11.1 Simple Harmonic Motion Hooke’s Law • Repeated motion is “periodic motion”. • Like a pendulum, back and forth over the same path. At equilibrium position, speed reaches maximum • As the mass is pulled away, it displaces the spring to a certain distance, x = ? • This exerts a force towards the equilibrium position.
At (b) force = 0 and so does displacement. • Acceleration also equals “0”. • However, speed is at max due to momentum. • At (c) due to momentum, mass overshoots equilibrium and compresses spring the other way.
At maximum displacement, spring force and acceleration reach a maximum • Beyond equilibrium the force and acceleration increase, however speed slows down. • At maximum displacement, speed is 0 but acceleration and force are at max. Then, back and forth (oscillates). • Friction brings the vibrating mass to rest (damping).
In simple harmonic motion, restoring force is proportional to displacement • This pushing and pulling is sometimes called “restoring force”, it is directly proportional to the displacement of a mass. • Determined by Robert Hooke in 1678… • The negative sign means that the spring force is opposite the direction the mass is displaced. Harmonic Motion Animation
k is the spring constant and is based on the stiffness of the spring in N/m. • The motion of the vibrating mass is an example of “simple harmonic motion” and any periodic motion that is a result of a restoring force. Spring Constant Animation
Practice AHooke’s Law • A mass of 0.55 kg, attached to a vertical spring, stretches the spring -0.020 m from its original equilibrium position. • What is the spring constant? Answer Given: m = 0.55 kg g = 9.81 m/s2 x = 0.020 m Unknown: k = ? 270 N/m
A stretched or compressed spring has elastic potential energy • A bent bow is the same as a stretched spring. • Stretched or compressed springs store elastic potential energy (PE). • Once released the PE becomes KE, or moving arrow!
The Simple Pendulum • The swinging motion of a pendulum is periodic vibration. • A simple pendulum consists of a mass called a bob which is attached to a fixed string. The restoring force of a pendulum is a component of the bob’s weight • If the restoring force is proportional to the displacement, then the pendulum’s motion is simple harmonic. • Any displacement from equilibrium can be resolved with both the x and y components. Simple Pendulum Animation
For small angles, the pendulum’s motion is simple harmonic • When the maximum angle for displacement θ is relatively small (<150), sin θ is approximately equal in radians. • Pendulum's motion is an excellent approximation of simple harmonic motion. • Because a simple pendulum vibrates with simple harmonic motion, many of our earlier conclusions for a mass-spring system apply here.
Gravitational potential increases as a pendulum’s displacement increases • This diagram shows how a pendulum’s mechanical energy changes as the pendulum oscillates. • As the pendulum swings toward equilibrium, it gains kinetic energy and losses potential energy.
Questions1. Repeated motion is _______ motion.2.When the mass/spring is at maximum displacement, speed is 0 but acceleration and force are at _____.3. The motion of a vibrating mass is an example of “simple harmonic motion” and any periodic motion that is a result of a _________ force.4.The restoring force of a pendulum’s motion is _______5. As the pendulum swings toward equilibrium, it gains kinetic energy and losses ________ energy. periodic max restoring gravity potential
11.2 Measuring Simple Harmonic Motion Amplitude, Period, and Frequency • A moving trapeze always returns to the same displacement from equilibrium, this is the amplitude. • A pendulum’s amplitude can be measured by the angle between the pendulum’s equilibrium position and its maximum displacement. • For a mass-spring system, this is its stretched or compressed position.
Period and frequency measure time • From one side of max displacement to the other, one complete cycle, is period, T • If one complete cycle takes 20 seconds, then the period of motion is 20 s. • The number of complete cycles in a unit of time is frequency. • If it takes 20 s to complete one cycle, the frequency is 1/20 cycles or 0.05 cycles. • Frequency is s-1 or hertz (Hz) • So… -period is time per cycle -frequency is number of cycles per unit of time.
