1. Motion near an equilibrium position can be approximated by SHM
THIS LECTURE
SIMPLE HARMONIC MOTION (SHM)
Some examples of SHMs
Motion near an equilibrium position can be approximated by SHM

2. An example of SHM: mass on springk x
From Newton’s and Hooke’s law we derive
the differential equation
Solution:
Angular frequency
Amplitude (A) and phase constant (j) are determined by the initial conditions
Period

3. Water molecules O H
Water molecules can vibrate in a number of ways with well-defined frequencies

4. Water molecules O H
Water molecules can vibrate in a number of ways with well-defined frequencies

5. An example of SHM: mass on springs
From Newton’s and Hooke’s law we derive
the differential equation
Solution:
Angular frequency
Period

6. An example of SHM: mass on springs
From Newton’s and Hooke’s law we derive
the differential equation
Solution:
Angular frequency
Period

7. Which of these systems has the shortest T?
Period
Which of these systems has the shortest T?

8. Motion near an equilibrium position can be approximated by SHM
General motion x
Motion near an equilibrium position can be approximated by SHM

9. Motion near an equilibrium position can be approximated by SHM
Near a position of stable equilibrium, U can be approximated by a parabola, i.e. an harmonic potential U
x =0

10. SHM for a diatomic moleculeN2
Morse potential
r = distance between atoms
The Morse potential, named after physicist P.M. Morse, is a convenient model for the potential energy of a diatomic molecule.

11. SHM for a diatomic molecule
For r~r0

12. Problem: Particle of mass m sliding without friction in a spherical bowl of radius r
Show that for small amplitude oscillations, the motion can be approximated by a SHM
Determine T for small amplitude oscillations