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

SHM Quantities

Energy in SHO

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