Harmonic oscillator phase

In classical mechanics, a harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x according to Hooke's law:
where k is a positive constant.

If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
Oscillate with a frequency smaller than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator). Decay exponentially to the equilibrium position, without oscillations (overdamped oscillator).

If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator.Mechanical examples include pendula (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.

Simple harmonic oscillator

A simple harmonic oscillator is an oscillator that is neither driven nor damped. Its motion is periodic— repeating itself in a sinusoidal fashion with constant amplitude, A. Simple harmonic motion SHM can serve as a mathematical model of a variety of motions, such as a pendulum with small amplitudes and a mass on a spring. It also provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.

In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T, the time for a single oscillation, its frequency, f, the reciprocal of the period f = 1⁄T (i.e. the number of cycles per unit time), and its phase, φ, which determines the starting point on the sine wave. The period and frequency are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and velocity) of that system. Overall then, the equation describing simple harmonic motion.

Air Cooling

If your bike engine  gets heat then  how it gets cool?Many question we will get in the  mind?Many questions will be answered by discussing.As we know heat makes many good things and also bad things.When it is makes good we should utilize,When it creates lot problems in that time we have to eliminate it.some drivers drive vehicle without looking in to the heat gauze so ultimately it seize engine.But same time heat is required to burn the fuel.We can not burn fuel without by the criteria heat.In industry without monitoring the heat gauge boiler gets burst it causes many fatal.So heat is essential also but same time it is not required also.Really we confused about the heat and its relevant things.In  summer we feel heat we required cool air.This fulfilled by air cooler.

Air Cooling Engine

Direct air cooling used in  rotary radial engines used. The machine is made such way that it must cope up heat.The many parts of engines are  forgings of alloy instead of cast iron.Air pins are made in engine cylinder .Air enter in to these pins it makes cool.

In liquid cooling system also having air cool system.I will tell you how?

Excess heat cooled by liquid cooling system,but extra heat that will be cooled by air cool.

Cooling system In engine

In engine cooling makes major job. It removes the excess heat from the engine. It won’t remove all the heat from engine. It removes excess heat. Efficiency of engine depend on the cooling system also because it take out complete heat engine mileage reduces. If it reduced engine get seize. So it perform major job in engine. Automobile cooling system has 2 systems 1) Air cooling system 2) Fluid cooling system.

Liquid cooling will be used in many vehicles today. Air cooling system also used by in vehicles. Depend upon the heat generation in the engine cylinder. Air cooling system used by airplanes, motorcycles and lawn mowers. Heavy commercial vehicles like earth movers, Mines Lorries, Trucks.

Newton's law of cooling

Newton’s law of cooling states that “the rate of change of the temperature of an object is proportional to the difference in temperatures between the body and its surroundings.”

The law is given as the differential equation:

Where, Q = Thermal energy in joules
 h = Convection Heat Transfer Coefficient
A = Surface area of the heat being transferred
T = Temperature of the object's surface and interior (since these are the same in this approximation)
T_env = Temperature of the environment
ΔT(t) = T(t) − T_env is the time-dependent thermal gradient between environment and object

Example: ("coffee cooling problem”)

Suppose, You are having lunch at a restaurant. You place your order, and the waitress brings you your coffee much earlier than the rest of your meal. You want the coffee to stay warm until your meal arrives so you can have them at the same time. You always add cream to your coffee, but know that from Newton’s Law of Cooling equation that a hot object transfers heat to its surroundings at a rate proportional to the difference in temperature between the two. So your choice is to either add the cream to your coffee now, or add the cream to your coffee once your meal arrives. You think about the problem for a moment and come to a conclusion.

If you add the cream right away the temperature difference between the coffee and its surrounding air is brought closer together than between just the hot coffee without cream and restaurant air. A hot object cools at a rate that is faster when the difference between the temperatures of liquid and the surrounding air and cup is the greatest. Adding cool cream at the beginning slows down the cooling speed because it decreases the difference in temperature between the hot coffee and its surroundings. If you did not add cream right away the difference in temperatures of the hot coffee and restaurant air and cup is the greatest, so it would cool more rapidly and then when the cream would be added, it would cool even further. You add your cream to your coffee as soon you got it,and enjoy a nice hot cup of coffee when your meal arrives all thanks to Newton’s Law of Cooling to help you out.

Newton's Law of Cooling

Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings).

Newton's Law makes a statement about an instantaneous rate of change of the temperature. We will see that when we translate this verbal statement into a differential equation, we arrive at a differential equation. The solution to this equation will then be a function that tracks the complete record of the temperature over time. Newton's Law would enable us to solve the following problem.

