Transcript Simple Harmonic Motion

Simple Harmonic Motion
SHM Position, Velocity, and Acceleration
Springs and Simple Harmonic Motion
Equations of Motion
Conservation of Energy allows a calculation of the velocity
of the object at any position in its motion…
Energy in SHM
energy
velocity
Energy-time graphs
KE
PE
Total
Note: For a spring-mass system:
KE = ½ mv2  KE is zero when v = 0
PE = ½ kx2  PE is zero when x = 0 (i.e. at vmax)
Energy–displacement graphs
energy
KE
PE
Total
-xo
displacement
+xo
Note: For a spring-mass system:
KE = ½ mv2  KE is zero when v = 0 (i.e. at xo)
PE = ½ kx2  PE is zero when x = 0
Conservation of Energy For A Spring in Horizontal Motion
E =
E =
Kinetic + Elastic Potential
½ mv2 +
½ kx2
=
Constant
• At maximum displacement, velocity is zero and all energy is
elastic potential,
so total energy is equal to
½ kxo2
Simple Harmonic Motion - Energy
Ek (max) = 1/2mvo2
Ep (max) = 1/2kxo2
Where they happen
•Ek:
•Ep:
0
max
max
0
0
max
Potential energy in SHM
If
a = - ω2 x
then the average force applied trying to pull the
object back to the equilibrium position as it moves
away from the equilibrium position is…
F = - ½ mω2x
Work done by this force must equal the PE it gains
(e.g in the springs being stretched). Thus..
Ep (max) = ½ mω2xo2