1. Chapter 12 VibrationsandWaves

2. Chapter 12 Objectives Hooke’s Law Hooke’s Law Simple Harmonic Motion Simple Harmonic Motion Elastic Potential Energy Elastic Potential Energy Wave properties Wave properties –Frequency –Amplitude –Wavelength Wave Types Wave Types Pendulum Pendulum Superposition Superposition Wave Interference Wave Interference

3. 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 – 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 (-) because the spring is always providing a force opposite the motion of the mass

4. 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.

5. 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: hertz (Hz) which is equivalent to an inverse second. (s -1 )

6. 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.

7. 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

8. 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.

9. 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

10. 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

11. 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. +

12. 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.

13. 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

14. Period of a Pendulum A pendulum exhibiting simple harmonic motion will have a period that depends on: 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) 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) the faster gravity can pull the pendulum, the shorter the time it takes to complete its cycle (period) T = 22 (L/g)(L/g) 

15. Period of a Spring System The period of a mass oscillating in simple harmonic motion depends on The period of a mass oscillating in simple harmonic motion depends on – mass of the spring system more mass takes more time to accelerate more mass takes more time to accelerate and the longer the path (larger amplitude) it creates and the longer the path (larger amplitude) it creates – spring constant stiffer the spring, the less time it takes to accelerate 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2 (m/k)(m/k) 

16. 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. 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 Guitar strings The position where the string is fixed, or appears to be motionless, is called a node. 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.