It's a Great Day For Physics: Optical Instruments

Making things look bigger

When you use an optical instrument, whether it be something very simple like a magnifying glass, or more complicated like a telescope or microscope, you're usually trying to make things look bigger so you can more easily see fine details. One thing to remember about this is that if you want to make things look bigger, you're always going to use converging mirrors or lenses. Diverging mirrors or lenses always give smaller images.

When using a converging lens, it's helpful to remember these rules of thumb. If the object is very far away, the image will be tiny and very close to the focal point. As the object moves towards the lens, the image moves out from the focal point, growing as it does so. The object and image are exactly the same size when the object is at 2F, twice the focal distance from the lens. Moving the object from 2F towards F, the image keeps moving out away from the lens, and growing, until it goes to infinity when the object is at F, the focal point. Moving the object still closer to the lens, the image steadily comes in towards the lens from minus infinity, and gets smaller the closer the object is to the lens.

Note that similar rules of thumb apply for a converging mirror, too.

Multiple lenses

Many useful devices, such as microscopes and telescopes, use more than one lens to form images. To analyze any system with more than one lens, work in steps. Each lens takes an object and creates an image. The original object is the object for the first lens, and that creates an image. That image is the object for the second lens, and so on. We won't use more than two lenses, and we can do a couple of examples to see how you analyze problems like this.

A microscope

A basic microscope is made up of two converging lenses. One reason for using two lenses rather than just one is that it's easier to get higher magnification. If you want an overall magnification of 35, for instance, you can use one lens to magnify by a factor of 5, and the second by a factor of 7. This is generally easier to do than to get magnification by a factor of 35 out of a single lens.

A microscope arrangement is shown below, along with the ray diagram showing how the first lens creates a real image. This image is the object for the second lens, and the image created by the second lens is the one you'd see when you looked through the microscope.

Note that the final image is virtual, and is inverted compared to the original object. This is true for many types of microscopes and telescopes, that the image produced is inverted compared to the object.

Telescopes

A telescope needs at least two lenses. This is because you use a telescope to look at an object very far away, so the first lens creates a small image close to its focal point. The telescope is designed so the real, inverted image created by the first lens is just a little closer to the second lens than its focal length. As with the magnifying glass, this gives a magnified virtual image. This final image is also inverted compared to the original object. With astronomical telescopes, this doesn't really matter, but if you're looking at something on the Earth you generally want an upright image. This can be obtained with a third lens.

Note that the overall effect of the telescope is to magnify, which means the absolute value of the magnification must be larger than 1. The first lens (the objective) has a magnification smaller than one, so the second lens (the eyepiece) must magnify by a larger factor than the first lens reduces by. To a good approximation, the overall magnification is equal to the ratio of the focal lengths. With o standing for objective and e for eyepiece, the magnification is given by:

m = - f_o / f_e, with the minus sign meaning that the image is inverted.

Resolving power

The resolving power of an optical instrument, such as your eye, or a telescope, is its ability to separate far-away objects that are close together into individual images, as opposed to a single merged image. If you look at two stars in the sky, for example, you can tell they are two stars if they're separated by a large enough angle. Some stars, however, are so close together that they look like one star. You can only see that they are two stars by looking at them through a telescope. So, why does the telescope resolve the stars into separate objects while your eye can not? It's all because of diffraction.

If you look at a far-away object, the image of the object will form a diffraction pattern on your retina. For two far-away objects separated by a small angle, the diffraction patterns will overlap. You are able to resolve the two objects as long as the central peaks in the two diffraction patterns don't overlap. The limit is when one central peak falls at the position of the first dark fringe for the second diffraction pattern. This is known as the Rayleigh criterion. Once the two central peaks start to overlap, in other words, the two objects look like one.

The size of the central peak in the diffraction pattern depends on the size of the aperture (the opening you look through). For your eye, this is your pupil. A telescope, or even a camera, has a much larger aperture, and therefore more resolving power. The minimum angular separation is given by:

The factor of 1.22 applies to circular apertures like your pupil, a telescope, or a camera lens.

The closer you are to two objects, the greater the angular separation between them. Up close, then, two objects are easily resolved. As you get further from the objects, however, they will eventually merge to become one.

It's a Great Day For Physics: Heat and Temperature

Heat energy is most intense in substances whose molecules are moving rapidly in a very disorderly way. Such a substance will give up some of its heat to another substance whose molecules are less agitated. When this happens, the heat is said to “flow” from one substance to another (or from one body to another). The energy transfer is indicated by a change in temperature.

