How Things Work Midterm
5 Principles
1. Your position indicates exactly where you're located. 2. Your velocity measures the rate at which your position is changing with time. 3. Your acceleration measures the rate at which your velocity is changing with time. 4. To accelerate, you must experience a net force. 5. The greater your mass, the less acceleration you experience for a given net force. *
* You drop a marble from rest, and after 1 s, its velocity is 9.8 m/s (32 ft/s) in the downward direction. What was its velocity after only 0.5 s of falling?
4.9 m/s (16 ft/s) in the downward direction. A freely falling object accelerates downward at a steady rate. Its velocity changes by 9.8 m/s (16 ft/s) in the downward direction each and every second. In half a second, the marble's velocity changes by only half that amount, or 4.9 m/s (16 ft/s).
modes
A basic pattern of distortion or oscillation.
Inertia
A body in motion tends to remain in motion, a body at rest tends to stay at rest
springlike forces
A force that is proportional to displacement, consistent with Hooke's law.
** When two people swing a long jump rope, they can make it swing as a single arc or as two half-ropes arcing in opposite directions. To make the rope swing as two half-ropes, it must be turned faster or with less tension. Why?
A jump rope is essentially a vibrating string. The two-half-rope pattern is the second harmonic mode, with a vibrational frequency twice that of the normal fundamental arc. Why: Although it swings around in a circle, the jump rope is actually vibrating up and down at the same time it's vibrating forward and back. Together, these two vibrations create the circular motion. To make the rope vibrate in its second harmonic mode (as two half-ropes) without changing its tension, it must be swung twice as fast as normal.
stiffness
A measure of how rapidly a restoring force increases as the system exerting that force is distorted.
mechanical wave
A natural and often rhythmic motion of an extended object about its stable equilibrium shape or situation.
trough
A peak negative excursion of an extended system that is experiencing a wave.
crests
A peak positive excursion of an extended system that is experiencing a wave.
wavelength
A peak positive excursion of an extended system that is experiencing a wave.
vibrational antinode
A region of a vibrating object that is experiencing maximal motion.
vibrational nodes
A region of a vibrating object that is not moving at all.
oscillation
A repetitive and rhythmic movement or process that usually takes place about an equilibrium situation. A more general term for any repetitive process, and it can even apply to such nonmechanical processes as electric or thermal oscillations.
vibration
A spontaneous repetitive and rhythmic movement about an equilibrium position. Implies a fast oscillation in a mechanical system
higher-order vibrational modes
A vibrational mode that is more complicated than the fundamental mode and in which different parts of the extended system move in opposite directions.
standing wave
A wave in which all the nodes and antinodes remain in place.
longitudinal wave
A wave in which the underlying oscillation is parallel to the wave itself.
transverse wave
A wave in which the underlying oscillation is perpendicular to the wave itself.
traveling wave
A wave that moves steadily through space in a particular direction.
If you carry a U.S. penny (0.0025 kg) to the top of the Empire State Building (443.2 m, or 1453.7 ft), how much gravitational potential energy will it have?
About 11 J. The penny's gravitational potential energy is given by gravitational potential energy = 0.0025 kg · 9.8 N/kg · 443.2 m = 11 N · m = 11 J This 11-J increase in energy would be quite evident if you were to drop the penny. In principle, the penny could accelerate to very high speed (up to 340 km/h or 210 mph) and do lots of damage when it hit the ground. Fortunately, a falling penny actually tumbles through the air, which slows it to about 40 km/h or 25 mph.
If it takes you about 1.4 s to reach the water from the 10-m (32-ft) diving platform, how fast are you going just before you enter the water?
About 14 m/s (45 ft/s, 50 km/h, or 31 mph). The downward acceleration due to gravity is 9.8 m/s2 (32 ft/s2). You fall for 1.4 s, during which time your velocity increases steadily in the downward direction. Since you start with zero velocity, Eq. 1.2.2 gives a final velocity of final velocity = 9.8 m/s2 * 1.4 s = 13.72 m/s. Since the time of the fall is given only to two digits of accuracy (1.4 s could really be 1.403 s or 1.385 s), we shouldn't claim that our calculated final velocity is accurate to four digits. We should round the value to 14 m/s (45 ft/s).
