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