BPSC AE Paper 4 (General Engineering Science) 2012 Official Paper (Previous Year Paper)

Explanation:

Bernoulli's principle can be derived from the principle of conservation of energy.

Bernoulli's theorem states that, in a steady flow, the sum of all forms of energy in a fluid along a streamline is the same at all points on that streamline

Bernoulli equation represented in head form (the total energy per unit weight):

$\frac{P}{\rho g} + \frac{{{v^2}}}{{2g}} + Z = Constant$

where

$p\over{\rho g}$ = pressure head

$V^2\over 2g$= velocity head

Z = datum head

Explanation:

The continuity Equation is based on the principle of conservation of mass. For a fluid flowing through a pipe at all the cross-sections, the quantity of fluid per second is constant.

The continuity equation is given as ${ρ _1}{A_1}{V_1} = ;{ρ _2}{A_2}{V_2}$

Density ρ = C for incompressible fluid.

∴ Continuity equation for an incompressible fluid A1V1 = A2V2

Additional Information

Generalized equation of continuity. 

$\frac{{\partial \left( {ρ u} \right)}}{{\partial x}} + \frac{{\partial \left( {ρ \nu } \right)}}{{\partial y}} + \frac{{\partial \left( {ρ w} \right)}}{{\partial z}} + \frac{{\partial ρ }}{{\partial t}} = 0$

This equation can be written in vector form as,

Case 1: For steady flow $\frac{{\partial ρ }}{{\partial t}} = 0$ then the above equation will become,

$\frac{{\partial \left( {ρ u} \right)}}{{\partial x}} + \frac{{\partial \left( {ρ \nu } \right)}}{{\partial y}} + \frac{{\partial \left( {ρ w} \right)}}{{\partial z}} = 0$

Case 2: For Incompressible flow, ρ is constant, therefore the continuity equation of steady incompressible for three-dimensional flow is,

$\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0$

$\nabla .\vec V = 0$

For two dimensional flow

$\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0$

Explanation:

Kirchhoff ’s law:

• According to Kirchhoff’s law, the ratio of total emissive power to absorptivity is constant for all bodies which are in thermal equilibrium with the surroundings. Therefore, it means that the emissivity of a body is equal to its absorptivity.
• Let E be the total emissive power of the body and α be the absorptivity of the body.

Emissivity (ϵ) is the ratio of the emissive power of a non-black body to the black body.

ϵ = $\frac{E}{E_b}$

Kirchhoff’s law also holds for monochromatic radiation, for which

\(\frac{{{E_{\lambda 1}}}}{{{\alpha {\lambda 1}}}} = \frac{{{E{\lambda 2}}}}{{{\alpha {\lambda 2}}}} = \frac{{{E{b\lambda }}}}{{{\alpha {b\lambda }}}} = \frac{{{E{b\lambda }}}}{1}\)

∵ Absorptivity of a black body is one.

\(\therefore \frac{{{E_\lambda }}}{{{\alpha \lambda }}} = {E{b\lambda }} \Rightarrow {\alpha \lambda } = \frac{{{E\lambda }}}{{{E_{b\lambda }}}} = {\epsilon_\lambda }\)

Therefore, the monochromatic emissivity of a black body is equal to the monochromatic absorptivity at the same wavelength.

Eλ = αλEbλ

Important Points

Stefan-Boltzmann Law: The thermal energy radiated by a black body per second per unit area is proportional to the fourth power of the absolute temperature and is given by:

E ∝ T4

E = σT4

where σ = The Stefan – Boltzmann constant = 5.67 × 10-8 W m-2K-4

Wien’s displacement law: For a black body emissive spectrum, the wavelength λmax giving the maximum emissive power at a temperature is inversely proportional to the absolute temperature.

λmax = C/T

λmaxT= C

Wavelength = Speed of Light/Frequency

$\lambda~=~\frac{c}{v}$

Explanation:

In many real processes, it is found that the states during an expansion or compression can be described approximately by a relation of the form Pvn = constant,

where n is a constant called index of compression or expansion, P and v are the average value of pressure and specific volume for the system.

Compressions and expansions of the form Pvn = constant are called reversible polytropic process.

For the reversible polytropic process, single values of P and v can truly define the state of a system, dW = -Pdv.