If you're walking at a pace of 1 m per second, how many miles will you travel in an hour?
About 2.24 miles. There are many different units in this example, so we must do some converting. First, an hour is 3600 s, so in an hour of walking at 1 m per second you will have walked 3600 m. Second, a mile is about 1609 m, so each time you travel 1609 m you have traveled 1 mile. By walking 3600 m, you have completed 2 miles and are about one-quarter of the way into your third mile.
It's much easier to stop a bicycle traveling toward you at 5 kilometers per hour (3 miles per hour) than an automobile traveling toward you at the same velocity. What accounts for this difference?
An automobile has a much greater mass than a bicycle. To stop a moving vehicle, you must exert a force on it in the direction opposite its velocity. The vehicle will then accelerate backward so that it eventually comes to rest. If the vehicle is heading toward you, you must push it away from you. The more mass the vehicle has, the less it will acceler- ate in response to a certain force and the longer you will have to push on it to stop it completely. Although it's easy to stop a bicycle by hand, stopping even a slowly moving automobile by hand requires a large force exerted for a substantial amount of time.
harmonics
An integer multiple of the fundamental frequency of oscillation for a system. The second harmonic is twice the frequency of the fundamental, and the third harmonic is three times the frequency of the fundamental. In principle, harmonics can continue forever.
electric charge
An intrinsic property of matter that gives rise to electrostatic forces between charged particles. Electric charge is a conserved physical quantity. A specific charge can have a positive amount of electric charge (a positive charge) or a negative amount (a negative charge). The SI unit of electric charge is the coulomb.
Newton's First Law of Motion
An object that is not subject to any outside forces moves at constant velocity, covering equal distances in equal times along a straight-line path
anharmonic oscillator
An oscillator in which the restoring force on an object is not proportional to its displacement from a stable equilibrium. The period of an anharmonic oscillator depends on the amplitude of its motion. i.e. if you displace the pendulum too far, its restoring force ceases to be proportional to its displacement from equilibrium, and its period begins to depend on its amplitude. Since a change in period will spoil the clock's accuracy, the pendulum's amplitude must be kept small and steady. That way, the amplitude has almost no effect on the pendulum's period.
harmonic oscillator
An oscillator in which the restoring force on an object is proportional to its displacement from equilibrium. Its period of oscillation depends only on the stiffness of that restoring force and on its mass, not on its amplitude of oscillation (motion). Whether that amplitude is large or small, the period remains exactly the same. This insensitivity to amplitude is a consequence of its special restoring force, a restoring force that is proportional to its displacement from equilibrium. Overall, it completes a large cycle of motion just as quickly as it completes a small cycle of motion. i.e. pendulum
As you watch people walk off the diving board at a pool, you notice that it bends downward by an amount proportional to each diver's weight. Explain.
Answer: The diving board is behaving as a spring, bending downward in proportion to the weight of each diver. Why: The heavier the diver, the more the board bends downward before exerting enough upward force on the diver to balance the diver's weight.
To make the air in a soda bottle vibrate, you must blow across the bottle's mouth. Why doesn't blowing into its mouth work?
By blowing across the mouth, you let air that is already vibrating in the bottle redirect your breath so that it enhances the vibration. Blowing into the bottle's mouth merely compresses the air inside the bottle. Like the bow of a violin moving across its strings, your breath moving across the bottle's mouth enhances the air's vibration via resonant energy transfer. The spontaneous redirection of your breath when you blow across the bottle's mouth leads to rhythmic pushes that are perfectly synchronized with the air's vibration.
Speed
Distance / Time
quantized
Existing only in discrete units or quanta. Quantized physical quantities are observed only in integer multiples of the elementary quantum.