Example 1: The Big Pot of Soup As part of his summer job at a restaurant, Jim learned to cook up a big pot of soup late at night, just before closing time, so that there would be plenty of soup to feed customers the next day. He also found out that, while refrigeration was essential to preserve the soup overnight, the soup was too hot to be put directly into the fridge when it was ready. (The soup had just boiled at 100 degrees C, and the fridge was not powerful enough to accommodate a big pot of soup if it was any warmer than 20 degrees C). Jim discovered that by cooling the pot in a sink full of cold water, (kept running, so that its temperature was roughly constant at 5 degrees C) and stirring occasionally, he could bring that temperature of the soup to 60 degrees C in ten minutes. How long before closing time should the soup be ready so that Jim could put it in the fridge and leave on time

Here a bit of care is needed: Clearly if the soup is hotter than the water in the sink  , then the soup is cooling down which means that the derivative  should be negative. (Remember the connection between a decreasing function and the sign of the derivative ?). This means that the equation we need has to have the following sign pattern:

Nitrogen and carbon cycle

Introduction:

CARBON CYCLE-

There are more compounds of carbon, than of all the other elements taken together except hydrogen. This wide variety of carbon compounds is essential for the existence of the complex molecules of life, For example, carbohydrates, fats, proteins, vitamins and nucleic acids. It is extremely important that carbon atoms transfer form the living to the non-living forms and vice-versa. This is not only linked to transfer of energy but also to basic processes by which life survives on the earth. The exchange of carbon between the living and non-living thing centres around two processes, namely respiration and photosynthesis, and one compound, carbon dioxide. This Cycle of carbon in nature is called Carbon cycle.

Nitrogen Cycle-

Nitrogen is essential for all the living things. Proteins and nucleic acids, which are essential for growth and good health, contain nitrogen. Like carbon, there is a global cycle for nitrogen, which is known as the Nitrogen cycle. Nitrogen atoms are cycled between various form of life, and between the atmosphere and the soil, by a series of interlinked chemical changes. Animals feed on plants and other animals for their requirement of nitrogen for making proteins. Most plants obtain the nitrogen they require from the soil. In soil, nitrogen is present as nitrates, which are soluble salts of nitric acid. The solubility of nitrates is of great importance. Plants absorb nitrates from aqueous solutions through their roots. Nitrates come to the soil from the atmosphere with rain water. In the atmosphere, at the time of lightning, nitrogen and oxygen combine to form oxides of nitrogen, which, in turn, form nitrates. Nitrates also enter the soil from the decay of dead plants and animals. Nitrogen-fixing bacteria are found in the soil, which can convert the nitrogen in air directly in to nitrates. Some plants are also capable of fixing atmospheric nitrogen because their roots have such nodules that contain nitrogen-fixing bacteria. These plants are leguminous, known as legumes. Beans plant is an example of a leguminous plant.

Zinc -carbon Battery

Introduction :
Zinc -carbon battery is also called as dry cell or dry battery.  Zinc -carbon battery is a battery which is packaged in zinc that can  serves as both a negative terminal and a container.  Zinc -carbon battery was developed from wet Leclanche cell.   In Zinc-carbon battery carbon rod or the graphite rod is the positive terminal which is surrounded by manganese dioxide and carbon powder mixture.   Zinc chloride and aluminium chloride solution is used as an electrolyte.  The original ammonium chloride variety is improved by Zinc chloride cells.  They are commonly termed as “General purpose” batteries.   These are the primary batteries.

Construction of Zinc -carbon Battery :

Figure:   Cross section zinc-carbon battery.
The above figure is the cross sectional picture of zinc-carbon battery.
The zinc can is the container in the zinc-carbon battery which is the negative terminal which contains a layer of NH₄Cl with aqueous paste of ZnCl₂ separated by a paper layer from a mixture of manganese oxide and a powdered carbon which is packed around the carbon rod.The carbon rod is slightly porous which allows to escape out the accumulated gas retaining the water for the electrolyte. The ratio of manganese dioxide and carbon powder in the cathode paste affects the cell’s characteristics. If the carbon powder is more then there will be decrease in the internal resistance but the capacity is improved by more manganese dioxide.
Reactions involved in zinc-carbon battery:
In a zinc- carbon battery, the zinc container which is a negative terminal. Here zinc undergo oxidation as follows,
Zn(s) → Zn²⁺(aq) + 2 e^-.
A graphite rod which is surrounded by a powder containing manganese (IV) oxide is the positive terminal. Here the manganese (IV) oxide mixed with the carbon which increases the electrical conductivity. The reaction is as follows,
2MnO₂(s) + H₂ (g) → Mn₂O₃(s) + H₂O(l) , here H₂ is obtained from ammonium salt.
2NH₄⁺(aq) + 2 e^- → H₂(g) + 2NH₃(aq)
In this reaction, the manganese is reduced from an oxidation station (+4) to (+3).
The overall reaction of the zinc carbon battery is
Zn(s) + 2MnO₂(s) + 2NH₄⁺ (aq) → Mn₂O₃(s) + Zn(NH₃)₂²⁺(aq) + H₂O(l).

