Holt Physics Chapter 11 Vibrations and Waves

Simple Harmonic Motion Simple Harmonic Motion – vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium.  Springs, pendulums, etc… The spring force always pushes or pulls the mass back toward its original equilibrium position (sometimes called a restoring force).  See figure 12-1, page 438

Vibrations and Waves Simple Harmonic Motion

Example: gravity acting on a mass hanging from a string. Example: gravity acting on a mass hanging from a spring. *

When the restoring force is linearly proportional to the amount of the displacement from equilibrium, the force is said to be a Hooke’s Law force.

Hooke’s Law For mass-spring systems Formula F elastic = -kx Spring force=-(spring constantXdisplacement)  The negative sign in the equation refers to the force being opposite to the displacement.  k is a measure of stiffness and the unit is N/m Remember, stretching or compressing a spring stores elastic potential energy that can be converted to kinetic energy.

Book Example, page 366 If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant? m=0.55 kgx=-2.0cm = m g=9.81m/s 2 k=?

m=0.55 kgx=-2.0cm = m g=9.81m/s 2 k=? F elastic = |mg| F elastic = -kx mg=-kx k=-mg/x k=-(0.55kg)(9.81m/s 2 )/(-.020m) k=270N/m If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant? F elastic FgFg

Example 2. How much force will a vertical spring with a spring constant of 350 N/m exert when it is stretched to 22 cm past its equilibrium point?

Example 3. When a 1.5 kg mass is suspended from a spring hanging vertically, How much does it stretch if the spring constant Is 98N/m? Remember: The spring force (F E ) is equal and opposite to the weight (F w ).

Practice page 367

Simple Harmonic Motion, pg. 371

 Some basic elements of simple harmonic motion  At equilibrium position, velocity = maximum  At maximum displacement, the spring force and acceleration = maximum  This motion repeats itself in simple harmonic motion.

See page 371 for a comparison of spring-mass systems and pendulum motion.  The pendulum stores gravitational potential energy, and the spring-mass system stores elastic potential energy.  See page 370

Measuring Simple Harmonic Motion Amplitude – maximum displacement from equilibrium position.  Unit is radians or meters Period (T) time it takes to complete one cycle of motion.  Unit is seconds Frequency (f) is the number of cycles per second. T=1/f or f=1/T  Unit for f is s -1 or Hertz (Hz)

The time for one cycle is the period, T. The number of cycles per second is the frequency.

Calculating the Period of a Simple Pendulum T = 2  L/g  L = length of pendulum  g = free-fall  T = Period of pendulum  Amplitude and mass do not affect the period of a pendulum

Measuring Simple Harmonic Motion Example 1. A chandelier exhibiting simple harmonic motion has a period of 1.3 s. How long is the cord it hangs from?

Example 2. An astronaut on a foreign planet notes that a pendulum with a length of 2.5 m has a period of 1.4 s. Find g on this planet.

Example 3. Find the period (T) and the frequency (f) for wave (B) below. From the image, we see that the period, or time to complete one cycle, is 0.60 s.

Example 4. Find the period (T) and the frequency (f) for wave (b) below. From the image, we see that the period, or time to complete one cycle, is 0.80 s.

Homework: Practice page 375 and Formative Assessment pg. 371

 Proof of Earth’s rotation?  Foucault Pendulum

Mass-Spring Systems The period of a mass-spring system depends on mass and spring constant. T = 2  m/k

Book example The body of a 1275 kg car is supported on a frame by four springs. Two people riding in the car have a combined mass of 153 kg. When driven over a pothole in the road, the frame vibrates with a period of s. For the first few seconds, the vibration approximates simple harmonic motion. Find the spring constant of a single spring. m total = 1275kg = 357kg/wheel 4 wheels 4 m car =1275 kgm people = 153 kg 4 wheels T=0.840sk=?