Newton's Third Law of Motion
For every force that one object exerts on a second object, there is an equal but oppositely directed force that the second object exerts on the first object.
Normal forces
Forces that are directed exactly away from surfaces (perpendicular)
neutral
Having zero net electric charge.
You are pushing a child on a playground swing. If you exert a 50-N (11-lbf) force on him as he is swinging away from you, how much force will he exert back on you?
He will exert 50 N (11 lbf) on you. Whenever you exert a 50-N force on an object, whether it's moving or stationary, it will exert a 50-N force back on you. There are no exceptions. If that object is a friend, it doesn't matter whether she is stationary or moving or wearing roller skates or even sound asleep; she will push back with 50 N of force. She has no choice in the matter. Similarly, if someone pushes on you, you will feel yourself pushing back. That's how Newton's third law works.
sound
In air, sound consists of density waves, patterns of compressions and rarefactions that travel outward from their source at the speed of sound.
* You drop a marble from rest, and after 1 s, it has fallen downward a distance of 4.9 m (16 ft). How far had it fallen after only 0.5 s?
It had fallen about 1.2 m (4 ft). While a freely falling object's velocity changes steadily in the downward direction, its change in height is more complicated. When you drop the marble from rest, it starts its descent slowly but picks up speed and covers the downward distance faster and faster. In the first 0.5 s, it travels only a quarter of the distance it travels in the first 1 s, or about 1.2 m (4 ft).
The gift you are about to unwrap is electrically neutral. You tear off the clingy wrapper and find that it has a large negative charge. What charge does the gift itself have, if any?
It has a large positive charge equal in amount to the wrapper's negative charge. Why: Since charge is a conserved physical quantity, the wrapper and gift must remain neutral overall even after you separate them. The wrapper's negative charge must be balanced by the gift's positive charge.
You're in your spacecraft on the surface of the moon. Before getting into your suit, you weigh yourself and find that your moon weight is almost exactly one-sixth your Earth weight. What is the moon's acceleration due to gravity?
It is about 1.6 m/s2 (5.3 ft/s2). acceleration due to gravity is proportional to an object's weight: acceleration due to gravity = weight * mass Your mass doesn't change in going to the moon, so any change in your weight must be due to a change in the acceleration due to gravity. Since your moon weight is one-sixth of your Earth weight, the moon's acceleration due to gravity must be one-sixth that of Earth, or about 1.6 m/s^2.
You are moving books to a new shelf, 1.20 m (3.94 ft) above the old shelf. The books weigh 10.0 N (2.25 lbf) each, and you have 10 of them to move. How much work must you do on them as you move them? Does it matter how many you move at once?
It takes 120 N · m (88.6 ft · lbf), no matter how many you lift at once. To keep each book from accelerating downward, you must support its weight with an upward force of 10.0 N. You must then move it upward 1.20 m. The work you do pushing upward on the book as it moves upward is given by: work = force · distance = 10.0 N · 1.20 m = 12.0 N · m. It takes 12.0 N · m of work to lift each book, whether you lift it together with other books or all by itself. The total work you must do on all 10 books is 120 N · m.
You're hosting a party in your third-floor apartment. When the first 10 guests begin standing in your living room, you notice that the floor has sagged 1 centimeter in the middle. How far will the floor sag when 20 guests are standing on it? when 100 guests are standing on it?
It will sag 2 centimeters and 10 centimeters (assuming that the floor doesn't break). A floor, like most suspended surfaces, behaves like a spring. Your floor distorts 1 cm before it exerts an upward restoring force equal to the weight of 10 guests. It will thus distort 2 cm before supporting 20 guests and 10 cm before supporting 100 guests. While this distortion should be within the elastic limit of the floor beams, it may cause the plaster and paint to crack. If the beams break, the floor will collapse.
charges
Objects, particularly small particles, that carry electric charge.
Vector quantity
Quantity that consists of both a magnitude and a direction
Scalar quantity
Quantity that contains only an amount
To measure the distance from Earth to the moon, scientists bounce light from reflectors placed on the moon by the Apollo astronauts. Light travels at a constant speed. How can a measurement of light's travel time to and from the moon be used to determine the distance from Earth to the moon?
