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 massFelastic = -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 15o

• 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

• PEg 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 = amplitude2

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