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Lecture 3-4C: Energy Diagrams
Reminder: PVA Diagrams po
sitio
n
time
positive slope ⇒ positive velocity
positive curvature (increasing slope) ⇒ positive acceleration
zero slope ⇒ zero velocity
Position-velocity-acceleration diagrams allow us to visualize and graphically compute 1D motion
Energy Diagram of a Spring
E
xeq xi position
Energy Diagram of a Spring
E
xeq xi x position
Energy Diagram of a Spring
E
position xeq xi x
Energy Diagram of a Spring
E
position xeq xi
Energy Diagram of a Spring
E
position xeq xi
Energy Diagram of a Spring
E
position xeq xi
Energy Diagram of a Spring
E
position xeq xi
Energy Diagram of a Spring
E
position xeq
Not
Allo
wed
xi
Not
Allo
wed
Allowed
Constraints on 2D motion
REPLACE WITH KOTSON_8012_FALL2008.AVI
Energy Diagram of a Spring
E
position xeq xi x
PE slope < 0
Energy Diagram of a Spring
E
position xeq xi x
PE slope > 0
Energy Diagram of a Spring
E
position xeq xi x
PE slope = 0
Force and Potential Energy Definition of Potential Energy in 1D:
In 2D and 3D:
where is the gradient operator; in rectangular coordinates:
Energy Diagram of a Spring
E
position xeq xi
Curvature ccccccc
⇒ stable about equilibrium point
The Tipping Pencil
θ Center of mass = L/2
h L/2
The potential energy of a pencil of mass m and height L standing on its end is just:
PE = mgh where h is the height of the center of mass:
h = (L/2) cos θ ⇒ PE = mg (L/2) cos θ
Energy Diagram of a Tipping Pencil
E
angle θeq = 0º
Curvature ccccccc
⇒ unstable about equilibrium point
Van der Waals Potential
E
interatomic separation
The van der Waals or Lennard potential approximately describes molecular bonding energies.
Form of potential allows us to identify bound and unbound states and radii where attraction or repulsion dominates
bound
unbound
attraction repulsion
Energy Dissipation
E
position xeq xi x1 x2 x3
images from Institute of Physics https://tap.iop.org/vibration/shm/306/page_46606.html and wikipedia https://en.wikipedia.org/wiki/Damping#/media/File:Damped_spring.gif
Summary Energy diagrams allow us to visualize the
potential and total energy of system, and calculate:
KE = E - PE constraints on motion conservative forces as negative of PE slope stability/instability based on PE curvature
Dissipative forces can be represented as a lowering the total energy "bar" over time
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