Vibrations & Waves Chapter 11

Simple Harmonic Motion Periodic motion = repeated motion Good example of periodic motion is mass on a spring on a frictionless surface

Points of mass on a spring Force is always toward equilibrium Force & acceleration are zero at equilibrium Speed is greatest at equilibrium Momentum of mass continues it moving past equilibrium Force & acceleration are greatest at maximum displacement Force is referred to as a restoring force Damping: friction slows movement of mass-spring system

Hooke’s Law Restoring force is directly proportional to displacement of mass F elastic = -kx Negative shows that force is always opposite direction of displacement k is the spring constant; units are N/m; Greater the k the stiffer the spring

Simple Harmonic Motion Describes any periodic motion that is the result of a restoring force that is proportional to displacement Always a back and forth motion over the same path

Springs & Elastic PE When you stretch a spring, you store energy This energy is elastic PE When spring is released, object attached to it has KE This energy is conserved

Simple Pendulums Applies only for angles under 15 o Mass called a bob attached to end of massless string X-component of gravity is restoring force of a pendulum; y-component & tension of string balance out PE g increases as pendulum’s displacement increases

Internet Sites Mass on a Spring Mass on a Spring #2

Measuring Simple Harmonic Motion Amplitude: max displacement from equilibrium position; angle in pendulums; amount of compression/stretch in spring Period (T): time for one complete cycle Frequency (f): # of cycles in a unit of time; cycles per second; measured in Hertz (Hz) T and f are reciprocals of each other

Factors affecting Period of a Pendulum Period of pendulum depends on length of string and gravity For small amplitudes, mass & amplitude do not affect period Formula on bottom of p. 377

Period of mass-spring system Period of mass-spring depends on mass of object & spring constant Restoring force in Hooke’s law is dependent on displacement, not mass Increasing mass on spring increases inertia but not restoring force Heavier masses have longer periods Larger spring constants provide greater force, greater acceleration, shorter periods Formula on page 380

Properties of Waves Medium is not carried along with a wave Particles of medium are disturbed but return to original position after energy passes through Medium: material wave disturbance travels through Mechanical waves require a medium

Wave Types Pulse waves are composed of a single pulse If source of a wave is undergoing SHM, it produces a sine wave Transverse waves have medium moving at right angles to the direction the wave is moving

Parts of a Wave Crest: highest pt above equilibrium Trough: lowest pt below equilibrium Amplitude: distance from equilibrium to crest or trough Wavelength: distance wave travels in 1 cycle; represented by lambda ( 

Longitudinal Waves Has wave motion & particle motion in same direction (parallel) Compression: compressed region of wave; similar to crest; high density region Rarefaction: stretched region of wave; similar to trough; low density region

Speed of a Wave Source of all wave motion is vibrating obj Frequency of wave = frequency of source Frequency: # of wave passing in a given amt of time Period: time required for 1 complete vibration (time for 1 wavelength)

Speed of a Wave v = f Speed is constant for a medium Speed changes only when wave changes mediums or properties of the medium change Even if f is changed, will change to make product (v) the same

Waves Transfer Energy Medium remains in basically same location Energy is transferred by wave from 1 place to another Matter vibrates & transfers energy; matter does not change location Amplitude & energy are directly related Energy transfer = amplitude 2

Wave Interactions Waves are disturbances of matter (not matter) so they can occupy the same space Superposition: combination of 2 overlapping waves Figure 15, page 389

Interference Displacements in the same direction produce constructive interference –Waves maintain their original properties after interference –Resultant wave is the sum of the amplitude –Add amplitudes together to get larger waves See Figure 16, page 390

Interference Displacements in opposite directions produce destructive interference –Occurs when crests & troughs meet –Resultant wave is difference between pulses –Resultant has smaller amplitude If 2 waves have equal amplitude, difference is zero & it is complete destructive interference

Interference Interference patterns are for both transverse & longitudinal waves

Standing Waves Standing wave: resultant wave pattern that appears to be stationary; composed of alternating constructive/destructive interference Nodes: pts on standing waves where there is complete destructive interference Antinodes: constructive interference