[Concept] : Archimedes Principle

Archimedes was the son of an astronomer. He had traveled to Alexandria, Egypt, a place of great learning, where he studied the works of some other mathematicians, like Euclid and Conon. Archimedes was friends with King Hieron II of Syracuse. Archimedes helped his friend King Hieron II by creating machines for the king’s army. The pulley was one of these inventions, but Archimedes thought the study of mathematics was the most important thing he could do. Sometimes Archimedes got so busy thinking about mathematics that he forgot to take a bath and his servants would have to force him to go to the public baths.

Kings don’t like to be tricked. I’m sure you know this from all the stories you read when you were small. King Hieron II of Syracuse, was no exception. He worried that the people who made his crown charged him the price of using solid gold but instead they tricked him and used gold mixed with silver which costs less.

King asked the scientists to test his crown which is supposed to be made of pure gold. So Archimedes also thought of this crown testing. When Archimedes was at the public bath he noticed that when he climbed in to a soaking bath the water level went up. Archimedes knew he could use this knowledge to test whether King Hieron’s crown was made of solid gold. He was so excited about this new idea and he wanted to tell the king. He jumped out of the bath and shouted “Eureka!” which, is what you should shout whenever you have a great idea! Then Archimedes ran naked through the streets of Syracuse to tell the King his new idea.

This is how most of the principles will evolve so many of them have no scientific reason to evolve.

Principle:  is a law of physics stating that the upward buoyant force exerted on a body immersed in a fluids equal to the weight of the fluid the body displaces. In other words, an immersed object is buoyed up by a force equal to the weight of the fluid it actually displaces.

buoyancy = weight of displaced fluid

Explanation: If an immersed object displaces 1 kilogram of fluid, the buoyant force acting on it is equal to the weight of 1 kilogram (as a kilogram is unit of mass and not of force, the buoyant force is the weight of 1 kg, which is approximately 9.8 Newtons.) It is important to note that the term immersed refers to an object that is either completely or partially submerged. If a sealed 1-liter container is immersed halfway into the water, it will displace a half-liter of water and be buoyed up by a force equal to the weight of a half-liter of water, no matter what is in the container.

If such an object is completely submerged, it will be buoyed up by a force equivalent to the weight of a full liter of water (1 kilogram of force). If the container is completely submerged and does not compress, the buoyant force will equal the weight of 1 kilogram of water at any depth, since the volume of the container does not change, resulting in a constant displacement regardless of depth. The weight of the displaced water, and not the weight of the submerged object, is equal to the buoyant force.

Formula:

• The weight of the displaced fluid is directly proportional to the volume of the displaced fluid.
• The weight of the object in water is less than the weight of object in air.

Suppose a rock’s weight is measured as 10 newtons when suspended by a string in a vaccum with gravity acting upon it. Suppose that when the rock is lowered into water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyant force: 10 − 3 = 7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor. It is generally easier to lift an object up through the water than it is to pull it out of the water.

Therefore we can modify Archimedes statement as

apparent immersed weight =weight ­– weight of displaced fluid

This can be expanded like

Density/ Density of fluid =weight/ weight of displaced fluid

Density/ Density of fluid  = weight/weight – apparent immersed wgt

This formula is used for example in describing the measuring principle of a dasymeter   and of hydrostatic weighing.

Limitations: This principle will not consider surface tension effects.

[Concept] : Laser and Hologram

A laser beam can cut through steel as easily as knife cuts through butter. A laser is a device that produces a powerful beam of light. The word ‘laser’ stands for Light Amplification by Stimulated Emission of Radiation.

All lasers produce coherent light. Coherent light is very pure, which means that all the light waves have the same wavelength, they are all “in step” with one another, and they are all travelling in exactly the same direction. Laser light can be used to create three dimensional photographs called holograms.

 How a laser works

The heart of a laser is a material called lasing medium. The lasing medium is given energy, usually by an electric current or by light from device called a flash tube. The atoms of the lasing medium absorb energy and then give it out again as waves of coherent light. The light reflects back and forth between two mirrors, becoming more and more intense, until it emerges from one of the mirrors (which is only partly reflective) as a laser beam.

Application of lasers

Lasers have many uses, because they produce a powerful beam of uniform light that will not spread out over long distances and that can be directed very precisely. Lasers are used to read supermarket bar codes, play compact disks, guide weapons, and send signals along optical fibres.

