Transcript Chapter 13
Chapter 13 Vibrations and Waves Chapter 13 Objectives • • • • • • • • • • Hooke’s Law Simple Harmonic Motion Elastic Potential Energy Velocity vs Position Harmonic Motion vs Circular Motion Wave properties – Frequency – Amplitude – Wavelength Wave Types Pendulum Superposition Wave Interference Hooke’s Law • 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 – Fs = -kx • k is spring constant • x is displacement from rest position of spring • (-) 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. – 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. – Must be back and forth along same path. • Also called periodic motion. Wave Properties • Amplitude – maximum distance object travels away from rest point •A – units: meters • Period – time it takes to complete one full cycle of motion •T – units: seconds • Frequency – the number of cycles per unit of time • number of waves past a given a point in one second – Inverse of = T-1 the Period units: inverse seconds (s-1) Elastic Potential Energy • Remember elastic potential energy can be found – PEelastic = ½ kx2 • k is called the spring constant – units: N/m • 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 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. v v v=0 E = KE E = KE + PEelastic E = PEelastic v E = KE + PEelastic x=0 Velocity vs Position of a Spring • 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 • 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. PEi + KEi = PEf + KEf ½kA2 = ½kx2 + ½mvf2 vf = k/m (A2 – x2) Simple Harmonic vs Uniform Circular • The period for uniform • circular motion is the amount of time necessary for one whole circle. The amplitude is the radius. In a circle 2πA T= v • The angular velocity of circular motion is equivalent to angular frequency for harmonic motion. – = T-1 = 1/2π k/m Simple Harmonic Motion T = 2π m/ k = 2π = k/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. With that established, we should be able to identify where an object is at any position along the wave. Maximum Amplitude Position of Object (Not Distance) = 2π x = A cos(t) Time Elapsed Pendulum • A pendulum also exhibits simple θ 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 L mg sin θ • 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. mg Wave Types • 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. More Wave Properties • 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. – Typically measured from crest-to-crest. λ A A λ Velocity vs Frequency and Wavelength • The velocity of a wave can be found very simply by remember what velocity is measuring. – distance over time v= v= Δx Δt λ T v=λ Remember that T-1 is the same as . 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. + Types of Interference • Constructive interference occurs • Destructive 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. • • when two waves meet that are out of phase. This will typically make a smaller wave. If the two waves are 180o out of phase, then the waves will cancel each other out.