PLANE STRESS AND STRAIN (1)

A REVIEW ON FORCE:

A force exerted on a body can cause a change in either the shape or the motion of the body. The unit of force is the newton*, N. No solid body is perfectly rigid and when forces are applied to it, changes in dimensions occur. Such changes are not always perceptible to the human eye since they are so small.

The three main types of mechanical force that can act on a body are:

(i) tensile

(ii) compressive and;

(iii) shear

Tensile is a force that tends to stretch a material.A tensile force, i.e. one producing tension, increases the length of the material on which it acts.

Compression is a force that tends to squeeze or crush a material.A compressive force, i.e. one producing compression, will decrease the length of the material on which it acts.

Shear is a force that tends to slide one face of the material over an adjacent face. A shear force can cause a material to bend,slide or twist.

 

Image result for tensile shear force

 

 

Advertisements

Radioactivity(1)

The Nucleus.

Strong nuclear force holds neutrons and protons together to form a nucleus

Review of Atomic Terms •

Nucleons – particles found in the nucleus of an atom – neutrons and protons

  • Atomic Number (Z) – number of protons in the nucleus
  • Mass Number (A) – sum of the number of protons and neutrons
  • Isotopes – atoms with identical atomic numbers but different mass numbers
  • Nuclide – each unique atom.

KEYNOTE: Each Isotope has its own characteristic half-life. The rate of decay for an isotope is constant. It is unaffected by pressure, Temperature, Magnetic and Electric field, Chemical Reaction…

Radius Of the Nucleus.

Image result for radius of a nucleus

  Where A is the Atomic Mass, Fm is femtometer, it stands for 10 raised to the power of -15

The diameter of a Nucleus is D = 2r

Radioactivity.

Radioactivity was discovered in 1896 by the French physicist, Henri Becquerel working in Paris. In 1896, Henri Becquerel discovered, almost by accident, that uranium can blacken a photographic plate, even in the dark.

Radioactivity is the process by which nuclei emit particles and rays These penetrating particles and rays are called radiation.

Radioactivity is a phenomenon that occurs naturally in a number of substances. Atoms of the substance spontaneously emit invisible but energetic radiations, which can penetrate materials that are opaque to visible light. The effects of these radiations can be harmful to live cells but, when used in the right way, they have a wide range of beneficial applications, particularly in medicine.

Unstable isotopes can become stable by releasing different types of particles.

  • This process is called radioactive decay and the elements which undergo this process are called radioisotopes/radionuclides.

Therefore, Radioactive decay is a process by which the nuclei of a nuclide emit α, β or γ rays. A few naturally occurring isotopes and all of the man-made isotopes are unstable.

Uranium emits very energetic radiation – it is radioactive.

 

Black Body Radiation

A black body is an idealised physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence

Blackbody radiation” or “cavity radiation” refers to an object or system which absorbs all radiation incident upon it and re-radiates energy which is characteristic of this radiating system only, not dependent upon the type of radiation which is incident on it. The radiated energy can be considered to be produced by standing wave or resonant modes of the cavity which is radiating.

A simple example of a black body radiator is the furnace. If there is a small hole in the door of the furnace heat energy can enter from the outside. Inside the furnace, this is absorbed by the inside walls. The walls are very hot and are also emitting thermal radiation.

The amount of radiation emitted in a given frequency range should be proportional to the number of modes in that range. The best of classical physics suggested that all modes had an equal chance of being produced and that the number of modes went up proportional to the square of the frequency.

Stars approximate blackbody radiators and their visible colour depends upon the temperature of the radiator. The curves show blue, white, and red stars. The white star is adjusted to 5270K so that the peak of its blackbody curve is at the peak wavelength of the sun, 550 nm. From the wavelength at the peak, the temperature can be deduced from the Wien displacement law.

Wien’s Displacement Law

When the temperature of a blackbody radiator increases, the overall radiated energy increases and the peak of the radiation curve moves to shorter wavelengths. When the maximum is evaluated from the Planck radiation formula, the product of the peak wavelength and the temperature is found to be a constant.

For a blackbody radiator, the temperature can be found from the wavelength at which the radiation curve peaks.

How does a blackbody look like?

A blackbody is usually made of two parts linked through a cable:

  • emissive head: including an emissive surface coated with a highly emissive coating or a cavity. An accurate temperature sensor is inserted into the emissive surface or cavity measuring the temperature of the source in real time. Depending on the temperature range the emissive head is also equipped with heating and/or cooling means.
  • electronic controller: which brings power to the heating/cooling meanwhile acquiring and displaying the temperature of the emissive surface in real time. The electronic controller is equipped with accurate servo control loop for high-stability temperature regulation.