Metal cutting

A powerful infrared laser beam can generate enough heat to cut through metals or to weld (join) them together by melting them.

Laser surgery

Surgeons can control lasers with great precision to burn away cancer cells or delicately trim the lens of an eye to improve a person’s sight.

Light show

Laser beams always follow a straight line, so they can be used to produce stunning visual effects at rock concerts and other special events.

Holograms

Holograms are photographs that appear three dimensional. This effect is produced by taking a photograph using two different sets of light waves from a laser beam. Holograms have many uses because they allow people to see an object from different angles.

Making holograms

To make a hologram, a laser beam is split into two parts, one called an object beam and the other a reference beam. Only the object beam reflects off the object that is to be photographed. Both beams strike a plate of photographic film, where they interfere and create a three dimensional looking image.

Famous personality

Theodore Maiman

In 1953, US physicist Charles Townes invented a device called a maser, which produced microwaves. In 1960, his fellow US physicist Theodore Maiman used the principle of Townes’ device to build a laser. Maiman’s laser used a ruby crystal as the lasing medium.

[Concept] : Optics

Ever wondered why the beautiful rainbow occurs?  Have you ever seen water on road (and it being disappearing when you go close) on a hot summer day? Ever wondered how our eye works? Heard the stories of Archimedes destroying enemy ships by focusing sunlight using lenses over enemy ship?

Well, all the answers lie in optics- one of the most fascinating topics in physics.

What is light?

Well in common terms, anything visible to our eyes is called light. Technically light is a part of a very wide electromagnetic spectrum of which a very small part which our eyes can sense is termed as visible light.

Also light has a very interesting history on what it is. First newton showed that it was a particle (photon) and he and other scientists proved the laws  that a ray of light follows. But later in the nineteenth century English scientist Thomas Young and French scientist Fresnel showed that light is a wave by showing the interference. Later Maxwell proved that light is an electromagnetic wave by calculating  velocity of electromagnetic wave and showing that it is same as velocity of light. By then it was thoroughly established that light is a wave. But at the end of nineteenth century neared some serious problems came up with the wave nature of light- it couldn’t explain some phenomenon like the black body radiation and the photoelectric effect. ____ explained the black body radiation and photoelectric effect by treating light as a particle. Simultaneously a new field of physics -quantum physics emerged. Now scientists were facing a serious dilemma- light showed interference which was explained  by the wave nature of light while the blackbody radiation and photoelectric effect were explained only by the particle theory- then what was light. Here quantum theory helped which stated that light( this may be extended to matter also )in nature has dual character- it is both simultaneously wave and particle. So finally we came to know what is light- it is both

Some basic laws followed by light:

Light is a ray i.e, it travels in a straight line.

• Laws of reflection
  • Angle of incidence is equal to the angle of reflection.
  • The incident ray, the reflected ray and the normal to the surface, all lie in the same plane.

As shown in the figure  is the angle of incidence and  is the angle of reflection where PO is the incident ray and QO is the reflected ray.

• Laws of refraction
  •  The incident ray, the reflected ray and the normal to the surface, all lie in the same plane
  •  Product of refractive index and sine of angle of ray with the normal of interface is constant. This is called Snell’s  law. This law has many implications. If there are several medium side by side and you want to find out the angle of refraction of any medium (lets say the fifth medium) you can use this method easily instead of finding each angle sequentially. Just compare with the product of sine angle and refractive index of  any known medium.

The refraction of light is a consequence of light changing its velocity in different medium. also the product of velocity of light in a medium and its refractive index in constant.

You might have heard that the wavelength of light changes in a medium. This is  to change  its velocity. Velocity is the product of wavelength and the frequency.

Have you ever wondered why the wavelength of light changes to change the velocity y and not the frequency. This is due to fact that energy of light  is the product of planck’s constant and frequency. As the energy of light is constant in any medium the frequency remains constant.

The reflection and refraction has  a widespread application in our daily life. It can be seen almost everywhere- from simple mirrors in our houses to huge ones used in astronomical telescopes, from optical fibre cable to microscopes and to polaroid glasses with which you watch 3d movies. All of these are based on one or more of many effects that light follows. Lets see some of them briefly:

Total Internal Reflection: It is based on the Snell’s law . When a light ray moving from an optically denser to rarer  medium is greater than certain angle called critical angle the whole ray is reflected with no transmission hence there is no loss of light( In normal conditions light is always partially reflected or refracted). Due to this, optical fibres are used in high speed data transmission as there is minimal loss of data.

 is the critical angle

Mirage:  Almost all of you must have seen a mirage on a road on a hot day. It is formed as the layers of air near the road are hotter than the air above hence light is refracted sequentially and you see water on road. You must have also heard of stories of travelers seeing water in a desert. They actually see the reflection of blue sky on the desert due to same effect as on road.