Temperature, therefore, is not the same thing as heat—although the two words are often used interchangeably. Temperature can be defined as the degree of intensity of hotness or coldness. “Hotness” and “coldness,” however, are comparative terms. A flame, for example, is hot when compared with ice but cold when compared with the sun. This definition of temperature, therefore, is vague and unscientific, although it does convey the correct impression that temperature is a measure of relative intensity rather than of quantity.

A more specific definition is: temperature is the ability of one body to give up heat energy to another body. A hot body becomes cooler, and a cold body becomes warmer, as long as heat is flowing from one to the other. The hot body has a greater ability to give up heat and therefore has a higher temperature. After a time the two bodies may reach a condition of heat equilibrium, or balance of heat intensity. Then, heat flow ceases. At the point of equilibrium both bodies can be said to be at the same temperature.

Measurement of Temperature

Temperature is measured by means of instruments called thermometers. Several temperature scales have been devised for relating the hotness and coldness of bodies to fixed temperatures, such as the freezing point and boiling point of water. On most temperature scales, the unit of temperature is called a degree. The Kelvin scale is an exception; its unit of temperature is the kelvin.

The Fahrenheit, Celsius (or centigrade), and Reaumur scales are used in the range of temperatures important for human comfort, laboratory experiments, and industrial processes.

The Rankine scale and the Kelvin scale are based on the concept of absolute zero; all temperature readings on these scales are positive numbers. The Kelvin scale is widely used in scientific work. The Rankine scale is used primarily by British and American engineers.

Absolute Zero

Experiments have shown that every 1° C. increase or decrease in temperature causes the pressure exerted by a gas to increase or decrease at the constant rate of 1/273.15 of its pressure at 0° C. This means that at -273.15° C. an ideal (theoretical) gas would exert no pressure at all. Since experiments with real gases have shown a clear relation between pressure and temperature, zero pressure would indicate that the ideal gas had lost all its ability to give up heat. Its molecules would be absolutely motionless. This is impossible—molecules are always agitated, to some extent—and therefore the absolute zero of temperature remains a theoretical concept. The concept is, however, a useful one, for it gives a base point to which all temperature measurements may be referred, in positive numbers.

The idea that absolute zero can never be reached is sometimes considered important enough to be called the third law of thermodynamics. Scientists have succeeded in cooling substances to within a small fraction of a degree above absolute zero. The study of the behavior of substances at very low temperatures is called cryogenics.

High Temperatures

Absolute zero is the lower limit for temperature, but there is no upper limit. The hottest substances known are ionized gases in certain stars, with temperatures of a billion degrees or more.

Measurement of Heat

The heat released or absorbed in a physical or chemical process can be measured with an instrument called a calorimeter. Commonly used units for measuring heat are the calorie and the British thermal unit, or Btu. Heat is also measured in such other units as the joule (the unit of energy in the SI, or metric system).

Heat travels by conduction, convection, or radiation, or a combination of these methods.

Conduction

As molecules move about they frequently bump into each other. According to the second law of thermodynamics, the faster-moving (hotter) molecules will give up some of their heat energy to the slower-moving (colder) molecules whenever collisions occur. The newly heated molecules then will be able to pass on a share of heat to molecules possessing less heat. The process continues in the direction away from the hottest molecules. By this means, heat is conducted (led) from the warm to the cool parts of a substance, or into a cool body that is in contact with a warm body. The heating of an iron rod, as shown in the drawing How Heat Travels, illustrates both: the heat spreads through the rod and also warms the handle that holds the rod.

Substances vary in their ability to conduct heat. Air and water are rather poor conductors. Most metals conduct heat rapidly. Asbestos conducts heat so poorly that it is used as a heat insulator.

Convection

When a fluid (a liquid or gas) is heated, the portion of the fluid nearest the heat source will expand as it gains energy. In expanding, this portion becomes less dense (lighter) and is pushed upward by cooler, heavier portions of the surrounding fluid. The displacement brings the cooler portions nearer the heat source, and they in turn gain energy, become lighter, and are pushed upward. The resulting movements, or currents (called convection currents), distribute heat from the source throughout the fluid.

Heat will continue to travel by convection as long as temperature differences exist within the fluid. Examples of convection are the movement of warm air in a room and the circulation of water in a kettle placed over a fire.