Application Of Black Body Radiation

The main applications are of course IR sensors calibration and their specifications measurement.

Infrared sensors are electro-optical devices that convert thermal radiation received into an electrical signal to give a temperature measurement. These systems can be used in many applications: in the security sector, for surveillance purposes, and the civilian sector, for industrial non-contact thermal monitoring. From the basic pyrometers to more complex IR imaging systems, these devices have to be accurately tested, during the development phase, as well as during manufacturing and commissioning. They have to be calibrated prior to delivery and then periodically in the field.

Optics

When an object is dropped in still water, the circular wave fronts that are produced move out from the contact point over the two‐dimensional surface. A light source emits light uniformly in all directions of the three‐dimensional world. The wavefronts are spherical, and the direction of motion of the wave is perpendicular to the wavefront, as depicted in the figure below.  This straight line path shown by the arrow is called a ray. Depicting light as rays in ray diagrams provides a method to explain the images formed by mirrors and lenses.

Image result wey dey for waves are perpendicular to spherical wave form

Far from the source, the curvature of the wavefront is small, so the wavefront appears to be a plane. Then, the light rays will be nearly parallel. Rays from the sun are considered to be parallel when reaching the earth.

The law of reflection

Most visible objects are seen by reflected light. There are few natural sources of light, such as the sun, stars, and a flame; other sources are man‐made, such as electric lights. For an object to be visible, light from a source is reflected off the object into our eyes (except in the special case of phosphors).

Image result wey dey for Laws of reflection

NOTE: Vision is the result of light reflected from the object.

As shown above, light strikes a mirror and is reflected. The original ray is called the incident ray, and after reflection, it is called the reflected ray. The angles of the incident and reflected rays are always measured from the normal. The normal is a line perpendicular to the surface at the point where the incident ray reflects. The incident ray reflected ray, and normal all lie in the same plane perpendicular to the reflecting surface, known as the plane of incidence. The angle measured from the incoming ray to the normal is termed the incident angle. The angle measured from the outgoing ray to the normal is called the reflected angle. The law of reflection states that the angle of incidence equals the angle of reflection. This law applies to all reflecting surfaces.

TYPES OF REFLECTION

Light undergoes either diffuse or regular reflection. 

Image result wey dey for diffuse and clear reflection

Diffuse reflection occurs when light reflects from a rough surface.Regular reflection is reflected from a smooth surface, such as a mirror. The reflected rays are scattered in diffuse reflection. This scattering is because the local direction of the normal to the surface is different for the different rays. By contrast, in regular reflection, the reflected light rays are orderly because each local region of the surface has a normal in the same direction.

Thermal Expansion

THERMAL EXPANSION.

Thermal expansion is a small, but not always insignificant effect. Typical coefficients are measured in parts per million per kelvin (10-6/K).

Almost all materials expand on heating, the most famous exception being water, which contracts as it is warmed from 0 degrees Celsius to 4 degrees. when water at 00C is heated, it decreases in volume until it reaches 40C. Above 4oC water behaves normally and expands in volume as the temperature is increased. This is referred to as the anomalous behaviour of water.

This is actually a good thing, because as freezing weather sets in, the coldest water, which is about to freeze, is less dense than slightly warmer water, so rises to the top of a lake and the ice begins to form there( on the surface). For almost all other liquids, solidification on cooling begins at the bottom of the container.

Because of the unusual behaviour of water, it is hard for any large body of water to freeze completely. This is made possible, by the layer of ice on the surface, acting as an insulator to reduce the flow of heat out of the water into the cold air.

NOTE: Water expands as it freezes to ice, and this is why pipes break when the water inside them freezes.

Linear Expansivity of Materials.

The coefficient of linear expansion,α is defined as the increase in length per unit length per degree change in temperature, hence

α = ΔL/(LoΔT)

where ΔL is the change in length, Lo is the original length, and ΔT(Tf – Ti) is the change in temperature.

The change in Length,ΔL(L – L0), can be written as:

ΔL = αL0ΔT

where ΔT is the difference between the original temperature and the temperature T, while ΔL is the change in length of the material, and is also equal to how much an Object expands or contracts.

It is customary to choose the standard length Lo to be the length as measured at 0oC.

