Chapter 11: Vibrations and Waves

• Periodic Motion – any repeated motion with regular time intervals

• Simple Harmonic Motion – vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium

SHM

• Speed is max at equilibrium

• Speed is least at max displacement

• Max force is at max displacement

• Max acceleration is at max displacement

Simple Harmonic Motion

Hooke’s LawFelastic = -kx

Spring force= -(spring constant x displacement)SI unit for k = N/m

(-) signifies the direction of the spring force is always opposite the direction of the mass’s displacement from

equilibrium

Example

• If a mass of 0.55kg attached to a vertical spring stretches the spring 2.0cm from its original equilibrium position, what is the spring constant?

Example

• Suppose the spring in the previous example is replaced with a spring that stretches 36 cm from its equilibrium position. What is the spring constant Is this spring stiffer or less stiff than the one

on the original example?

Simple Pendulum• Weight (force of gravity) is

the restoring force• SHM for small angles only

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Bellringer 12/3

• A 76N crate is hung from a spring (k=450N/m). How much displacement is caused by the weight of this crate?

Measuring Simple Harmonic Motion

• Amplitude – maximum displacement from equilibrium

• Period – (T) – time it takes to complete one cycle

• Frequency (f) – number of cycles per unit of time (Hertz = s-1 )

Amplitude

Measuring SHM

• Frequency and period are inversely related

f 1T

or T 1f

Example

• The reading on a metronome indicates the number of oscillations per minute. What are the frequency and period of the metronome’s vibrations when the metronome is set at 180?

Pendulum in SHM

• Period of a simple pendulum depends on pendulum length and free-fall acceleration Length is measured from center of mass of

the bob and the pivot point

Period of a Simple Pendulum in SHMT = 2π L

√ ag Period = 2 π x sqrt of (length divided by free fall)

Example

• You need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extending from the ceiling almost touches the floor and that its period is 12s. How all is the tower?

Mass/Spring in SHM

• Period of a mass spring system depends on mass and spring constant

Period of a Mass-Spring System in SHMT = 2π m √ k

Period = 2 π x sqrt of (mass divided by spring constant)

Example

• A child swings on a playground swing with a 2.5m long chain. What is the period of the child’s motion? What is the frequency of vibration?

Example

• The body of a 1275kg car is supported on a frame by four springs. Two people riding in the car have a combined mass of 153kg. When driven over a pothole in the road, the frame vibrates with a period of 0.840s. For the first seconds, the vibration approximates simple harmonic motion. Find the spring constant of a single spring.

Bellringer 12/4

• A spring of spring constant 30.0N/m is attached to different masses, and the system is set in motion. Find the period and frequency of vibration for masses of the following magnitudes: 2.3kg 15kg 1.9kg

Properties of Waves

• Wave – motion of disturbance that transmits energy

• Medium – physical environment through which a disturbance can travel

• Mechanical waves – requires a medium• Electromagnetic waves – do not require a

medium

• Pulse wave – single traveling pulse• Periodic wave – continuous pulse waves• Sine wave – source vibrates with SHM

Wave Types

• Transverse wave – particles vibrate perpendicularly to the direction the wave is traveling Wavelength (λ)– distance the wave travels

along its path during one cycle

Wave Types

• Longitudinal wave – particles vibrate parallel to the direction the wave is traveling Crests – coils are compressed (high density and

pressure) Troughs – coils are stretched (low density and

pressure)

Period and Frequency

• Wave frequency – number of waves that pass a given point in a unit of time

• Period – time needed for a complete wavelength

Wave Speed

• Speed of a mechanical wave is constant for any given medium

Wave Speedv = f λ

Speed of wave = frequency x wavelength

Example

• A piano string tuned to middle C vibrates with a frequency of 262 Hz. Assuming the speed of sound in air is 343m/s, find the wavelength of the sound waves produced by the string.

Example

• A piano emits frequencies that range from a low of about 28Hz to a high of about 4200Hz. Find the range of wavelengths in air attained by this instrument when the speed of sound in air is 340m/s.

Bellringer 12/5

• A tuning fork produces a sound with a frequency of 256 Hz and a wavelength in air of 1.35m. What value does this give for the speed of

sound in air? What would be the wavelength of this same

sound in water in which sound travels at 1500m/s

• Constructive interference – two or more waves are added together to form a resultant wave

• Destructive interference – two or more waves with displacements on opposite sides of equilibrium position are added together to make a resultant wave

Reflection

• At a free boundary – waves are reflected

• At a fixed boundary – waves are reflected and inverted

Free End Reflection

Fixed End Reflection

Standing waves

• Standing wave – results when two waves with same frequency, wavelength, and amplitude travel in opposite directions and interfere Nodes – point in standing wave where it

maintains zero displacement Antinodes – point in standing wave, halfway

between two nodes, at which largest displacement occurs

Standing Waves

Standing Waves

Example

• A wave of amplitude 0.30m interferes with a second wave of amplitude 0.20m. What is the largest resultant displacement that may occur?