Transcript Chapter 13
Chapter 12 Vibrations and Waves Chapter 12 Objectives • • • • • • • • Hooke’s Law Simple Harmonic Motion Elastic Potential Energy 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 – x will be negative when the spring is stretched – x will be positive when the spring is compressed • (-) 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 the Period = T-1 units: hertz (Hz) which is equivalent to an inverse second. (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 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 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. 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 Period of a Pendulum • A pendulum exhibiting simple harmonic motion will have a period that depends on: – length of the pendulum, L • longer the pendulum, longer the arc (larger amplitude) – So it would take more time to complete one cycle – acceleration due to gravity • the faster gravity can pull the pendulum, the shorter the time it takes to complete its cycle (period) T = 2 (L/g) Period of a Spring System • The period of a mass oscillating in simple harmonic motion depends on – mass of the spring system • more mass takes more time to accelerate • and the longer the path (larger amplitude) it creates – spring constant • stiffer the spring, the less time it takes to accelerate – It only depends on these two things because gravity remains constant and those are the only two values that could vary in the system. T = 2 (m/k) Standing Waves • A standing wave is a wave that can be sent down a medium and returned in such a way that it appears to stand still. – One end is free to oscillate and the other is attached to a fixed point. • • Guitar strings The position where the string is fixed, or appears to be motionless, is called a node. – The large part of the wave is called the antinode.