Then the above equation can be written as

L = Lo (1 +αΔT)

where L is the new Length.

Examples on linear Expansivity

1.An iron of length 50m and at a temperature of 600C is heated to 700C. Calculate its new length .[linear expansivity of iron = 1.2 x 10-5/k]

Solution

Lo=50m, Ti= 600oC, Tf= 700oC, L = ?

L = Lo (1 +αΔT) ,

Substituting the given values into the equation,

L = 50 ( 1 + 1.2 x 10-5 x ( 700 – 600))

L = 50 (1 + 1.2 x 10-5 x 100)

L = 50 ( 1 + 0.0012), 50( 1.0012)

     = 50( 1.0012)

L = 50.06m

Heat (1)

What is Heat And Temperature:

Heat is a form of energy that could alter the value of temperature. Heat refers to a quantity of energy transferred between two bodies. It is a measure of the total kinetic energy of the atoms or molecules in a body.

The units it is measured is in Joules (J) or kiloJoules (kJ). The heat content of a body will depend on its temperature, its mass, and the material it is made of.

Heat energy is always transferred from an object at high temperature to one at a lower temperature.

Temperature is not the same as heat, Temperature measures the degree of hotness of a body. It doesn’t depend on the mass or the material of an object. It can be thought of as a measure of the average kinetic energy of the atoms or molecules in a body.

As the temperature decreases, the kinetic energy of the particles will decrease. This means that as the temperature of a body decreases, heat energy of the body also decreases.

Thermal equilibrium is a state that two objects reach the same temperature by exchanging heat.

Zeroth law of thermodynamics: Zeroth law of thermodynamics states that If two objects are in thermal equilibrium with a third, then they are in thermal equilibrium with each other.

NOTE: The “third body” in a practical situation is just the thermometer.

This means that if two separate systems or objects are each in thermal equilibrium with a third system or object, then all three systems are in thermal equilibrium with each other and thus have the same temperature.

The thermometer works on the principle of Zeroth law of thermodynamics. The thermometer is, in fact, measuring its own temperature. With Zeroth law, we know that the measured temperature is also the temperature of the object(In this case the human body) that has the thermal equilibrium with the thermometer.

Image result for zeroth law of thermodynamics

If system A and system B are each in thermal equilibrium with system C, then system A and B are in thermal equilibrium with each other.

TEMPERATURE SCALES

Temperature is measured using a variety of temperature scales. The most commonly used are:

The Celsius Scale (°C). This scale puts the freezing point of water at Oo C and the boiling point

of water at 100o C. The temperatures in between are divided up into 100 units (degrees).

The disadvantages of this scale are:

• There may be temperatures below Oo C.

The pressures and volumes of gases do not change in proportion to Celsius temperature.

2. The Kelvin Scale (K). This scale has absolute zero as the zero point on it’s scale. The size of the degree is the same as a Celsius degree.

Advantages:

• There are no negative temperatures.

• Pressures and volumes of gases will change in proportion to Kelvin temperature.

Absolute zero is 273 degrees below Oo C.

3. The Fahrenheit Scale(F). Fahrenheit is a thermodynamic temperature scale, where the freezing point of water is 32 degrees Fahrenheit (℉) and the boiling point 212℉ (at standard atmospheric pressure).

NOTE: There are many different types of thermometer used for measuring temperature e.g. mercury, alcohol, bi-metallic strip, thermocouple, electrical resistance, brightness thermometer etc.

CONVERTING TEMPERATURE SCALES.

Degree Celsius to Fahrenheit.

TF = 1.8 * ToC + 32

TF is the value in Fahrenheit and ToC is the value in Celsius.

b. Degree Celsius to Kelvin.

TK = ToC + 273.15

Examples.

Convert 100oC to kelvin.

Solution.

ToC = 100oC, TK = ?

Therefore, Tk = ToC + 273.15 becomes; TK = 100 + 273.15

Thus; TK = 373.15K

2. Convert 212oF to Celsius.

TF =212oF, ToC = ?

TF = 1.8 * ToC + 32

Substituting the given values into the equation,we have;

212 = 1.8 * ToC + 32

212 – 32 = 1.8 * ToC

180 = 1.8 * ToC

ToC = 1801.8

Therefore, ToC = 100oC

Note: To convert degree fahrenheit to Kelvin, first convert to Celsius, before converting to Kelvin using the formulae above.

Simple Harmonic Motion(1)

WHAT IS SIMPLE HARMONIC MOTION (SHM)?