Since light travels at a constant speed, the distance it travels is equal to its speed times its travel time. If you know the travel time and the speed, you can determine that distance. Why: Many distance measurements are made by measuring time. Surveyors routinely use light's travel time to measure distances. Decorators and architects often use sound's travel time to measure the distances between walls. In general, the motion of an object at constant velocity can be used either to measure the distance traveled if you know the elapsed time or to measure the elapsed time if you know the distance traveled.
** Helium in a toy balloon has the same stiffness as ordinary air, but its density and inertia are smaller. How does this difference affect the speed of sound in helium?
Sound travels faster in helium than it does in ordinary air. With its reduced density and inertia, helium vibrates faster than air when the two gases carry sound waves of equal wavelengths. Since the speed at which sound of a specific wavelength travels is proportional to the frequency of that sound, the wave speed in helium is greater than that in air.
Position
Specific point in space
Period of a pendulum
T = 2π√(1ength of pendulum/acceleration due to gravity the period of a pendulum is equal to two pi times the square root of the length of the pendulum divided by the acceleration due to gravity
coulomb
The SI unit of electric charge. One coulomb is about 1 million times the charge you acquire by rubbing your feet across a carpet in winter.
When pushing a child on a playground swing, you normally push her forward as she moves away from you. What happens if you push her forward each time she moves toward you?
The amplitude of her motion will gradually decrease so that she comes to a stop. To keep her swinging, you must make up for the energy she loses to friction and air resistance. By pushing her forward each time she moves away from you, you do work on her and increase her energy. However, when you push her as she moves toward you, she does work on you and you extract some of her energy. You are then slowing her down rather than sustaining her motion.
Out in deep space, far from any celestial object that exerts significant gravity, would an astronaut weigh anything? Would that astronaut have a mass?
The astronaut would have zero weight but would still have a normal mass. Weight is a measure of the force exerted on the astronaut by gravity. Far from Earth or any other large object, the astronaut would experience virtually no gravitational force and would have zero weight. But mass is a measure of inertia and doesn't depend at all on gravity. Even in deep space, it would be much harder to accelerate a school bus than to accelerate a baseball because the school bus has more mass than the baseball.
elementary unit of electric charge
The basic quantum of electric charge, equal to about 1.6×10^−19C.
Why must a sharpshooter or an archer aim somewhat above her target? Why can't she simply aim directly at the bull's-eye to hit it?
The bullet or arrow will fall in flight, so she must compensate for its loss of height. To hit the bull's-eye, the sharpshooter or archer must aim above the bull's-eye because the projectile will fall in flight. Even if the target is higher or lower than the sharpshooter, the projectile will fall below the point at which she is aiming. The longer the bullet or arrow is in flight, the more it will fall and the higher she must aim. As the distance to the target increases, the flight time increases and her aim must move upward.
Energy
The capacity to do work. Energy has no direction. It can be hidden as potential energy.
You toss a coin straight up, and it rises well above your head. At the moment the coin reaches its peak height, what is its velocity? Is that velocity constant or changing with time? Is the coin's acceleration constant or changing with time?
The coin's velocity is momentarily zero at its peak, but that velocity is changing with time. The coin's acceleration, however, is constant—the acceleration due to gravity. Once the coin leaves your hand, it's a falling object and constantly accelerates downward at the acceleration due to gravity. Because it begins its fall traveling upward, it rises at a gradually decreas- ing speed, is momentarily motionless at its peak, and then descends at a gradually increasing speed.
When you step on the surface of a spring bathroom scale, you can feel it move downward slightly. How is the distance that the scale's surface moves downward when you step on it related to the weight it reports?
The distance the scale's surface moves downward is proportional to the weight it reports. The scale's spring is connected to its surface by levers so that as the surface moves downward, the spring distorts by a proportional amount. The spring's distortion is reported on the dial. Thus the dial's reading is proportional to the surface's downward movement.
neutrons
The electrically neutral subatomic particles that, together with protons, make up atomic nuclei.