Radiation

All bodies continually give off energy in the form of rays. The rays may be composed of particles or waves. Heat rays, called infrared radiation, are electromagnetic waves that resemble light waves but have somewhat longer wavelengths.

A body emits heat rays as a result of the vibration of its molecules. As the rays are emitted, the molecules lose some of their energy. When another body absorbs the rays, its molecules become more agitated and the body thus gains heat energy. Heat rays can travel through a vacuum. Infrared radiation from the sun, for example, passes through empty space to reach the earth.

The amount of radiation a body will absorb varies with the material out of which it is made and the nature of its surface. In general, materials with a dark or rough surface will absorb more infrared radiation than materials with a white or shiny surface.

The main changes that substances undergo when they are heated are (1) increase in temperature, (2) change of state, and (3) expansion. Each of these changes depends on properties that differ from one substance to another. The rate of temperature change depends on the specific heat of the substance. Change of state—from solid to liquid, or from liquid to gas—occurs only when the latent heat requirements of the substance have been met. Expansion of solids and liquids takes place in accordance with the coefficient of expansion of the substance.

Specific Heat

The amount of heat required to raise the temperature of a unit mass of a substance by a certain amount is called that substance's specific heat. It is expressed as a ratio to the specific heat of water, which by definition is 1.

Latent Heat

The amount of heat energy that must be absorbed or released by a given quantity of a substance to bring about a complete change of state in the substance. As the latent heat is absorbed or released, the temperature of the substance remains the same. Latent heat is usually referred to in terms of the type of change of state involved.

Heat of Fusion is the latent heat needed to change a substance from a solid to a liquid. The change is called melting.

Heat of Vaporization is the latent heat needed to change a substance from a liquid to a gas (vapor). The change is called boiling if the vapor forms within the liquid, or evaporation if the vapor forms only at the surface.

Heat of Sublimation. At atmospheric pressure some substances when heated change directly from a solid to a gas. This change of state is called sublimation, and the latent heat needed for it is called the heat of sublimation.

Heat of Condensation is the latent heat given up by a substance in changing from a gas to a liquid. The change of state is sometimes called liquefaction. For any given substance, the heat of condensation is equal to the heat of vaporization.

Heat of Solidification is the latent heat given up by a substance in changing from a liquid to a solid or, for those substances that undergo sublimation, in changing from a gas into a solid. In the first case, the heat of solidification is equal to the heat of fusion and the change of state is commonly called freezing. In the second case, the heat of solidification is equal to the heat of sublimation.

Expansion

As a substance gains heat energy, its molecules push farther apart, causing the substance to occupy more space. This increase in size is called expansion. Solids and liquids expand in the same way. Gases are subject to different laws.

It's a Great Day For Physics: Elasticity and Hooke's Law

Elasticity, the property of a substance that enables it to recover its original shape and size after it has been stretched, squeezed, or bent. All substances are elastic in one way or another.

Gases and liquids have elasticity of volume. They have the ability to expand to their original volume after a compressive force has been removed. They also expand or contract to their original volume after being heated or cooled.

Solids have elasticity of form. They tend to resume their original shapes after being deformed by bending, twisting, pulling, or pressure. Some solids, such as putty and modeling clay, are plastic, or relatively inelastic. Others, such as rubber and steel, are very elastic. All solids can be deformed beyond their elastic limit—the point at which they will no longer resume their original form, even if the deforming force is removed.

Hooke's Law, formulated by the 17th-century English scientist Robert Hooke, is the basic law of elasticity. It states that the strain (tendency to deform) of a body is proportional to the stress (deforming force) applied to the body.

Hooke's Law states that "the extension of a helical spring is directly proportional to the weight applied, provided the elastic limit of the spring is not exceeded."

You may also see it written as: "Hooke's Law states that in an elastic material strain is proportional to stress and the point at which a material ceases to obey Hooke's Law is known as its elastic limit."

What this means is if you have a helical spring or an elastic material and apply a weight to it will stretch by a certain amount. If you remove the weight and apply another which is twice as heavy then the spring will stretch twice as far as it did the first time. If you remove that weight and apply a weight that is three times heavier than the first one then the spring will stretch three times further than it did the first time. This is where the "directly proportional to the weight applied" or "strain is proportional to stress" bit is relevant and importantly each time you remove a weight the spring returns to its unstressed size. However this cannot go on forever. There will come a point when the spring is stretched too far and it cannot return to its original state or in fact snaps. At this point the "elastic limit" has been exceeded and Hooke's Law no longer applies.

