Chapter 13 VibrationsandWaves

Chapter 13 Objectives Hooke’s Law Hooke’s Law Simple Harmonic Motion Simple Harmonic Motion Elastic Potential Energy Elastic Potential Energy Velocity vs Position Velocity vs Position Harmonic Motion vs Circular Motion Harmonic Motion vs Circular Motion Wave properties Wave properties –Frequency –Amplitude –Wavelength Wave Types Wave Types Pendulum Pendulum Superposition Superposition Wave Interference Wave Interference

Hooke’s Law The simplest type of vibrational motion is a mass attached to a spring moving without any frictional forces. The simplest type of vibrational motion is a mass attached to a spring moving without any frictional forces. –No friction –No air resistance The force provided by the spring is The force provided by the spring is – F s = -kx k is spring constant k is spring constant x is displacement from rest position of spring x is displacement from rest position of spring (-) because the spring is always providing a force opposite the motion of the mass (-) because the spring is always providing a force opposite the motion of the mass

Simple Harmonic Motion Simple harmonic motion occurs when the net force acting in the direction of motion follows Hooke’s Law. Simple harmonic motion occurs when the net force acting in the direction of motion follows Hooke’s Law. –That is no frictional forces present and… –The force is proportional to the displacement but opposite in direction Basically with simple harmonic motion, the motion will repeat a cycle of back and forth forever. Basically with simple harmonic motion, the motion will repeat a cycle of back and forth forever. –Must be back and forth along same path. Also called periodic motion. Also called periodic motion.

Wave Properties Amplitude Amplitude –maximum distance object travels away from rest point A –units: meters Period Period –time it takes to complete one full cycle of motion T –units: seconds Frequency Frequency –the number of cycles per unit of time number of waves past a given a point in one second number of waves past a given a point in one second –   Inverse of the Period   = T -1  units: inverse seconds (s -1 )

Elastic Potential Energy Remember elastic potential energy can be found Remember elastic potential energy can be found –PE elastic = ½ kx 2 k is called the spring constant k is called the spring constant –units: N/m x is the distance the spring is stretched or compressed away from its resting point x is the distance the spring is stretched or compressed away from its resting point The energy is only stored in a spring when it is either stretched or compressed. The energy is only stored in a spring when it is either stretched or compressed. The potential energy in a spring is always positive. The potential energy in a spring is always positive. –That is because x is squared.

How to Use Elastic Potential Energy Be sure to identify what types of energy are present at each position of the problem. Be sure to identify what types of energy are present at each position of the problem. x = 0 v E = KE v E = KE + PE elastic E = PE elastic v = 0 v E = KE + PE elastic

Velocity vs Position of a Spring The velocity of an object attached to a spring can be found by knowing its position. The velocity of an object attached to a spring can be found by knowing its position. –Granted the velocity will be the same at two positions coming in or going out coming in or going out Energy must be conserved Energy must be conserved –So the stored energy at the maximum position should be equal to the total kinetic and elastic potential energy at any other point in the process. PE i + KE i = PE f + KE f ½kA 2 = ½kx 2 + ½mv f 2 v f = k / m (A 2 – x 2 ) 

Simple Harmonic vs Uniform Circular The period for uniform circular motion is the amount of time necessary for one whole circle. The period for uniform circular motion is the amount of time necessary for one whole circle. The amplitude is the radius. The amplitude is the radius. The angular velocity of circular motion is equivalent to angular frequency for harmonic motion. The angular velocity of circular motion is equivalent to angular frequency for harmonic motion. –  T =T = 2πA v In a circle Simple Harmonic Motion T =T =2π2π  m/km/k  = T -1 = 1/2π= 1/2π  k/mk/m  = 2π  =  k/mk/m

Position, Velocity, and Acceleration vs Time By relating angular velocity to angular frequency, we can consider this to be our judge of how “fast” the wave is traveling. By relating angular velocity to angular frequency, we can consider this to be our judge of how “fast” the wave is traveling. With that established, we should be able to identify where an object is at any position along the wave. With that established, we should be able to identify where an object is at any position along the wave. x = A cos(  t)   = 2π  Maximum Amplitude Time Elapsed Position of Object (Not Distance)

Pendulum A pendulum also exhibits simple harmonic motion under certain conditions. A pendulum also exhibits simple harmonic motion under certain conditions. –The force must follow Hooke’s Law by being proportional to the displacement at all times –The initial angle of displacement must be less than 15 degrees The restoring force to maintain simple harmonic motion acts tangential to the path of the swing. The restoring force to maintain simple harmonic motion acts tangential to the path of the swing. –That force is the component of the weight of the object that is tangent to the circular path of the pendulum. L θ mg mg sin θ mg cos θ

Wave Types A transverse wave is a wave that its particles move perpendicular to the overall motion of the wave. A transverse wave is a wave that its particles move perpendicular to the overall motion of the wave. A longitudinal wave is a wave in which its particles move in the same direction as the overall motion of the wave. A longitudinal wave is a wave in which its particles move in the same direction as the overall motion of the wave.

More Wave Properties Besides calculating the amplitude and frequency of a wave, we can also calculate the wavelength. Besides calculating the amplitude and frequency of a wave, we can also calculate the wavelength. The wavelength (λ) of a wave is the distance between two successive points on the wave. The wavelength (λ) of a wave is the distance between two successive points on the wave. –Typically measured from crest-to-crest. λ A λ A

Velocity vs Frequency and Wavelength The velocity of a wave can be found very simply by remember what velocity is measuring. The velocity of a wave can be found very simply by remember what velocity is measuring. –distance over time v = ΔxΔx ΔtΔt λ T Remember that T -1 is the same as . v =  λ λ

Superposition Principle If two or more waves are moving through a medium, the resultant wave is found by adding together the displacements of the individual waves point by point. If two or more waves are moving through a medium, the resultant wave is found by adding together the displacements of the individual waves point by point. +

Types of Interference Constructive interference occurs when two waves meet that are in phase. Constructive interference occurs when two waves meet that are in phase. –Waves that are in phase have crests and valleys that line up exactly. This type will make a bigger wave. This type will make a bigger wave. Destructive interference occurs when two waves meet that are out of phase. Destructive interference occurs when two waves meet that are out of phase. This will typically make a smaller wave. This will typically make a smaller wave. If the two waves are 180 o out of phase, then the waves will cancel each other out. If the two waves are 180 o out of phase, then the waves will cancel each other out.