Ramps for handicap entrances to buildings are often quite long and may even involve several sharp turns. A shorter, straighter ramp would seem much more convenient. What consideration leads the engineers designing these ramps to make them so long?
The engineers must limit the amount of force needed to propel a wheelchair steadily up the ramp. The steeper the ramp, the more force is required. A person traveling in a wheelchair on a level surface experiences little horizontal force and can move at constant velocity with very little effort. However, climbing a ramp at constant velocity requires a substantial uphill force equal in magnitude to the downhill force from gravity. The steeper the ramp, the more uphill force is needed to maintain constant velocity. A 12 to 1 grade (12 meter of ramp surface for each meter of rise in height) is the accepted limit to how steep such a long ramp can be.
electrostatic forces
The force experienced by a charged particle in the presence of other charged particles.
pitch
The frequency of a sound.
A common way to determine the tension in a cord is to pluck it and listen for how fast it vibrates. Why does this technique measure tension?
The frequency of a string's fundamental vibrational mode increases with its tension. Any cord that is drawn taut from its ends will exhibit natural resonances like those in a violin string. The tauter the string, the higher will be the frequencies of those resonances.
subatomic particles
The fundamental building blocks of the universe, from among which atoms and matter are constructed.
Coulomb's Constant
The fundamental constant of nature that determines the electrostatic forces two charges exert on one another. Its measured value is 8.988×10^9 N·m^2/C^2 one of the physical constants found in nature
resonant energy transfer
The gradual transfer of energy to or from a natural resonance caused by small forces timed to coincide with a particular part of each oscillatory cycle.
When you accidentally strike a chandelier with a broom, this hanging lamp begins to twist back and forth with a regular period. What determines its period of oscillation?
The hanging chandelier is a harmonic oscillator. Its supporting cord opposes any twists by exerting a restoring force on the chandelier. Once you twist it away from its equilibrium orientation, the chandelier oscillates back and forth with a period determined only by the cord's torsional stiffness (its stiffness with respect to twists) and the chandelier's rotational mass. As with any harmonic oscillator, the amplitude of the chandelier's motion doesn't affect its period.
Bowling balls come in various masses. Suppose that you try bowling with two different balls, one with twice the mass of the other. If you push on them with equal forces, which one will accelerate faster and how much faster?
The less massive ball will accelerate twice as rapidly. An object's acceleration is inversely proportional to its mass. acceleration = net force / mass If you push on both bowling balls with equal forces, then their accelerations will depend only on their masses. Doubling the mass on the right side of this equation halves the acceleration on the left side. That means that the more massive ball will accelerate only half as quickly as the other ball.
Coulomb's Law
The magnitudes of the electrostatic forces between two objects are equal to the Coulomb constant times the product of their two electric charges divided by the square of the distance separating them. If the charges are like, then the forces are repulsive. If the charges are opposite, then the forces are attractive. F = (k*q1*q2)/r^2
amplitude
The maximal displacement of an oscillator away from its equilibrium position.
Mass
The measure of your inertia/matter, the resistance to changes in velocity.
Work
The mechanical means for transferring energy. work = force * distance W = Fd
timbre
The mixture of tones in an instrument's sound that are characteristic of that instrument.
Newton's Second Law of Motion
The net force exerted on an object is equal to that object's mass times its acceleration. The acceleration is in the same direction as the net force.
frequency
The number of cycles completed by an oscillating system in a certain amount of time. The SI unit of frequency is the hertz. frequency = 1 / period
Sometimes a tone from an instrument or sound system will cause some object in the room to begin vibrating loudly. Why does this happen?
The object has a natural resonance at the tone's frequency, and sympathetic vibration is transferring energy to the object. Energy moves easily between two objects that vibrate at the same frequency. A note played on one instrument will cause the same note on another instrument to begin playing. Even everyday objects will exhibit sympathetic vibration when the right tone is present in the air.
