Analogous Oscillations of a Pendulum and Liquid in a U-Tube
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
The simple harmonic motion for a liquid in a U-tube and a simple pendulum as proposed by Newton in his Principia Mathematica is animated in this Demonstration, with similar design and constraints (no friction) as in Newton's original work. All units are imperial. The energy level plot on the left is the water's energy; however, the timing is also for the pendulum. The clock 
starts with motion already in progress at the displacement position for maximum kinetic energy for both the water and the pendulum. The assumed initial condition would be pressure on side 
of the tube, causing displacement from equilibrium of 
and 
to a variable distance 
at 
and 
; then the pressure is released to allow the water to move.
Contributed by: Conrad Benulis (January 2016)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The snapshots show PE maximum, PE equal to KE, and KE maximum.
The "Period (sec)" display is for both the pendulum and the water. The quantity 
is half the length of the water column and the full length of the pendulum support line. The displacement of the water above or below equilibrium ( and ) is represented by . Constants used: 
, water density 
, and radius of water column 
.
Generally, it's best to fix the length and displacement and then animate the time. To animate or , make their animation rates much slower than the clock 
.
References
[1] I. Newton, Philosophiæ Naturalis Principia Mathematica, 3rd ed., Book II, Sec. VIII: Proposition XLIV, Theorem XXXV, 1726.
[2] MIT Deptartment of Physics, Physics 8.01, "Worked Example W13D1-1: U-Tube Oscillations Solution." (Can be found here.)
Permanent Citation
Motions of a Simulated Damped Harmonic Oscillator
Teun Zijp
Creep and Stress Relaxation for Four-Element Viscoelastic Solids and Liquids
Mark D. Normand and Micha Peleg
Drag and Gravity Forces on a Falling Sphere
Conrad A. Benulis
Forces on a Partially Submerged Gate
Rachael L. Baumann
Active Shock Absorbers
Enrique Zeleny
Manometers
Rachael L. Baumann
Buoyancy of a Balloon
Zachary Hebebrand and James Drost
Fracture Element in Rheological Models
Mark D. Normand and Micha Peleg
Falling Body with Zero, Linear, or Quadratic Air Resistance
Anne Tabor-Morris
A Contact Element in Rheological Models
Mark D. Normand and Micha Peleg