Simple harmonic motion is the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke’s Law. The motion is sinusoidal in time and demonstrates a single resonant frequency.

KEY POINT: *The vibration of a guitar string is an example of simple harmonic motion*

When a musician strums a guitar, the vibration of the strings creates sound waves that human ears hear as music. When a guitar string is plucked, it moves a certain distance, depending on how hard the guitar player strums. The string returns to its starting point and travels nearly the same distance in the opposite direction. The vibrational energy of the string is dissipated in the form of sound. This causes the distance the string moves, or the amplitude of the vibrations, to decrease gradually. The volume of the sound fades until the string eventually falls silent.

 

SIMPLE HARMONIC EQUATION

The motion equation for simple harmonic motion contains a complete description of the motion, and other parameters of the motion can be calculated from it.

The total energy for an undamped oscillator is the sum of its kinetic energy and potential energy, which is constant at

Simple harmonic oscillators work because the force acts in the opposite direction to the displacement. As the pendulum moves away from the area immediately below the peg it is hanging on, the force no longer acts in the opposite direction to the displacement.DISPLACEMENT, VELOCITY AND ACCELERATIONThe displacement of a simple harmonic oscillator is:

{\displaystyle x=A\cos {\omega t}}

Velocity  is the rate of change of displacement, so:

{\displaystyle v={\frac {dx}{dt}}=-A\omega \sin {\omega t}}

Acceleration is the rate of change of velocity, so:

{\displaystyle a={\frac {dv}{dt}}={\frac {d^{2}x}{dt^{2}}}=-A\omega ^{2}cos{\omega t}=-\omega ^{2}x}

TIME PERIOD 

The time period of an oscillation is the time taken to repeat the pattern of motion once. In general:

{\displaystyle T={\frac {2\pi }{\omega }}}

However, depending on the type of oscillation, the value of ω (angular velocity) changes. For a mass on a spring:

{\displaystyle \omega ={\sqrt {\frac {k}{m}}}}

For a pendulum:

{\displaystyle \omega ={\sqrt {\frac {g}{l}}}}

ANGULAR VELOCITY

Angular velocity in circular motion is the rate of change of angle. It is measured in radians per. second. Since 2π radians is equivalent to one complete rotation in time period T:

{\displaystyle \omega ={\frac {2\pi }{T}}=2\pi f}

 

 

 

 

TYPES OF RADIOACTIVE DECAY

The major types of Radioactive decay are;

1. Alpha Radiation/Decay alpha particle

During Alpha decay an atom spits out two protons and two neutrons from its nucleus. This little bundle is called an “alpha particle.”

  • Alpha decay usually happens in larger, heavier atoms.
  • The symbol looks like Helium because Helium-4 has the same number of protons and neutrons as an alpha particle (no electrons, though).
  • Since Alpha particles have two protons and no electrons, they have a net charge of 2+.
  • During Alpha radiation an atom’s proton count drops by two, and we know what that means – a NEW element is formed!
  • Alpha radiation can be stopped by PAPER.

NOTE: One of the main sources of alpha particles is the element radon, which is a gas found in many rocks. If a lot of radon is inhaled, it can get in your lungs and damage lung tissue. Some homes are even equipped with radon detectors to warn you if the levels are getting too high.

2. Beta Radiation/Decay beta particle

A neutron is a proton with an electron attached. In beta decay a neutron sends its electron packing, literally ejecting it from the nucleus at high speed. The result? That neutron turns into a proton!

  • Beta decay increases an atom’s electron count by 1 (notice the 1- in the symbol).
  • During Beta radiation an atom’s proton count grows by one. Once again, NEW element!
  • Beta radiation can be stopped by WOOD.

NOTE: Beta emission is when a high speed electron (negative charge) leaves the nucleus. Beta emission occurs in elements with more neutrons than protons, so a neutron splits into a proton and an electron. The proton stays in the nucleus and the electron is emitted.

3. Gamma Radiation/Decay gamma particle

Gamma rays is electromagnetic radiation similar to light. Gamma decay does not change the mass or charge of the atom from which it originates. Gamma is often emitted along with alpha or beta particle ejection.

  • Gamma radiation can be stopped by LEAD.

NOTE: The emission of gamma rays does not alter the number of protons or neutrons in the nucleus but instead has the effect of moving the nucleus from a higher to a lower energy state (unstable to stable). Gamma ray emission frequently follows beta decay, alpha decay, and other nuclear decay processes.