A trampoline is hazardous with several children on it because a child landing on one side of its surface can launch skyward a second child standing on the other side of the surface. How does a downward impact on one side of the trampoline produce a sudden rise of the other side?
The off-center impact causes the surface to vibrate in its noncircular overtone modes. The simplest such mode has its two sides moving in alternate directions. The trampoline is essentially a drumhead, and the children are riding its vibrational modes. Off-center impacts can cause the surface to vibrate in its overtone modes, and these can toss the children in unexpected directions.
superposition
The overlapping of two or more waves so that their amplitudes add together and they form a combined wave.
nucleus
The positively charged central component of an atom, containing most of the atom's mass and about which the electrons are arranged. Plural is nuclei.
protons
The positively charged subatomic particles found in atomic nuclei.
Why does a moving hockey puck continue to slide across an ice rink even though no one is pushing on it?
The puck coasts across the ice because it has inertia. A hockey puck resting on the surface of wet ice is almost completely free of horizontal influences. If someone pushes on the puck, so that it begins to travel with a horizontal velocity across the ice, inertia will ensure that the puck continues to slide at constant velocity.
simple harmonic oscillator
The regular, repetitive motion of a harmonic oscillator. The period of simple harmonic motion doesn't depend on the amplitude of oscillation.
If you drop a metal rod on the floor, end first, you hear a high-pitched tone. What's happening?
The rod is vibrating as a harmonic oscillator, with its two halves first approaching one another and then moving apart. The metal rod vibrates in the same manner as a quartz crystal. The body of the rod exerts restoring forces on its two halves. After hitting the floor, these halves move toward and away from one another rapidly, emitting the tone that you hear. Because it's a harmonic oscillator, the frequency (and pitch) of the tone doesn't change as the amplitude of motion decreases.
If you pull the basket of a hanging grocery store scale downward 1 cm, it reports a weight of 5 N (about 1.1 lbf) for the contents of its basket. If you pull the basket downward 3 cm, what weight will it report?
The scale will read 15 N (about 3.3 lbf). The scale's dial is simply reporting the position of its basket. The dial is calibrated so that a 1-cm drop in the basket indicates that the spring is pulling up on it with a force of 5 N. Since the spring's restoring force is described by Hooke's law, a 3-cm drop in the basket means that the spring is exerting an upward force of 15 N on the basket.
fundamental vibrational mode
The slowest and often broadest vibration that an extended object can support.
Why does an acoustic guitar have a sound box?
The sound box transfers the vibrational energy of the strings to the air. Guitar strings are too narrow to push effectively on the air and emit sound. They do better by transferring their energy to the body of the acoustic guitar so that its flat surfaces can push on the air. An electric guitar avoids the need for a sound box by converting the string's vibrations directly into electric currents and from there into movements of an audio speaker.
wave velocity
The speed and direction of the moving crests of a wave.
speed of sound
The speed at which sound's compressions and rarefactions travel in a medium such as air or water.
Although water is about 800 times as dense as ordinary air, water is also about 15,000 times as stiff and its sound vibrations therefore have increased frequencies relative to those in air. When two sound waves have equal wavelengths, the wave in water has a frequency about 4.3 times greater than the wave in air. What is the speed of sound in water?
The speed of sound in water is about 1420 m/s (4700 ft/s). The wave speed in water must be 4.3 times the wave speed in air. Since the sound waves have a wave speed of about 331 m/s in air, they must have a speed of about 331 m/s times 4.3 or 1420 m/s in water.
A child swinging on a swing set travels back and forth at a steady pace. What determines the period of the child's motion?
The strength of Earth's gravity and the length of the swing's chains. A child swinging on a swing set is a form of pendulum. As with any pendulum, the child's period of motion is determined only by the strength of gravity and the length of the pendulum. In this case, the length of the pendulum is approximately the length of the swing's supporting chains. Thus, a tall swing has a longer period than a short swing.
net electric charge
The sum of all charges on an object, both positive and negative. Positive charges increase the net charge and negative charges decrease it. Net charge can be negative.
