Oscillations and Waves Periodic Motion Simple Harmonic Motion Connections between Uniform Circular Motion and Simple Harmonic Motion The Period of a Mass on a Spring Energy Conservation in Oscillatory Motion The Pendulum
Simple Harmonic Motion Simple harmonic motion occurs when the net force along the direction of motion obeys Hooke s law when the net force is proportional to the displacement from the equilibrium point and is always directed toward the equilibrium point.
Not all periodic motions over the same path can be classified as simple harmonic motion.
The motion of an object suspended from a vertical spring is also simple harmonic. In this case the force of gravity acting on the attached object stretches the spring until equilibrium is reached and the object is suspended at rest.
The following three concepts are important in discussing any kind of periodic motion: The amplitude A is the maximum distance of the object from its equilibrium position. The period T is the time it takes the object to move through one complete cycle of motion. The frequency f is the number of complete cycles or vibrations per unit of time, and is the reciprocal of the period ( f = 1/T ).
The acceleration of an object moving with simple harmonic motion: Constant- Acceleration Equations Don t Apply
Day 2
When an object moving in simple harmonic motion is at its maximum displacement from equilibrium, which of the following is at a maximum? (a) velocity (b) acceleration (c) kinetic energy. Pre-Class
ELASTIC POTENTIAL ENERGY
Potential and Kinetic Energy
Conservation of Mechanical Energy with conservative forces with non-conservative forces Rotational KE must also be included in situations where there is torque.
Velocity as a Function of Position Solve for v:
Finding maximum velocity, minimum velocity, and maximum force. a) Max Velocity: When x = 0 b) Min Velociy: x = A c) When x = A because acceleration is greatest.
Problem A 13 000-N car starts at rest and rolls down a hill from a height of 10.0 m. It then moves across a level surface and collides with a light spring-loaded guardrail. (a) Neglecting any losses due to friction, and ignoring the rotational kinetic energy of the wheels, find the maximum distance the spring is compressed. Assume a spring constant of 1.0 x 10 6 N/m. (b) Calculate the maximum acceleration of the car after contact with the spring, assuming no frictional losses. (c) If the spring is compressed by only 0.30 m, find the change in the mechanical energy due to friction.
Relating SHM to Circular Motion
Period and Frequency of Circular Shadow
Period of a Mass/Spring System Notice frequency not dependent on Force. Measured in Hertz...Hz
A 1.30 x 10 3 kg car is constructed on a frame supported by four springs. Each spring has a spring constant of 2.00 x 10 4 N/m. If two people riding in the car have a combined mass of 1.60 x 10 2 kg, find the frequency of vibration of the car when it is driven over a pothole in the road. Find also the period and the angular frequency. Assume the weight is evenly distributed.
Day 4
Pre-Class
Quote of the Day A cement mixer collided with a prison van on I-95. Motorists are asked to be on the lookout for sixteen hardened criminals.
Daily Objective SWBAT utilize the quantities of mass, amplitude, and length IOT determine the period, linear frequency, and angular frequency of a Simple Pendulum in Simple Harmonic Motion.
The Simple Pendulum
What does the period depend upon?
A desktop toy pendulum swings back and forth once every 1.0 s. How long is this pendulum? (Answer 0.25 m)
What is the free-fall acceleration at a location where a 6.00 m long pendulum swings through exactly 100 cycles in 492 s? (Answer 9.79 m/s 2 )
P R O B L E M You need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extending from the ceiling almost touches the floor and that its period is 12 s. How tall is the tower?
Day 5
Quote of the Day Bert: Ernie, you have a banana in your ear. Ernie: What was that, Bert? I m sorry. I couldn t hear you. I had a banana in my ear.
Practice Problems
1. A person in a rocking chair completes 12 cycles in 21 s. What are the period and frequency of the rocking? (T = 1.75 s ; f =. 57 Hz )
2. You rev your car s engine to 2700 rpm (rev/min (a) What are the period and frequency of the engine? ( Ans. T =.022 sec; f = 45 Hz) (b) If you change the period of the engine to 0.044 s how many rpms is it doing? (Ans. 1364 rpm )
3.
4. A 0.46-kg mass attached to a spring undergoes simple harmonic motion with a period of 0.77 s. What is the force constant of the spring? (Ans. 30.1 N/m)
5. When a 0.50-kg mass is attached to a vertical spring, the spring stretches by 15 cm. How much mass must be attached to the spring to result in a 0.75-s period of oscillation? (Ans..47 kg )
6. A simple pendulum of length 2.5 m makes 5.0 complete swings in 16 s. What is the acceleration of gravity at the location of the pendulum? (Ans. 9.64 m/s 2 )
Day 6
Quote of the Day Last week I saw my psychiatrist. I told him, Doc, I keep thinking I m a dog. He told me to get off his couch.
DAMPED OSCILLATIONS
WAVE MOTION
A wave is the motion of a disturbance medium a physical environment through which a disturbance can travel Note that the medium does not actually travel with the waves. After the waves have passed, the water returns to its original position. mechanical wave a wave that requires a medium through which to travel Not all wave propagation requires a medium. Electromagnetic waves, such as visible light, radio waves, microwaves, and X rays, can travel through a vacuum.
Now imagine that you continue to generate pulses at one end of the rope. Together, these pulses form what is called a periodic wave. Whenever the source of a wave s motion is a periodic motion, such as the motion of your hand moving up and down repeatedly a periodic wave is produced.