Rainbow: Rainbow is also formed due to total internal reflection of the sun rays and hence dispersion by the water droplets presents in the sky after the rain. Have you ever noticed that rainbow is always formed in the direction against the sun?

Interference: If i ask you a question that if you superimposed two light rays over each other what’ll happen? You might think that ( and it happens in general conditions) the intensity will double and there’ll be more light. But you’ll be amazed to know  that under proper conditions two light sources can add up to darkness!!!!(called destructive interference). It can also add up to more than twice the intensity(constructive interference)

To achieve this the two light sources must be in same phase. You might understand it better it by the following figure:

At the last last the story that Archimedes burned down ships of enemy might not be true as many scientists believe that it is not possible. As to achieve this mirrors of very high quality would be required which was not made that time. Also both the ships would have been in constant motion and focussing the sunlight over one point  would have been extremely difficult!!!!!!!! Infact flaming arrows or catapult throwing fire would have been much easily burned out the ships. He actually built many other things like ship shaker to destroy enemy ships.

[Concept] : Conservation Law

If the laws of forces acting on a system’s particles and the state of the system at a certain initial moment are known, the motion equations can help predict the subsequent behaviour of the system, i.e. find its state at any moment of time. However, an analysis of a system’s behavior by the use of the motion equations requires so much effort, that a comprehensive solution seems to be practically impossible.

Under these circumstances the following question natural­ly comes up: are there any general principles following from Newton’s laws that would help avoid these difficulties by opening up some new approaches to the solution of the problem.

It appears that such principles exist. They are called conservation laws.

As it was mentioned, the state of a system varies in the course of time as that system moves. However, there are some quantities, state functions, which possess the very important and remarkable property of retaining their values constant with time. Among these constant quantities, energy, momentum and angular momentum play the most significant role. These three quantities have the important general property of additivity: their value for a system com­posed of parts whose interaction is negligible is equal to the sum of the corresponding values for the individual con­stituent parts (incidentally, in the case of momentum and angular momentum additivity holds true even in the pres­ence of interaction). It is additivity that makes these three quantities extremely important.

By way of explanation, the energy conservation law is associated with uniformity of time, while the laws of conservation of momentum and angular momentum with uniformity and isotropy of space respectively. This implies that the conservation laws listed above can be derived from Newton’s second law. We shall not, however, discuss this problem in more detail.

The laws of conservation of energy, momentum and angu­lar momentum fall into the category of the most fundamental principles of physics, whose significance cannot be overes­timated. These laws have become even more significant since it was discovered that they go beyond the scope of mechanics and represent universal laws of nature. In any case, no phenomena have been observed so far which do not obey these laws. They are among the few most general laws underlying contemporary physics.

The conservation laws do not depend on either the paths of particles or the nature of acting forces. Consequently, they allow us to draw some general and essential conclusions about the properties of various mechanical processes without resorting to their detailed analysis by means of motion equations. For example, as soon as it turns out that a cer­tain process is in conflict with the conservation laws, one can be sure that such a process is impossible and it is no use trying to accomplish it.

Since the conservation laws do not depend on acting forces, they may be employed even when the forces are not known. In these cases the conservation laws are the only and indispensable instrument.

Even when the forces are known precisely, the con­servation laws can help substantially to solve many problems of motion of particles. Although all these problems can be solved with the use of motion equations (and the conserva­tion laws provide no additional information in this case), the utilization of the conservation laws very often allows the solution to be obtained in the most straightforward and elegant fashion, obviating cumbersome and tedious calcula­tions. Therefore, whenever new problems are ventured, the following order of priorities should be established: first, one after another conservation laws are applied and only having made sure that they are inadequate, the solution is sought through the use of motion equations.

We shall begin examining the conservation laws with the energy conservation law, having introduced the notion of energy via the notion of work.

Let a particle travel some arbitrary path under the action of the force F. In the general case the force F may vary during the motion, both in magnitude and direction. Let us consider the elementary displacement , during which the force F can be assumed constant and along the direction of displacement.