Net force
The sum of all forces acting on an object Fnet = mass * acceleration
You are standing on the sidewalk, watching a train coast eastward at constant velocity. Your friend is riding in that train. In her inertial frame of reference, the sweater in her lap is motionless. Describe the sweater's motion in your inertial frame of reference.
The sweater is coasting eastward at constant velocity. Although both of you agree that the sweater is not accelerating and that it is moving according to Newton's first law, you disagree on its specific velocity. She sees the sweater at rest, while you see it coasting eastward at constant velocity. Your viewpoints are equally valid.
period
The time required to complete one full cycle of a repetitive motion. T = 1/f
electrons
The tiny negatively charged particles that make up the outer portions of atoms and that are the main carriers of electricity and heat in metals.
You're planning to construct a bungee-jumping amusement at the local shopping center. If you want your customers to have a 5-s free-fall experience, how tall will you need to build the tower from which they'll jump? (Don't worry about the extra height needed to stop people after the bungee pulls taut.)
The tower should be about 122 m (402 ft or as high as a 40-story building). As they fall, the jumpers will travel downward at ever increasing speeds. Since the jumpers start from rest and fall downward for 5 s, final height = initial height − 1/2 · 9.8 m/s^2 · (5 s)^2 = initial height − 122.5 m. The downward acceleration is indicated here by the negative change in height. At the end of 5 s, the jumpers will have fallen more than 122 m (402 ft) and will be traveling downward at about 50 m/s (160 ft/s). The tower will need additional height to slow the jumpers down and begin bouncing them back upward. Clearly, a 5-s free fall is pretty unrealistic. Try for a 2- or 3-s free fall instead.
Trains spend much of their time coasting along at constant velocity. When does a train accelerate forward? backward? leftward? downward?
The train accelerates forward when it starts out from a station, backward when it arrives at the next station, to the left when it turns left, and downward when it begins its descent out of the mountains. Whenever the train changes its speed or its direction of travel, it is accelerating. When it speeds up on leaving a station, it is accelerating forward (more forward-directed speed). When it slows down at the next station, it is accelerating backward (more backward-directed speed or, equivalently, less forward-directed speed). When it turns left, it is accelerating to the left (more leftward-directed speed). When it begins to descend, it is accelerating downward (more downward-directed speed).
sympathetic vibration
The transfer of energy between two natural resonances that share a common frequency of oscillation.
As you ride upward in an elevator at a constant velocity, what two forces act on your body and what is the net force on you?
The two forces are your downward weight and an upward support force from the floor. They balance, so that the net force on you is zero. Whenever anything is moving with constant velocity, it's not accelerating and thus has zero net force on it. Although the elevator is moving upward, the fact that you are not accelerating means that the car must exert an upward support force on you that exactly balances your weight. You experience zero net force.
If you blow across a soda bottle, it emits a tone. Why does adding water to the bottle raise the pitch of that tone?
The water shortens the column of moving air inside the bottle and increases the frequency of its fundamental vibrational mode. Why: A water bottle is essentially a pipe that is open at only one end. It has a fundamental vibrational mode with a frequency that is half that of an open pipe of equal length. As you add water to the bottle, you shorten the effective length of the pipe and raise its pitch.
A typical singing voice can cover a range of about two octaves—for example, from C4 to C6. How broad is this range of frequencies?
There is a factor of 4 in frequency between the lowest and the highest notes that the typical voice can sing. Since notes separated by an octave are separated by a factor of 2 in frequency, notes separated by two octaves are separated by a factor of 4.
Inertial frame of reference
Viewpoint of an inertial object — an object that is not accelerating and that moves according to Newton's first law.
Do any of these objects have energy they can spare: a compressed spring, an inflated toy balloon, a stick of dynamite, and a falling ball?
Yes, they all do. Each of these four objects can easily do work on you and thereby give you some of its spare energy. It does this work by pushing on you as you move in the direction of that push.
