Vibrations and Waves

Extreme Example Tacoma Narrows Bridge Tacoma Narrows Bridge It stood for only 3 months before…. It stood for only 3 months before….

Vibrations and waves SHM SHM Oscillation Oscillation Periodic Periodic f= -KX F= Restoring force K= spring constant X= displacement

Equilibrium Position Equilibrium Position Displacement Displacement Amplitude Amplitude Period-T, time needed for 1 full cycle (sec) Period-T, time needed for 1 full cycle (sec) Frequency- f, # of cycles per second (Hz) Frequency- f, # of cycles per second (Hz)

f= 1/T T= 1/f *Car Spring Example, pg. 311 Section 11-2 draws heavily from chpt. 6 Read on your own pg

Section 11-3, read the deriving of formula… Section 11-3, read the deriving of formula… T s = 2  m/k Formula for Period of Spring m= mass k= spring constant

Not direct relationship! Not direct relationship! T m/k Mass must QUADRUPLE to double period Mass must QUADRUPLE to double period We can Substitute… We can Substitute… f = 1/T = f = 1/2  k/m

What are T & f of the car example on pg.311 after hitting a bump? Assume shock absorbers are poor so car really oscillates a lot. What are T & f of the car example on pg.311 after hitting a bump? Assume shock absorbers are poor so car really oscillates a lot. T = 2  √m/k = 6.28 √ 1400kg/ 6500N/m =.92sec F = 1/.92 = 1.09Hz

A small insect (.3g) is caught in a spider web (mass-less). The web vibrates at 15Hz A small insect (.3g) is caught in a spider web (mass-less). The web vibrates at 15Hz a. estimate the value of K for the web b. find f for an insect of mass.1g.

f = 1/2  √K/m K = (2  f) 2 m = (6.26 x 15) 2 (3 x kg) 2.7 N/m Sub in for M 1 x kg f = 26Hz

Simple Pendulum What determines T for a Pendulum? What determines T for a Pendulum? Mass of bob Amplitude of swing Length of string Gravity of location × ×

Formula T p = 2  √ L / g F = 1 / T = ½  √ g / L For pendulums to be said to be in SHM the displacement angle has to be small- less than 15°

Damped Harmonic Motion Amplitude of a swinging spring on a pendulum slowly decreases in time until the oscillations stop all together. Amplitude of a swinging spring on a pendulum slowly decreases in time until the oscillations stop all together. X Time

Damping is caused by air friction and internal friction within the system Damping is caused by air friction and internal friction within the system Common dampening systems are shock absorbers and door closing mechanisms. Common dampening systems are shock absorbers and door closing mechanisms.

A = over damped (too slow) B= Critically damped C=under damped (still oscillating) C B A x time

Resonance… Forced Vibrations Objects have natural resonant frequencies. When vibrations are put on an object that are at the natural resonant frequency, an increase in amplitude is observed. Objects have natural resonant frequencies. When vibrations are put on an object that are at the natural resonant frequency, an increase in amplitude is observed.

If pushing is random- swing just bounces around If pushing is random- swing just bounces around But if you push with frequency equal to Natural Resonant Frequency(NRF) you can get a big amplitude But if you push with frequency equal to Natural Resonant Frequency(NRF) you can get a big amplitude Swing

examples… Tuning Fork Guitars

Wave Motion mechanical waves for now… mechanical waves for now… -OVER HEAD- -OVER HEAD- Is the velocity of a wave moving along a cord the same as the velocity of a particle of the cord? Is the velocity of a wave moving along a cord the same as the velocity of a particle of the cord? NO – velocities are different so are directions…

Waves don’t carry matter, carry matter,theytransferenergy

The parts of a single frequency wave

Types of waves Wave Motion Transverse Compression Rare factions Longitudinal S- waves Earth quake waves P- waves; only P travels through liquid

Water Waves Water Waves

Wave Formula V= f Velocity=wavelength x frequency

Reflection of Waves Wave fronts Plane waves Law of reflection

Wave Interference Principle of superposition Destructive interference Constructive Interference In phase or Out of phase

Standing Waves: Resonance Standing Waves Nodes Antinodes Natural frequencies Fundamental frequencies Overtones z53w_k_j_A#!

Refraction and Diffraction