The action of the force F over the displacement dr is characterized by a quantity equal to the scalar product  and called the elementary work of the force F over the dis­placement . It can also be presented in another form:

where a is the angle between the vectors F and ,  is the elementary path, and Fs is the projection of the vector F on the vector  (Fig. 37).

Thus, the elementary work of the force F over the dis­placement  is

The quantity is algebraic: depending on the angle between the vectors F and , or on the sign of the projection F, of the vector F on the vector , it can be either positive or negative, or, in particular, equal to zero (when either of force and displacement vanishes or when force is perpendicular to the displacement).

Integrating the expression over all elementary sections of the arbitrary  path from point 1 to point 2, we find the work of the force F over the given path.

The expression can be graphically illustrated. Let us plot F, as a function of the particle position along the path. The elementary work  is seen to be numerically equal to the area of the shaded portion enclosed by the graph and the x-axis. Here the area of the figure lying over the x axis is taken with the plus sign (it corre­sponds to positive work) while the area of the figure lying under the x axis is taken with the minus sign (it corresponds to negative work).

Let us consider a few examples involving calculations of work.

The work of the elastic force , where r is the radius vector of the particle A relative to a stationary point. Let us displace the particle A experiencing the action of that force along an arbitrary path from point 1 to point 2. Assume that point 1 and 2 are at distances a and b from the stationary point above respectively. We shall first find the elementary work performed by the force F over the elementary displacement :

Elementary work for displacing the particle by a small distance  :

Now integrating the above equation from a to b on both sides with respect to r gives:

Similarly, the reader is expected to try for columbic and frictional force. Columbic force is inversely proportional to the square of the radius vector and frictional force is a constant with respect to radius vector.

[Concept] : Basic Type of Thermometer

Concept: A thermometric property like length of a wire, resistance of a metallic and volume of a gas proportional to temperature.

In this article let X0, X100, Xt be the parameters at ice point, steam point and unknown temperature, respectively. So

t = (X0-Xt/X100-X0)1000C

Thermometer: The instrument used to measure the temperature is called is called Thermometre.

Classification: The classification is depending upon the physical property of the substance that varies with temperature.

1. Liquid Thermometres: These thermometers are based on the thermal expansion of liquids.

Principle: The increase in volume of the liquid in the glass bulb is directly proportional to increase in the temperature.

If l0, l100, lt are the parameters of length. Then temperature

t = (l0-lt/l100-l0)1000C

Range: For mercury thermometer upto 3000C.

2. Constant volume gas Thermometer: These thermometers are based on the thermal expansion of gases at constant volume.

Principle: The increase in pressure of a gas at constant volume is directly proportional to increase in the temperature.

If P0, P100, Pt are parameters of pressure. Then temperature

t = (P0-Pt/P100-P0)1000C

Range: Constant hydrogen gas thermometers has different range with different bulbs used

With Platinum bulb: -2000C to 5000C.

With Porcelain bulb: 11000C.

3. Constant pressure gas thermometer: These thermometers are based on the thermal expansions of gases at constant pressure.

Principle: The increase in volume of a gas at constant pressure is directly proportional to increase in its temperature.

If V0, V100, Vt are parameters of volume. Then temperature

t = (V0-Vt/V100-V0)1000C

With air as a thermometric substance can be used to measure temperature upto 6000C.

4. Resistance Thermometers: These thermometers are based on the variation of electric resistance of metals with temperature. These thermometers usually employ platinum as the thermometric substance.

Principle:  The increase in resistance of a platinum wire is directly proportional to increase in its temperature.

If R0, R100, Rt are parameters of resistance. Then temperature

t = (R0-Rt/R100-R0)1000C

Range: -2720C to 12000C.

Questions:

1. The resistances of a platinum resistance thermometer at lower and upper fixed points are 3.50ohms and 3.65ohms respectively. Then find the resistance of platinum wire at -370c? Ans: 3.55ohms

2. The readings corresponding to the ice and steam points for a certain pressure gas thermometer are 500cc and 545cc. If reading at certain temperature be 510cc then find that temperature? Ans: 22.220C

[Concept] : Gravitation

“Everything that goes up always comes down”

You might of heard of this quote. But this quote is not entirely true!!!

If you throw an object with velocity of 11.2km/s it’ll never come back!!!! This velocity is called the escape velocity of the earth. In fact every planet has a escape velocity which depends on its gravitational pull.