Who is Robert Hooke? He was a brilliant scientist, yet for some reason is relatively unknown.

Robert Hooke was born on the 18th July 1635, in Freshwater, on the Isle of Wight. The son of John Hooke, who taught him at home in his early years. Robert soon showed a keen mind, being a quick learner and showing great manual dexterity, making mechanical toys when he was a boy. He went to Westminster School when he was thirteen, and from there on to Oxford where he would meet many of the great scientists of the day. There he impressed with his abilities in constructing equipment and designing experiments and in 1658 he became assistant to Robert Boyle. He was friends with, worked with and sometimes argued with many scientists of note such as Christian Huygens, Christopher Wren, Robert Boyle, Antony van Leeuwenhoek and Isaac Newton. Regarding arguing Hooke and Newton had a somewhat tempestuous working relationship culminating in a bust up regarding their views on gravity. In 1662 Hooke was named Curator of Experiments of the Royal Society of London. He died in London on March 3, 1703.

It's a Great Day for Physics: Newton's Law

Next to E = mc², F = ma is the most famous equation in physics. Yet many people remain perplexed by this relatively simple algebraic expression. It's actually a mathematical representation of Sir Isaac Newton's second law of motion, one of the great scientist's most important contributions. The "second" implies that other laws exist, and, luckily for students and trivia hounds everywhere, there are only two additional laws of motion. All three are presented here:

1. Every object persists in its state of rest or uniform motion­ in a straight line unless it is compelled to change that state by forces impressed on it.
2. Force is equal to the change in momentum per change in time. For a constant mass, force equals mass times acceleration.
3. For every action, there is an equal and opposite reaction.

A.    The First Law

Let's restate Newton's first law in everyday terms:

An object at rest will stay at rest, forever, as long as nothing pushes or pulls on it. An object in motion will stay in motion, traveling in a straight line, forever, until something pushes or pulls on it.

The "forever" part is difficult to swallow sometimes. But imagine that you have three ramps. Also imagine that the ramps are never-endingly long and smooth. You let a marble roll down the first slope, which is set at a slight incline. The marble speeds up on its way down the ramp. Now, you give a gentle push to the marble going uphill on the second ramp. It slows down as it goes up. Finally, you push a marble on a ramp that represents the middle state ­between the first two -- in other words, a ramp that is perfectly horizontal. In this case, the marble will neither slow down nor speed up. In fact, it should keep rolling. Forever.

According to Newton's first law, the marble on that bottom ramp should just keep going. And going.

Physicists use the term inertia to describe this tendency of an object to resist a change in its motion. The Latin root for inertia is the same root for "inert," which means lacking the ability to move. So you can see how scientists came up with the word. What's more amazing is that they came up with the concept. Inertia isn't an immediately apparent physical property, such as length or volume. It is, however, related to an object's mass. To understand how, consider the sumo wrestler and the boy shown below.

Which person in this ring will be harder to move? The sumo wrestler or the little boy?

Let's say the wrestler on the left has a mass of 200 kilograms, and the boy on the right has a mass of 30 kilograms. Remember the object of sumo wrestling is to move your opponent from his position. Which person in our example would be easier to move? Common sense tells you that the boy would be easier to move.

You experience inertia in a moving car all the time. In fact, seatbelts exist in cars specifically to counteract the effects of inertia. Imagine for a moment that a car at a test track is traveling at a speed of 40 km/hour. Now imagine that a crash test dummy is inside that car, riding in the front seat. If the car slams into a wall, the dummy flies forward into the dashboard. Why? Because, according to Newton's first law, an object in motion will remain in motion until an outside force acts on it. When the car hits the wall, the dummy keeps moving in a straight line and at a constant speed until the dashboard applies a force. Seatbelts hold dummies (and passengers) down, protecting them from their own inertia.

B.     The Second Law

You may be surprised to learn that Newton wasn't the genius behind the law of inertia. But Newton himself wrote that he was able to see so far only because he stood on "the shoulders of Giants." And see far he did. Although the law of inertia identified forces as the actions required to stop or start motion, it didn't quantify those forces. Newton's second law supplied the missing link by relating force to acceleration.

When a force acts on an object, the object accelerates in the direction of the force. If the mass of an object is held constant, increasing force will increase acceleration. If the force on an object remains constant, increasing mass will decrease acceleration. In other words, force and acceleration are directly proportional, while mass and acceleration are inversely proportional.