The period of a simple pendulum depends on pendulum length and free-fall acceleration • Simple pendulums and mass-spring systems vibrate with harmonic motion. • To calculate period (T) and frequency ƒ in(Hz), requires a separate formula. • A pendulum with the same length (L) but with bobs of two different masses or amplitude, the period will be the same. • If free-fall acceleration (gravity) changes, so will the period.
When two pendulums have different lengths but the same amplitude, the shorter pendulum will have a smaller arc to travel through. • Mass and amplitude do not affect the period for the same reason all objects fall at the same rate. • The reason for this is, the more you increase the amplitude, the more restoring force there is (even though it has a greater distance to cover)
Practice B Simple Harmonic Motion of a Simple Pendulum • 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 it has a period of 12 s. • How tall is the tower? Answer 36 m
Period of a mass-spring system depends on mass and spring constant • The period of a mass-spring system uses Hook’s Law. • Because heavier objects have more inertia, they take longer to speed up. • This causes them to have longer periods.
The greater the spring constant (k), the greater the force needed to stretch or compress the spring. • When force is great, so is the acceleration. • This makes the time required to make one cycle less. • So, stiff spring = short period. • Also, changing amplitude does not effect the period.
Practice C Simple Harmonic Motion of a Mass-Spring System • 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 0.840 s. • For the first few seconds, the vibration approximates simple harmonic motion. • Find the spring constant of a single spring. Answer 20,000 N/m
Questions1. A pendulum’s _________ can be measured by the angle between the pendulum’s equilibrium position and its maximum displacement.2. The number of complete cycles in a unit of time is known as _________.3. A pendulum with the same length but with bobs of two different masses or amplitude, the ______ will be the same.4. Mass and amplitude (do / do not) affect the period of a pendulum.5. The stiffer the spring, the (longer / shorter) the period. amplitude frequency period do not shorter
11.3 Properties of Waves Wave Motion • If there is a disturbance in a pond, you see a circular pattern move outward in all directions. • If there is a leaf floating near by, it moves a little but does not travel with the wave. Breaking waves are different
A wave is the motion of a disturbance • This disturbance causes the pattern to move out in a circular pattern. • Water in the pond is the medium through which the disturbance travels. • The medium (water) does not actually travel with the wave. • Sound waves require air as their medium. In space there is no sound. • Waves that require a material medium are called mechanical waves. Animation
Wave Types • A wave that consists of a single traveling pulse is called a pulse wave. • If you continue to generate pulses, this will create a periodic wave.
Sine waves describe particles vibrating with simple harmonic motion • Periodic waves can show simple harmonic motion on a string. • A wave whose source vibrates with simple harmonic motion is called a sine wave. • These are called sine waves because a graph of trigonometric function y = sin x produces this curve when plotted.
Vibrations of a transverse wave are perpendicular to the wave motion • When vibrations are perpendicular to the direction of the wave’s motion they are called transverse waves. • Displacement of a single particle as time passes creates a waveform.
Wave measures include crest, trough, amplitude, and wavelength • A wave can be measured in terms of its displacement from equilibrium. • The highest point is called the crest. • The lowest point the trough. • Remember, amplitude is a measure of maximum displacement from equilibrium. • The distance the wave travels in one cycle along its path is called wavelength, (λ).
Vibrations of a longitudinal wave are parallel to the wave motion • When the displacement of the medium vibrates parallel to the direction of wave motion is called a longitudinal wave. • Longitudinal waves can also be described by a sine curve. • The type of wave represented above is often called a density or pressure wave. • The crests are where the spring coils are compressed. Animation
Period, Frequency, and Wave Speed • Sound waves may begin with the vibrations of your vocal cords. The source of wave motion is a vibrating object. • When the vibrating particles of the medium complete one full cycle, one wavelength passes any given point. • Thus, wave frequency describes the number of waves that pass a given point in a unit of time. • The period of a wave is the time required for one complete cycle of vibration of the mediums particles.
Wave speed equals frequency times wavelength • We can now derive an expression for the speed of a wave in terms of its period or frequency. Animation • The speed of a mechanical wave is constant for any given medium. • Even though all sounds are different, they reach your ears at the same speed. • As a result, when the frequency increases its wavelength must decrease.
