Vibrations and Waves

Chapter 14

Physics Principles and Problems

Zitzewitz, Elliot, Haase, Harper, Herzog, Nelson, Nelson, Schuler and Zorn

McGraw Hill, 2005

Periodic Motion - motions that all repeat in a regular cycle.

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Simple Harmonic Motion

• During motion, object has one point where the net force is zero (equilibrium).

• Whenever the object if pulled away from equilibrium, the net force becomes non-zero and works to pull the object back to equilibrium.

• The force needed to restore equilibrium is directly proportional to the displacement of the object.

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• Period (T) - time for the object to complete one cycle of motion.

• Amplitude - maximum distance that the object moves away from equilibrium.

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• Hooke’s Law - The force exerted by a spring is equal to the spring constant times the distance the spring is compressed or stretched from its equilibrium point. F = -kx

• Potential energy in a stretched or compressed spring is equal to one half the product of the spring constant and the square of the displacement. PEsp = 1/2kx2

http://hyperphysics.phy-astr.gsu.edu/HBASE/imgmec/hook.gif

Period of a Pendulum is equal to two times pi times the square root of the length of the pendulum divided by acceleration due to

gravity. T = 2√l/g

http://dev.physicslab.org/img/14c7c356-6f6e-49e4-a953-2396343435fb.gif http://www.gailruby.com/simplependulum.gif

Resonance - the increase in the amplitude of a vibrating or oscillating object due to the application

of small forces during regular intervals.

• Pumping your legs when swinging.• Wind causing the Tacoma Narrows bridge collapse.

http://www.sciencenews.org/pages/sn_arc99/1_30_99/bridge.gif http://www.worsleyschool.net/science/files/tacoma/pic3.JPG

Wave - a disturbance that carries energy through matter or space (matter is not transferred).

http://moments.capturednature.com/wp-content/uploads/2007/01/water-ripples.jpg

• Transverse Wave - wave pulse (disturbance) vibrates perpendicular to the direction of the wave’s motion.

• Longitudinal Wave - wave pulse (disturbance) vibrates parallel to the direction of the wave’s motion.

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Wave Measurements

• Speed - the speed of a wave pulse is equal to the displacement of the wave peak divided by the amount of time. v = ∆d / ∆t

• Amplitude - the maximum displacement of the wave from its position of rest. For waves that have the same speed, the rate of energy transfer is proportional to the square of the amplitude.

• Wavelength - the shortest distance between points where the wave pattern repeats itself.

• Period - the time (T) it takes wave to complete one cycle (wavelength).

• Frequency - the number of complete oscillations it makes each second.

• The frequency of a wave is equal to the reciprocal of the period. f = 1 / T

• The wavelength is equal to velocity divided by the frequency. = v / f

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Wave Behavior

• Incident wave - initial wave that strikes a boundary.

• Reflected wave - returning wave after the collision of the incident wave with a boundary.

http://www.iop.org/activity/education/Teaching_Resources/Teaching%20Advanced%20Physics/Vibrations%20and%20Waves/Images%20300/img_mid_4472.gif

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Principle of Superposition

• When two waves exist in the same place in the medium at the same time the displacement of the medium is equal to sum of the two individual wave displacements.

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Wave Interference - the result of the superposition of two or more waves.

• Destructive - when two waves with opposite amplitudes meet. If these are equal amplitudes then the point that they meet (node) does not move.

• Constructive - when two waves with amplitudes in the same direction meet. Produces an a point where the largest amount of displacement occurs (antinode).

• Standing waves are produced through the constructive and destructive interference of waves. Increasing the frequency of oscillations increases the number of nodes and antinodes.

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Waves in 2 Dimensions

• When an incident wave strikes a barrier it will be reflected. The angles formed between the incident ray and the normal (draw line that is perpendicular to the barrier) and the reflected ray and the normal are equal.

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• Refraction is the change in direction of a wave at the boundary between two different media.

• For example waves approaching shore will always have the same frequency, however, their wavelength and velocity decreases. This will cause a change in direction between these two media.

http://www.gcsescience.com/Refraction-Water-Waves.gif