Vibrations & Waves
Electromagnetic Radiation Spectrum
• Link between electricity & magnetism
Harmonic Motion I
• Type of motion in which an object moves along a repeating path (e.g, mass on a spring)
• System exhibits oscillatory motion (oscillations occur)
• Describe the motion in terms of velocity, amplitude of motion & position/displacement from
an EQM point
Harmonic Motion II
• In addition introduce quantities that describe the repeating nature of the motion
o Period (T): repeat time of motion
o Frequency (f): number of oscillations (or cycles) that occurs in a unit of time
• Unit: Hertz (Hz). 1 Hz = 1 oscillation per second
Harmonic Motion III
• Systems that oscillate in a sinusoidal fashion are called simple harmonic oscillators & they
exhibit simple harmonic motion (SHM)
• For SHM there must be a restoring force acting (e.g, provided by spring, gravity for pendula)
• E.g’s of SHM: mass is attached to a spring (horizontal, vertical arrangements), simple
pendulum
SHM: horizontal mass-spring system
• If object vibrates/oscillates back & forth over the same path,
each cycle taking the same amount of time, the motion is
called periodic. Mass &spring system is a useful model for
periodic system
• Ignore spring mass & friction for now
Elastic Deformations of Solids
• A deformation is a change in the size/shape of an object
• Many solids are stiff enough that the deformation can’t be
seen with the human eye, a microscope/other sensitive device is required to detect the
change in size/shape
• When the contact forces are removed, an elastic object returns to its original shape & size
Hooke’s Law for Tensile & Compressive Forces
• Suppose we stretch a wire by applying tensile forces of magnitude F to each end. The length
of the wire increases from L to L + ΔL.
• Stress & Strain
o The fractional length change is called the strain, it is a dimensionless measure of a deg
o ee of deformation
, (Fractional length change)
o Force per unit area is called the stress:
(force per unit cross-sectional area)
Hooke’s Law
➢ According to Hooke’s Law, the deformation is proportional to the deforming forces as long
as they’re not too large:
➢ Constant k depends on the length & cross-sectional area of the object. A larger cross-
sectional area A makes k larger, a greater length L makes k smaller
➢ Constant of proportionally Y is called an Elastic modulus/Young’s modulus, Y has the same
units as those of stress (Pa/N/m2), since strain is dimensionless
➢ Young’s modulus can be thought of as an inherent stiffness of a material, it measures the
resistance of a material to elongation/compression
Beyond Hooke’s Law
o A ductile material continues to stretch beyond its ultimate tensile strength without breaking,
the stress decreases from the ultimate strength
o For a brittle substance, the ultimate strength & the breaking points are close together
SHM: horizontal mass-spring system
,o We assume the surface is frictionless
o There’s a point where a spring is neither stretched/compressed is the EQM position. We
measure displacement form that point (x=0)
o Force is exerted by the spring depends on the displacement:
- Hooke’s Law
o The negative sign on the force indicates that it is a restoring force – it is directed to restore
the mass to its EQM position
o k = spring constant
o Force is not constant (x varying ), so the acceleration is not constant – can’t use equations
for motion with constant acceleration
o Simple Harmonic Motion
o A nonlinear force can be approximated as a
linear restoring force for small
displacements
o A spring in its relaxed position…. We choose an object’s EQM position as the origin, (x=0)
o Fx=-kx
o Energy Analysis in SHM
- The oscillator shows as it approaches endpoints & gains speed as it approaches the
EQM point
- Total mechanical energy of the mass & spring is constant:
, E = K + U = constant
U = ½ kx2
E = ½ kx2 + ½ mv2
o Max. displacement of the body is the amplitude A. Total energy E at the endpoints is:
Etotal = ½ kA2
@ x=0: Etotal = ½ mv2
½ mv2m = ½ k. A2
o Acceleration in SHM:
Fx = -kx = max
ax(t) = -k/m. x(t)
am = k/m. A
Simple Harmonic Motion
▪ Any vibrating system where the restoring force is proportional to the negative of the
displacement is a simple harmonic motion (SHM), & is often called the simple harmonic
oscillator
▪ Displacement: measured from the EQM
point
▪ Amplitude: max. displacement
▪ A cycle: a full to- & fro- motion. This figure
shows a ½ cycle
▪ Period: the time required to complete 1
cycle
▪ Frequency: the number of cycles completed
per second
SHM: vertical mass-spring system
