Transcript Simple Harmonic Motion v20120109

Simple Harmonic Motion
(SHM)
(and waves)
• What do you think Simple Harmonic
Motion (SHM) is???
Defining SHM
• Equilibrium
position
• Restoring force
– Proportional to
displacement
• Period of Motion
• Motion is back &
forth over same
path
Describing SHM
• Amplitude
Θ
Fg
Describing SHM
• Period (T)
• Full swing
– Return to
original
position
Θ
Fg
Frequency
• Frequency- Number of times a SHM
cycles in one second (Hertz = cycles/sec)
• f = 1/T
SHM Descriptors
• Amplitude (A)
– Distance from
start (0)
• Period (T)
– Time for
complete swing
or oscillation
• Frequency (f)
– # of oscillations
per second
Oscillations
• SHM is exhibited by simple harmonic
oscillators (SHO)
• Examples?
Examples of SHOs
• Mass hanging from spring, mass driven by
spring, pendulum
SHM for a Pendulum
L
T  2
g
• T = period of
motion (seconds)
• L = length of
pendulum
• g = 9.8 m/s2
SHM Quantities
Energy in SHO
Energy in SHO
•
•
•
•
EPE = ½ k x2
KE = ½ m v2
E = ½ m v2 + ½ k x 2
E = ½ m (0)2 + ½ k A2
E = ½ k A2
• E = ½ m vo2 + ½ k (0)2
E = ½ m vo2
Velocity
• E = ½ m v2 + ½ k x2
• ½ m v2 + ½ k x2 = ½ k A2
• v2 = (k / m)(A2 - x2) = (k / m) A2 (1 - x2 / A2)
– ½ m vo2 = ½ k A2
– vo2 = (k / m) A2
• v2 = vo2 (1 - x2 / A2)
• v = vo √ 1 - x2 / A2
Damped Harmonic Motion
• due to air resistance and internal friction
• energy is not lost but converted into
thermal energy
Damping
• A: overdamped
• B: critically damped
• C: underdamped
Resonance
• occurs when the
frequency of an applied
force approaches the
natural frequency of an
object and the damping is
small (A)
• results in a dramatic
increase in amplitude