ratio of Sun's gravitational force to the Moon's ⇒
Solar Tidal Displacement
Results Refer to Excel Tables
Table with Mass*e^(t)
t = 6
t = 17
y = 0.2419e1.0224x R² = 0.9999
0
500
1000
1500
2000
2500
3000
0 5 10
Dis
pla
cem
en
t (m
)
Time t
Tidal Displacement
Expon.…
Conclusions 1. At t = 7, Tidal displacement exceeds record wave
height
2. At t = 17, Sun engulfs Earth (negative distance)
3. With the Sun tripled in size, tidal displacement is not
very significant
The Titanic has fallen
Extensions
Measure danger of capsizing due to size waves and length
of righting arm:
Formula for relating size of waves to a vessel’s GZ curve
and righting arm
Probability of recovering from capsizing