Download - Simple Harmonic Motion (SHM)
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
Fg
Θ
Describing SHM
• Period (T)
• Full swing– Return to
original position
Fg
Θ
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
• T = period of motion (seconds)
• L = length of pendulum
• g = 9.8 m/s2
2L
Tg
Energy in SHO
• EPE = ½ k x2
• KE = ½ m v2
• E = ½ m v2 + ½ k x2
• 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
• A: overdamped
• B: critically damped
• C: underdamped
Damping
• 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
Resonance