Sine waves describe particles vibrating with simple harmonic motion Single Point in a wave
Vibrations of a transverse wave are perpendicular to the wave motion transverse wave a wave whose particles vibrate perpendicularly to the direction the wave is traveling Creating Transverse Waves
Wave measures include crest, trough, amplitude, and wavelength
Vibrations of a longitudinal wave are parallel to the wave motion
Creating Longitudinal Waves Transverse/Longitudinal Waves
Wave speed equals frequency times wavelength so,
Day 7
Pre-Class A wave has a wavelength of 3.00 m. Calculate the frequency of the wave if it is (a) a sound wave and (b) a light wave. Take the speed of sound as 343 m/s and the speed of light as 3.00 x 10 8 m/s.
Quote of the Day I took the Magician's Exam once. It was loaded with trick questions!
THE SPEED OF WAVES ON STRINGS The linear density of a string, µ, is equal to the mass per unit length of the string. Ex. A string of length 3 m has a mass of.7 kg. Find its linear density.
The velocity that a wave travels along a stretched string depends on two things: the Force of tension in the string (F), and the linear density of the string ( µ ). Consider the tuning of instruments that have strings:
Velocity of a wave on a string: Ex. Find the velocity of a wave traveling down a string that has a tension of 35 N and a linear density of.1 kg/m.
A uniform string has a mass of 0.030 0 kg and a length 6.00 m. Tension is maintained in the string by suspending a block of mass 2.00 kg from one end. (a) Find the speed of a transverse wave pulse on this string. (b) Find the time it takes the pulse to travel from the wall to the pulley. Neglect the mass of the hanging part of the string.
INTERFERENCE OF WAVES The Superposition Principle: When two or more traveling waves encounter each other while moving through a medium, the resultant wave is found by adding together the displacements of the individual waves point by point.
REFLECTION OF WAVES
Waves Entering Different Medium
Day 8
PRODUCING A SOUND WAVE Compressions and Rarefactions Longitudinal Wave
Audible waves are longitudinal waves that lie within the range of sensitivity of the human ear, approximately 20 to 20 000 Hz. Infrasonic waves are longitudinal waves with frequencies below the audible range. Earthquake waves are an example. Ultrasonic waves are longitudinal waves with frequencies above the audible range for humans and are produced by certain types of whistles. Animals such as dogs can hear the waves emitted by these whistles.
ENERGY AND INTENSITY OF SOUND WAVES
The faintest sounds the human ear can detect at a frequency of 1 000 Hz have an intensity of about 1 x 10 12 W/m2. This intensity is called the threshold of hearing. The loudest sounds the ear can tolerate have an intensity of about 1 W/m2 (the threshold of pain).
Intensity in Decibels The sensation of loudness is approximately logarithmic in the human ear. A sound is perceived to be twice as loud when it has ten times the intensity level.
How many times more intense is a 50dB sound than a 20dB?
A noisy grinding machine in a factory produces a sound intensity of 1.00 x 10 5 W/m2. Calculate (a) the decibel level of this machine (b) the new intensity level when a second, identical machine is added to the factory. (c) A certain number of additional such machines are put into operation alongside these two machines. When all the machines are running at the same time the decibel level is 77.0 db. Find the sound intensity.
By how many decibels is the sound intensity level raised when the sound intensity is doubled?
Suppose a manufacturing plant has an average sound intensity level of 97.0 db created by 25 identical machines. (a) Find the total intensity created by all the machines. (b) Find the sound intensity created by one such machine. (c) What s the sound intensity level if five such machines are running?
SPHERICAL AND PLANE WAVES
Relationship of Intensities at Two Different Spherical Surfaces
Plane Waves
A small source emits sound waves with a power output of 80.0 W. (a) Find the intensity 3.00 m from the source. (Ans..707 W/ m 2 ) (b) At what distance would the intensity be one-fourth as much as it is at r = 3.00 m? (Ans. 6 m ) (c) Find the distance at which the sound level is 40.0 db. (Ans. 2.52 x 10 4 m )
Day 9
Quote of the Day "You buy it, you break it." - Piñata Store Policy
THE DOPPLER EFFECT Case 1: The Observer Is Moving Relative to a Stationary Source
Case 2: The Source Is Moving Relative to a Stationary Observer
General Case When the observer moves toward the source, a positive speed is substituted for v O ; when the observer moves away from the source, a negative speed is substituted for v O. Similarly, a positive speed is substituted for v S when the source moves toward the observer, a negative speed when the source moves away from the observer.
What is the frequency heard by a person driving at 15 m/sec toward a blowing factory whistle (f = 800 hz) if the speed of sound is 340.6 m/sec?
A source is moving at 18 m/s in air emitting a frequency of 1225 cycles per second. What is the frequency you would hear as the source approaches?
A source sound is detected from a departing whale (in salt water v=1530 m/s) at a frequency of 422 cycles per second. If this whale is known to make calls at a frequency of 425 cycles per second, how fast is the whale moving? 10.9 m/s
Day 10
Standing Waves
Animation
1. One string on a toy guitar is 34.5 cm long. a.what is the wavelength of its first harmonic? b. The string is plucked, and the speed of waves on the string is 410 m/s. What are the frequencies of the first three harmonics? Answers a.69.0 cm b.590 Hz, 1200 Hz, 1800 Hz
Day 11
Pipes Open at Both Ends
Same as for a string fixed at both ends.
Pipes Closed at One End
Only Odd Numbered Harmonics can be obtained.