Practice D Wave Speed • The piano string tuned to middle C vibrates with a frequency of 264 Hz. • Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound waves produced by the string. Answer 1.30 m
Waves transfer energy • Waves transfer energy by the vibration of matter rather than by the transfer of matter itself. • For this reason, waves are often able to transport energy efficiently. • The rate at which a wave transfers energy depends on the amplitude, the greater the amplitude, the more energy carried in a given time interval. • When the amplitude is doubled, the energy carried increases by a factor of 4.
Questions1. Waves that require a material medium are called __________ waves.2. A wave whose source vibrates with simple harmonic motion is called a _____ wave.3. When the displacement of the medium vibrates parallel to the direction of wave motion is called a __________ wave. 4. When the frequency increases, its wavelength must? 5. When the amplitude is doubled, the energy carried increases by a factor of ______. mechanical sine longitudinal decrease. four
11.4 Wave Interactions Wave Interference • When two waves come together, they do not bounce back like bumper boats. • With sound waves, you can distinguish the sounds of different instruments. • This is because sound waves (mechanical waves) are not matter but displacements of matter. • Two waves can occupy the same space at the same time. • As they pass through one another, they interact to form an interference pattern.
Displacements in the same direction produce constructive interference • When two pulses meet, a resultant wave forms. • The amplitude of the resultant wave is equal to the sum of the amplitudes of each pulse. • Summing the displacements of waves is known as the superposition principle. • After the two pulses pass, they have their original shape. • If the displacements are on the same side of equilibrium, when added together, we get constructive interference.
Displacements in opposite directions produce destructive interference • The following shows what happens when pulses are on opposite sides of equilibrium. • When the positive and negative displacements are added we get destructive interference. • When two pulses coincide, their resultant wave can have a displacement of zero. • This is known as complete destructive interference. Animation
The superposition principle is valid for longitudinal (compression) waves. • In a compression, particles move closer together, while rarefaction, particles spread apart. • When a compression and rarefaction interfere, there is destructive interference. • In the case of sound, these wave can cancel causing a lack of sound.
Reflection At a free boundary, waves are reflected • At a free boundary, the rope is free to move up and down sending the wave back in the same way. • This is called reflection.
At a fixed boundary, waves are reflected and inverted • When the pulse reaches the wall, the pulse exerts an upward force on the wall. • The wall in turn exerts an equal and opposite reaction force on the rope. • As a result, the pulse is inverted.
Standing Waves • Standing waves occur when a string is attached to one ridged end and shaken in a regular motion. • This will produce a wave of a certain frequency, wavelength, and amplitude traveling down the string. • As the waves reach the other end they are reflected back toward the oncoming waves. • If the string is vibrated at a certain frequency, a standing wave or resultant wave pattern appears on the string. • The standing wave consists of alternating regions of constructive and destructive interference.
Standing waves have nodes and antinodes • Four possible standing waves are shown below. • Points where complete destructive interference happen are called nodes. • Midway between two adjacent nodes, where the string vibrates with the largest amplitude are the antinodes. • In the second example below, on the right, shows where there are 3 nodes(N) and 2 antinodes(A). Animation
Only certain frequencies, and wavelengths, produce standing waves. • A standing wave can only be produced for any wavelength that allows both ends of the string to be nodes. • Example (b) is half a wave length, so to find wave length you multiply the string length by two (2L). • Example (c) is one wave length so we just use (L). • Example (d) has 4 nodes and 3 antinodes. We would have to go another half wavelength to have two full wave lengths. In this case, to find the length of one wave, we just multiply by (2/3L).
Questions1. Sound waves (mechanical waves) are not matter but _____________ of matter.2. In a compression, particles move closer together, while __________, particles spread apart.3. At a free boundary, the rope is free to move up and down sending the wave back in the same way, this is called _________.4. Midway between two adjacent nodes, where the string vibrates with the largest amplitude are the _________.5. If a standing wave is half a wave length, to find its wave length you multiply the string length by ____. displacements rarefaction reflection antinodes two