Advantages of Zinc-carbon Battery:

They are least expensive primary batteries.
The power drain is not too high.
Disadvantages of zinc-carbon battery:
These zinc-carbon batteries are not rechargeable which must be discarded.
Applications of zinc-carbon battery:
They are least expensive so they are used in remote controls, flashlights, clocks, transistor radios and many more.

Hydrogen Fuel Cell Economy

Introduction

Let us discuss about the hydrogen fuel cell economy. The fuel cell is combined with of hydrogen and oxygen. The fuel cell to create the electricity, heat and water. The working electric power energy produced by chemical reaction in hydrogen fuel cell. The cell in which by incineration of gaseous fuel, the incineration effect obtained is changed into electrical effect in single step. Those cells are called as the hydrogen fuel cell economy.

Explanation of Hydrogen Fuel Cell Economy

 The simple explanation of cell utilizing hydrogen fuel cell economy is given in following diagram.
                               
In a vessel two porous carbon disphrams are placed and concentrated aqueous solution of NaOH is filled in between. Both these disphrams work as inert electrodes. The electrode substitute as anode consists of combined powder of platinum and silver oxide as catalyst. When hydrogen gas from the anode and oxygen gas from the cathode are passed the next reactions take place at the electrodes and electric current is formed.
            
      
 Theoretically it can be expected that the efficiency of like this cells may be 100% but in reality the efficiency is about 70-75%. The potential of this cell is about the volt is 1.23. Next we see the advantage of hydrogen fuel cell economy.

Advantages of hydrogen fuel cell economy
There are many advantages of the hydrogen fuel cell economy as compared to other cells. There is no air pollution due to this hydrogen fuel cell economy. It does not produce noise and its efficiency is especially high as compared to electrical production by thermal power station.

           The American scientists used this type of cell in space shuttle during Apollo space program. In addition the steam produced during cell reaction was cooled and used the available water. Recently, use of such hydrogen fuel cell economy is increasing in foreign countries.

Galvanic cell potential

Introduction :

The potential of individual half cell cannot be measured . We can only measure the difference between the two half cell potentials that gives the EMF of the cell. According to convention, a half cell called the standard hydrogen electrode represented by Pt(s)│H₂(g)│H⁺(aq), is assigned a zero potential at all temperatures corresponding to the reaction
            H⁺(aq) + e^- →1/2 H₂(g)

The standard hydrogen electrode consists of a platinum electrode coated with platinum black. The electrode is dipped In an acidic solution and pure hydrogen gas is bubbled through it. The concentration of both the reduced and oxidized forms of hydrogen is maintained at unity. This implies that the pressure of hydrogen gas is one bar and the concentration of hydrogen ion in the solution is one molar. At 298 K the emf of the cell, standard hydrogen electrode || second half cell constructed by taking standard hydrogen electrode as anode and the other half cell as cathode, gives the reduction potential of the other half cell. If the concentrations of the oxidized and reduced form of the species in the right hand half cell are unity, then the cell potential is equal to standard electrode potential, E^-_R of the given half cell.
      E^- = E^-_R – E^-_L
As E^-_L  for standard hydrogen electrode is zero.
      E^- = E^-_R – 0 = E^-_R

Galvanic Cell Potential

Measurement of Galvanic cell potential:

The measured emf of the cell:

 Pt(s) | H₂ (g, 1 bar) | H⁺(aq, 1 M)||Cu²⁺(aq,1M)| Cu

Is 0.34 V and it is also the value for the standard electrode potential of the half cell corresponding to the reaction:

  Cu²⁺ (aq, 1M) + 2e^- →Cu(s)

Similarly, the measured emf of the cell:

Pt(s)│H₂(g, 1 bar)|H⁺(aq, 1M)||Zn²⁺(aq,1M)|Zn

Is -0.76 V corresponding to the standard electrode potential of the half cell reaction:

   Zn²⁺(aq, 1M) + 2e^- →Zn(s)

In view of this convention, the half cell reaction of the daniell cell can be given as:

Left electrode: Zn(s) → Zn²⁺(aq,1M) + 2e^-

Right electrode: Cu²⁺(aq, 1M) + 2e^- →Cu(s)

The overall reaction of the cell is the sum of the above two reactions and we obtain the equation:

Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s)

Emf of the cell = E⁰_cell = E⁰_R – E⁰_L

                                        = 0.34 – (-0.76) = 1.10 V

Summary of Galvanic Cell Potential

The Emf of galvanic cell potential is 1.10 V.