When you throw a baseball horizontally, you're not pushing against gravity. Are you doing any work on the baseball?
Yes. Any time you exert a force on an object and the object moves in the direction of that force, you are doing work on the object. Since gravity doesn't affect horizontal motion, the work you do on the baseball as you throw it ends up in the baseball as kinetic energy (energy of motion). As anyone who has been hit by a pitch can attest, a moving baseball has more energy than a stationary baseball.
You're opening a company that will export gourmet food from Earth to the moon. You want the package labels to be accurate at either location. How should you label the amount of food in each package—by mass or by weight?
You should sell by mass—for example, by kilogram or pound-mass. If you label your product by weight, you are specifying the force that gravity exerts on it near Earth's surface. When it's exported to the moon, such a product will weigh just 1/6 as much, and your company may be fined for selling underweight groceries. If you label the packages according to their masses, that labeling will remain correct no matter where you ship the packages. Mass is the measure of inertia and depends only on the object, not on its environment.
spring constant (k)
a measure of the stiffness of a spring small spring constants: "soft" springs, i.e. spring in your pen large spring constants: "firm" springs, i.e. for your automobile chassis
natural resonance
a mechanical process in which an isolated object's energy causes it to perform a certain motion over and over again. The rate at which this motion occurs is determined by the physical characteristics of the object. practical clocks are based on this particular type of repetitive motion.
kinetic energy
energy of motion
potential energy
energy stored in the forces between or within objects
second harmonic mode
half-string vibration which occurs at the second harmonic pitch
elastic limit
if you stretch a spring too far, it won't return to its original equilibrium length when you release it
The length of a pendulum is doubled. The period of the pendulum's oscillation
increases by a factor less than two
Formula for weight
mass x acceleration due to gravity w = mg
octave
most important interval in virtually all music is 2/1, tones that differ by a factor of 2 in frequency sound so similar to our ears that we often think of them as being the same
periodic motion
motion repeated in equal intervals of time. it is performed, for example, by a rocking chair, a bouncing ball, a vibrating tuning fork, a swing in motion, the Earth in its orbit around the Sun, and a water wave.
equilibrium length
natural length when you leave it alone (i.e. a spring)
conserved quantity
one that can't be created or destroyed but that can be transferred between objects or, in the case of energy, be converted from one form to another
tension
outward forces that act to stretch it
gravitational potential energy
potential energy related to an object's height, energy stored in the gravitational forces = mass x acceleration due to gravity x height U = mgh
Position of a falling ball formula
present position = initial position + initial velocity * time + 1/2 * acceleration*time^2 xf = xi + vi * t + 1/2a*t^2
Velocity of a falling object formula
present velocity = initial velocity + acceleration*time vf = vi + a * t
Hooke's law
restoring force = − spring constant * distortion F = − k * x The restoring force exerted by an elastic object is proportional to how far it has been distorted from its equilibrium shape. The stiffer the spring and the farther you stretch it, the harder it pulls back. (negative just signals the displacement is opposite)
restoring force
the force that the spring exerts on its end acts to restore that end to equilibrium
mechanical advantage
the process whereby a mechanical device redistributes the amounts of force and distance that go into performing a specific amount of mechanical work (i.e. on a ramp)
stable equilibrium
the spring's right end naturally returns to this equilibrium position if the bar disturbs it and then lets go
time
the temporal dimension in our universe that separates events from one another according to past, present, and future.
space
the three spatial dimensions in our universe that separate events from one another according to distances and directions.
a fifth
this is the interval 3/2 is pleasing to most ears and is common in Western music
wave speed formula
wave speed = wavelength * frequency s= (wavelength)v Broad waves that vibrate quickly travel fast.
equilibrium position
when the bar isn't pulling or pushing on the spring's right end, that end is in equilibrium at a particular location
A metal block is attached to a spring, and undergoes harmonic oscillation horizontally. When the velocity is at its maximum to the right, the acceleration of the block is
zero
equilibrium
zero net force