 

RADIOACTIVITY AND RADIATION

Radioactivity refers to the particles which are emitted from nuclei as a result of nuclear instability. Radioactivity, property exhibited by certain types of matter of emitting energy and subatomic particles spontaneously. It is, in essence, an attribute of individual atomic nuclei.

 

Radioactive decay, also known as nuclear decay or radioactivity, is the process by which the nucleus of an unstable atom loses energy by emitting radiation, including alpha particles, beta particles, gamma rays and conversion electrons.

Key Point: A material that spontaneously emits such radiation is considered radioactive.

A radioactive atom is one that spontaneously emits energetic particles or waves (known as radiation). This radiation is emitted when an unstable (i.e. radioactive) nucleus transforms to some other nucleus or energy level. Imagine a big ball made of magnets that’s spinning really fast. Sometimes a few pieces of the magnet will shoot out and hit the wall. That’s kind of what radiation is like. As it applies to nuclear energy, many materials created during the operation of a reactor are unstable. As they decay over varying lengths of time (from microseconds to hundreds of thousands of years), they emit energetic particles or waves. The energy carried by this radiation is often sufficient to cause damage to biological cells and is therefore a health risk. Thus, radiation is the primary cause of safety concerns related to nuclear energy.

 

SOURCES OF RADIATION

1. Smoke detectors

Smoke detectors make use of the isotope Americium-241. This isotope emits alpha-particles at energies up to 5.4 MeV. The energetic alpha particles are used to ionize air. Once the air is ionized, a small current runs through it. When smoke enters the chamber, the current experiences an increase in resistance and a circuit sounds the alarm.

2. Coal-burning power plants

Coal is an impure fuel, and it usually contains 1.3 ppm of uranium and 3.4 ppm of thorium (not to mention arsenic, mercury, and sulfur). When coal burns, these isotopes are emitted into the atmosphere, where they enter our ecosystem. This leads to the astounding fact that the population effective dose equivalent from coal plants is 100 times that from nuclear plants.

3. Nuclear weapon detonations

The hundreds of atmospheric nuclear weapons tests that occurred before they were banned by the 1963 Limited Test Ban Treaty left long-lived radioisotopes in the atmosphere. Some of these are still in the atmosphere and account for some of our daily dose.

4. Radon gas

This natural occurring gas comes from soil and is found throughout the world. It emits alpha particles, and can therefore damage DNA and lead to cancer if inhaled. The EPA recommends you check your house for radon gas.

5. Cosmic rays

Cosmic rays are energetic particles that originate outside of earth, in the sun, distant stars, galaxies, and supernovae. Most of these are protons. The atmosphere shields us from most cosmic rays, but during air travel, one will accumulate much higher dose.

Note: The first three are Man made, while the last two are Natural.

VECTORS

A vector is something that mathematicians and engineers use to represent or calculate a force that has direction. The vector will show the direction of the force and the size of the force as well. Sometimes vectors are represented by arrows, with length or thickness of the arrow representing force, and direction representing direction. Sometimes vectors are represented by sets of numbers.

Often vectors are used in spacecraft navigation. The engineers know where they want to go and use vectors to figure out what direction to thrust in to go there.

 

A Vector is a quantity possessing both magnitude and direction, represented by an arrow the direction of which indicates the direction of the quantity and the length of which is proportional to the magnitude.

Examples of vector quantitiee include displacement, velocity, acceleration, and force. Each of these quantities are unique in that a full description of the quantity demands that both a magnitude and a direction are listed. For example, suppose your teacher tells you “A bag of gold is located outside the classroom. To find it, displace yourself 20 meters.” This statement may provide yourself enough information to pique your interest; yet, there is not enough information included in the statement to find the bag of gold. The displacement required to find the bag of gold has not been fully described. On the other hand, suppose your teacher tells you “A bag of gold is located outside the classroom. To find it, displace yourself from the center of the classroom door 20 meters in a direction 30 degrees to the west of north.” This statement now provides a complete description of the displacement vector – it lists both magnitude (20 meters) and direction (30 degrees to the west of north) relative to a reference or starting position (the center of the classroom door). Vector quantities are not fully described unless both magnitude and direction are listed.

VECTOR DIAGRAM

Vector quantities are often represented by scaled vector diagram. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction.

The vector diagram depicts a displacement vector. Observe that there are several characteristics of this diagram that make it an appropriately drawn vector diagram.

  • a scale is clearly listed
  • a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a head and a tail.
  • the magnitude and direction of the vector is clearly labeled. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North).