Electromagnetic Radiation Spectrum
• Link between electricity & magnetism
Harmonic Motion I
• Type of motion in which an object moves along a repeating path (e.g, mass on a spring)
• System exhibits oscillatory motion (oscillations occur)
• Describe the motion in terms of velocity, amplitude of motion & position/displacement from
an EQM point
Harmonic Motion II
• In addition introduce quantities that describe the repeating nature of the motion
o Period (T): repeat time of motion
o Frequency (f): number of oscillations (or cycles) that occurs in a unit of time
• Unit: Hertz (Hz). 1 Hz = 1 oscillation per second
Harmonic Motion III
• Systems that oscillate in a sinusoidal fashion are called simple harmonic oscillators & they
exhibit simple harmonic motion (SHM)
• For SHM there must be a restoring force acting (e.g, provided by spring, gravity for pendula)
• E.g’s of SHM: mass is attached to a spring (horizontal, vertical arrangements), simple
pendulum
SHM: horizontal mass-spring system
• If object vibrates/oscillates back & forth over the same path,
each cycle taking the same amount of time, the motion is
called periodic. Mass &spring system is a useful model for
periodic system
• Ignore spring mass & friction for now
Elastic Deformations of Solids
• A deformation is a change in the size/shape of an object
• Many solids are stiff enough that the deformation can’t be
seen with the human eye, a microscope/other sensitive device is required to detect the
change in size/shape
• When the contact forces are removed, an elastic object returns to its original shape & size
Hooke’s Law for Tensile & Compressive Forces
• Suppose we stretch a wire by applying tensile forces of magnitude F to each end. The length
of the wire increases from L to L + ΔL.
• Stress & Strain
o The fractional length change is called the strain, it is a dimensionless measure of a deg
o ee of deformation
, (Fractional length change)
o Force per unit area is called the stress:
(force per unit cross-sectional area)
Hooke’s Law
➢ According to Hooke’s Law, the deformation is proportional to the deforming forces as long
as they’re not too large:
➢ Constant k depends on the length & cross-sectional area of the object. A larger cross-
sectional area A makes k larger, a greater length L makes k smaller
➢ Constant of proportionally Y is called an Elastic modulus/Young’s modulus, Y has the same
units as those of stress (Pa/N/m2), since strain is dimensionless
➢ Young’s modulus can be thought of as an inherent stiffness of a material, it measures the
resistance of a material to elongation/compression
Beyond Hooke’s Law
o A ductile material continues to stretch beyond its ultimate tensile strength without breaking,
the stress decreases from the ultimate strength
o For a brittle substance, the ultimate strength & the breaking points are close together
SHM: horizontal mass-spring system
,o We assume the surface is frictionless
o There’s a point where a spring is neither stretched/compressed is the EQM position. We
measure displacement form that point (x=0)
o Force is exerted by the spring depends on the displacement:
- Hooke’s Law
o The negative sign on the force indicates that it is a restoring force – it is directed to restore
the mass to its EQM position
o k = spring constant
o Force is not constant (x varying ), so the acceleration is not constant – can’t use equations
for motion with constant acceleration
o Simple Harmonic Motion
o A nonlinear force can be approximated as a
linear restoring force for small
displacements
o A spring in its relaxed position…. We choose an object’s EQM position as the origin, (x=0)
o Fx=-kx
o Energy Analysis in SHM
- The oscillator shows as it approaches endpoints & gains speed as it approaches the
EQM point
- Total mechanical energy of the mass & spring is constant:
, E = K + U = constant
U = ½ kx2
E = ½ kx2 + ½ mv2
o Max. displacement of the body is the amplitude A. Total energy E at the endpoints is:
Etotal = ½ kA2
@ x=0: Etotal = ½ mv2
½ mv2m = ½ k. A2
o Acceleration in SHM:
Fx = -kx = max
ax(t) = -k/m. x(t)
am = k/m. A
Simple Harmonic Motion
▪ Any vibrating system where the restoring force is proportional to the negative of the
displacement is a simple harmonic motion (SHM), & is often called the simple harmonic
oscillator
▪ Displacement: measured from the EQM
point
▪ Amplitude: max. displacement
▪ A cycle: a full to- & fro- motion. This figure
shows a ½ cycle
▪ Period: the time required to complete 1
cycle
▪ Frequency: the number of cycles completed
per second
SHM: vertical mass-spring system