Gravitation is one of the most important branch of physics. It is due to gravitational pull  of sun that we live today. Gravity is everywhere on universe. Gravitational force is one of the basic force of physics.

English physicist Sir Isaac Newton discovered the laws of gravitation.(He had discovered the laws of motion earlier and argued that there must be some force acting on moon by earth else it would follow a straight line path( from the first law of motion) rather than a circular path around the earth). He stated the universal law of gravitation as:

Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to

the square of the distance between them.

Mathematically if two masses of mass m1 and m2 are separated by a distance r with each other the force of gravitation between them is

F = (Gm1m2)/ r2

where G is the universal gravitation constant. It has the value 6.67x 10-11 m3kg-1s-2. It is interesting to note that the universal gravitational constant was discovered much later in the eighteenth century by Henry Cavendish. Before that the constant was used as unity( same as the second law of motion) but for that gravitational mass was used in place of real mass.

To calculate the the gravitational constant of the earth Cavendish used two small spheres with same mass fixed to the ends of a light horizontal rod suspended by thin wire. Then he placed two large sphere with similar mass near to the small spheres. The attractive

force between smaller and larger spheres causes the rod to rotate and twist the

wire suspension. He measured the angle of rotation by the deflection of a light beam reflected from a mirror attached to the vertical suspension wire. With repeated attempts and proper calculations he calculated the gravitational constant of the earth. (Actually his actual  experiment was to measure the density of the earth with respect to water with the help of the gravitational interactions he measured).

Calculating the acceleration due to gravity of the earth:

After calculating the force due to the earth now lets calculate the acceleration with which the earth attracts every body:

from Newton’s second law

F = ma   where m is the mass of the body and a is acceleration

therefore

F = ma = (GMem)/R2

where, Me is the mass of the earth & R is the radius of earth.

therefore                                                                                                a = (GMe)/R2

putting values of the gravitational constant, mass of the earth and the radius of the earth

we get the acceleration due to gravity of earth as  9.81 ms-2. Please note here that this value is valid for objects on or near to the surface of the earth.

You might think that the astronaut in the space station are in gravity free state. This is not true. They are in state of free- fall due to the centrifugal force acting on them as they are orbiting around the earth which cancels the gravitational force.

Also there is a common misconception between the mass and weight of an object. Mass is an inherent virtue of an object while weight is the product of  acceleration due to gravity and mass. Due to this your weight will be different on different planet. Also your weight will be slightly different on equator in comparison to that of on the poles as earth is slightly flattened near the pole. (Hence the radius slightly decreases thereby increasing the acceleration due to gravity.

Lets calculate the escape velocity of the earth:

By using the law of  conservation of energy  one can easily calculate the escape velocity of the earth. Let v be the escape velocity of the earth, m be the mass of the object thrown and Me the mass of the earth

The kinetic energy of the object at earth is mv2/2. Its potential energy is (-GMem)/R where R is the radius of the earth. Its final potential energy is zero and we take the final kinetic energy also be zero. (If we throw by velocity greater than the escape velocity final kinetic will not be zero.

therefore

mv2/2  +  (-GMem)/r  = 0 + 0

v =  after putting the values the escape velocity comes out to be 11.2 km per sec.

It is to be noted here that the rockets launched in the space are not thrown with the escape velocity but they have fuel which with which they are able to move up even with less velocity.

To spaceships (like the voyager) rockets first take the ship in upper atmosphere and then give them the boost to escape velocity as there the escape velocity is less than that on earth.

Laws of Planetary motion (Kepler’s laws) :

Kepler gave the laws of planetary motion with the help of the data collected by Tycho Brahe (under whom he worked ). It is really amazing to know that these people collected data without the help of any telescope . Lets see the laws:

• Law of orbits :All planets rotate around the sun in elliptical orbits with sun at any one of the focal points. The closest distance between the planet and sun is called perihelion and the farthest distance is called aphelion.
  
• Law of equal areas: The radius vector drawn from the Sun to the planet sweeps out equal area in equal interval of time.
  With this law we can conclude that the speed with which the planet revolves around the sun is not constant. When it is nearer to the sun its speed is more than when it is far away from the sun. This is to keep the area swept by the line constant. (see the figure).
  
• Law of harmonies: The square of time period of revolution of a planet is proportional to cube of the semi-major axis of its elliptical orbit.
  From this law we can conclude that  the ratio T2/R3 is a constant for any planet or satellite