T = 2  m/k Square both sides…and solve for k k=4  2 m/T 2 k=4  2 357kg/(0.84s) 2 k=2.00X10 4 N/m

Practice and Formative Assessment pg. 377

Example 1. A mass of 0.50 kg is attached to a spring and is set into vibration with a period of 0.42 s. What is the spring constant of the spring?

Example 2. A 0.60 kg mass attached to a vertical spring stretches the spring 0.25 m. A)Find the spring constant. B)The system is now placed on a horizontal surface and set to vibrate. Find the period. A) B)

What is a wave ? A definition of a wave:  A wave is a traveling disturbance that transports energy but not matter. Examples:  Sound waves (air moves back & forth)  Water waves (water moves up & down)  Light waves (do not require a medium)  Mechanical waves require a medium  Electromagnetic waves (like light) do not

Measuring Waves Frequency (f) – number of cycles per second Period – time in seconds required for one complete cycle of motion Velocity(v) – speed of wave in m/s Formulas  v = /T  f = 1/T  v = f

Wave Properties Wavelength Wavelength: The distance between identical points on the wave. Amplitude A l Amplitude: The maximum displacement A of a point on the wave. A

Example 1. A transverse wave has a wavelength of 0.80 m and a period of 0.25 s. Find the speed of the wave.

Example 2. A transverse wave has a wavelength of 40 cm and a frequency of 30.0 Hz. Find the speed of the wave.

Example 3. An astronaut broadcasts radio waves with a speed (c) of 3.00 x 10 8 m/s and a frequency (f) of 92.0 MHz. Calculate the wavelength of these waves.

Example 4. A certain laser emits light of wavelength 633 nm. What is the Frequency of this light in a vacuum?

Homework pg. 383 practice and 384 formative assessment

Wave Types Pulse wave – single, non-periodic disturbance. Periodic wave – a wave whose source is some kind of periodic motion.

Transverse: The medium oscillates perpendicular to the direction the wave is moving.  Water (more or less)  String waves Longitudinal: The medium oscillates in the same direction as the wave is moving.  Sound

Wave Interactions Because mechanical waves are not matter, but are displacements of matter, they can occupy the same space at the same time. Superposition – combination of two overlapping waves

Wave Interactions Interference – interaction of waves as a result of them passing through each other.

When the displacements are in the same direction, this produces Constructive Interference (see figure 4.3, pg. 386) The resultant amplitude of the resultant wave is the sum of the amplitudes of the individual waves.  Each wave maintains it’s own characteristics after the interference

When the displacements are in the opposite direction, this produces Destructive Interference (see figure 4.4, pg. 387) The resultant amplitude of the resultant wave is the difference between the two pulses.  If the difference is zero, then it is called Complete Destructive Interference (fig 4.5)  Again, each wave maintains it’s own characteristics after the interference

Wave Motion Waves and springs animation. Wave pulse traveling down string, striking “fixed” boundary. Wave pulse traveling down string, striking “free” boundary.

Reflection At a free boundary (fig 4.6a pg. 388) waves are reflected. At a fixed boundary (fig 4.6b pg. 388) waves are reflected and inverted.

Fixed Boundary…

Free Boundary…

pulse travels down string/slinky reflects (& inverts) meets another pulse on the way back

Standing Waves A standing wave is produced when two waves with the same amplitude, wavelength and frequency travel in opposite directions and interfere (fig. 4.7, pg. 389). Node – point on a standing wave that always undergoes complete destructive interference and therefore is stationary. Antinode – a point in a standing wave half-way between two nodes at which the largest amplitude occurs.

If both ends of the string are fixed, a standing wave results. If the string is “infinitely” long and the up-and-down pulses repeat, we get a traveling wave.

Only certain frequencies will produce standing wave patterns.  The ends have to be nodes, so this includes a wavelength of 2L (twice the length of the string or tube where the vibration is taking place), L, and 2/3 L. (see figure 4.8, pg. 390)

Formative Assessment pg. 390

Review Problems