Technically, Newton equated force to the differential change in momentum per unit time. Momentum is a characteristic of a moving body determined by the product of the body's mass and velocity. To determine the differential change in momentum per unit time, Newton developed a new type of math -- differential calculus. His original equation looked something like this:

F = (m)(Δv/Δt)

where the delta symbols signify change. Because acceleration is defined as the instantaneous change in velocity in an instant of time (Δv/Δt), the equation is often rewritten as:

F = ma

The equation form of Newton's second law allows us to specify a unit of measurement for force. Because the standard unit of mass is the kilogram (kg) and the standard unit of acceleration is meters per second squared (m/s²), the unit for force must be a product of the two -- (kg)(m/s²). This is a little awkward, so scientists decided to use a Newton as the official unit of force. One Newton, or N, is equivalent to 1 kilogram-meter per second squared.

So what can you do with Newton's second law? As it turns out, F = ma lets you quantify motion of every variety. Let's say, for example, you want to calculate the acceleration of the dog sled shown below.

If you want to calculate the acceleration, first you need to modify the force equation to get a = F/m. When you plug in the numbers for force (100 N) and mass (50 kg), you find that the acceleration is 2 m/s².

Now let's say that the mass of the sled stays at 50 kg and that another dog is added to the team. If we assume the second dog pulls with the same force as the first (100 N), the total force would be 200 N and the acceleration would be 4 m/s².

           

Finally, let's imagine that a second dog team is attached to the sled so that it can pull in the opposite direction.

If two dogs are on each side, then the total force pulling to the left (200 N) balances the total force pulling to the right (200 N). That means the net force on the sled is zero, so the sled doesn’t move.

This is important because Newton's second law is concerned with net forces. We could rewrite the law to say: When a net force acts on an object, the object accelerates in the direction of the net force. Now imagine that one of the dogs on the left breaks free and runs away. Suddenly, the force pulling to the right is larger than the f­orce pulling to the left, so the sled accelerates to the right.

What's not so obvious in our examples is that the sled is also applying a force on the dogs. In other words, all forces act in pairs. This is Newton's third law.

C.     The Third Law

Newton's third law is probably the most familiar. Everyone knows that every action has an equal and opposite reaction, right? Unfortunately, this statement lacks some necessary detail. This is a better way to say it:

A force is exerted by one object on another object. In other words, every force involves the interaction of two objects. When one object exerts a force on a second object, the second object also exerts a force on the first object. The two forces are equal in strength and oriented in opposite directions.

Many people have trouble visualizing this law because it's not as intuitive. In fact, the best way to discuss the law of force pairs is by presenting examples. Let's start by considering a swimmer facing the wall of a pool. If she places her feet on the wall and pushes hard, what happens? She shoots backward, away from the wall.

Clearly, the swimmer is applying a force to the wall, but her motion indicates that a force is being applied to her, too. This force comes from the wall, and it's equal in magnitude and opposite in direction.

Next, think about a book lying on a table. What forces are acting on it? One big force is Earth's gravity. In fact, the book's weight is a measurement of Earth's gravitational attraction. So, if we say the book weighs 10 N, what we're really saying is that Earth is applying a force of 10 N on the book. The force is directed straight down, toward the center of the planet. Despite this force, the book remains motionless, which can only mean one thing: There must be another force, equal to 10 N, pushing upward. That force is coming from the table.

If you're catching on to Newton's third law, you should have noticed another force pair described in the paragraph above. Earth is applying a force on the book, so the book must be applying a force on Earth. Is that possible? Yes, it is, but the book is so small that it cannot appreciably accelerate something as large as a planet.

You see something similar, although on a much smaller scale, when a baseball bat strikes a ball. There's no doubt the bat applies a force to the ball: It accelerates rapidly after being struck. But the ball must also be applying a force to the bat. The mass of the ball, however, is small compared to the mass of the bat, which includes the batter attached to the end of it. Still, if you've ever seen a wooden baseball bat break into pieces as it strikes a ball, then you've seen firsthand evidence of the ball's force.

These examples don't show a practical application of Newton's third law. Is there a way to put force pairs to good use? Jet propulsion is one application. Used by animals such as squid and octopi, as well as by certain airplanes and rockets, jet propulsion involves forcing a substance through an opening at high speed. In squid and octopi, the substance is seawater, which is sucked in through the mantle and ejected through a siphon. Because the animal exerts a force on the water jet, the water jet exerts a force on the animal, causing it to move. A similar principle is at work in turbine-equipped jet planes and rockets in space.