The equation for the polytropic process: 

$P{v^n} = C \Rightarrow \frac{P}{{{\rho ^n}}} = C \Rightarrow \frac{{{P_1}}}{{{P_2}}} = {\left( {\frac{{{\rho _1}}}{{{\rho _2}}}} \right)^n}$

The general polytropic process is shown in the figure:

Additional Information

Explanation:

• Around 1850 Rudolf Clausius and William Thomson (Kelvin) developed the first law of thermodynamics, which states that the "total energy of an isolated system is constant".
• In 1848, Kelvin proposed an absolute temperature scale, now called " the Kelvin scale ".
• After further research, Kelvin formulated the second law of thermodynamics. This states that heat will not flow from a colder to a hotter body.
• Sadi Carnot is a french scientist who described the Carnot cycle, relating to the theory of heat engines.
• James Prescott Joule was an English physicist who established that the various forms of energy—mechanical, electrical, and heat—are basically the same and can be changed one into another. He forms the basis of the **law of conservation of energy.**​

Mistake Points

We have checked the Book PK Nag it says much work of joule is associated with the First Law of Thermodynamics not That the First Law of Thermodynamics was developed by Joule. This is the official question of UPRVUNL JE ME 25 Oct 2021 Official Paper (Shift 1) and UPRVUNL has considered the marked option as the correct answer.

Explanation:

Otto cycle:

The air-standard-Otto cycle is the idealized cycle for the spark-ignition internal combustion engines.

Otto cycle is the one which has two constant volume heat transfer processes and two adiabatic work transfer processes.

The Otto cycle 1-2-3-4 consists of the following four processes:

• Process 1-2: Reversible adiabatic compression of air

• Process 2-3: Heat addition at constant volume

• Process 3-4: Reversible adiabatic expansion of air

• Process 4-1: Heat rejection at constant volume

• During the constant volume process heat addition and heat rejection take place and no work transfer, because of this it is also known as a constant volume cycle.

• During the adiabatic processes [compressions/expansion] only work transfer taken place but no heat transfer occurs.

Important Points

Explanation:

Rankine cycle:

Rankine cycle is an idealised cycle for producing steam.

It is a reversible cycle which has two constant pressure and two isentropic processes.

There are four processes in the Rankine cycle:

Process 1–2: Isentropic compression

Working fluid is pumped from low to high pressure using a pump.

Process 2–3: Isobaric heat addition

The high-pressure liquid enters a boiler where it is heated at constant pressure by an external heat source to become a dry saturated or superheated vapour.

Process 3–4: Isentropic expansion

The dry saturated vapour expands through a turbine, generating power.

Process 4–1: Isobaric heat rejection

The wet vapour then enters a condenser where it is condensed at constant pressure and temperature to become a saturated liquid.

Explanation:

Centrifugal or radial pumps are simple and versatile constructions with a wide range of impeller design. The motion of water in it is from the centre towards the periphery. The motion is caused by the centrifugal force created in the pump as a result of the revolving motion of the working wheel.

• Water enters at the centre of the impeller. The water passes between the vanes and is pushed radically towards the casing and then onwards through the discharge nozzle. The vanes create an increase in both water velocity and pressure.
• The housing that encloses the rotating element and seals the pressurized liquid inside commonly called as Casing. The casing is of spiral shape and it terminates in a delivery pipe.
• Centrifugal pump acts as a reverse of an inward radial flow reaction turbine. This means that the flow in centrifugal pumps is in the radial outward directions. The flow enters the chamber along the axis of the chamber and is discharged radially.

Explanation:

Impulse turbine: 

• If the energy available at the inlet of the turbine is only kinetic energy, the turbine so known as impulse turbine.
• The available energy at the inlet is only kinetic energy if the inlet pressure and outlet pressure is the same equal to atmospheric pressure.
• Example: Pelton Turbine

Reaction Turbine: 

• At the inlet of the turbine, the water possesses kinetic energy as well as pressure energy
• Example: Francis Turbine, Kaplan Turbine

Short circuits in transformer windings and connections:

• These are high level faults and are likely to cause immediate damage.
• These short circuit faults occur due to the impulse voltage by lighting.
• These faults are detectable at the winding terminals due to the unbalances produced in the current and voltage.
• These types of faults includes line to ground faults or inter-turn faults at the HV or LV windings.

The correct answer is 'option 4'

Solution:

The overhead lines are open to the atmosphere and also pass through different environments viz. forests while other equipments are kept in substations where appropriate protection can be provided.

Additional Information There are two types of faults:

Open Circuit Faults:

The open-circuit fault happens due to the failure of one or two conductors. These faults take place in series with the line so referred to as a series fault. Such types of faults have a strong impact on the reliability of the system. The open-circuit fault is classified as:

1. Open Conductor Fault
2. Two conductors Open Fault
3. Three conductors Open Fault

Short Circuit Faults:

The short-circuit fault is commonly divided into symmetrical and asymmetrical types. These faults are further categorized as one of five types. In order of frequency of occurrence, they are:

Asymmetrical Faults:

Asymmetrical faults mandate the calculation of positive negative and zero sequence components separately.

• Single Line to Ground Fault: This type of fault occurs when you have one of the phases (A, B or C) is shorted with the ground.
• Line to Line Fault: This type of fault occurs when you have one of the phases (A, B or C) is shorted with the ground.
• Double Line to Ground Fault: This type of fault occurs when two phases are shorted with the ground together (A-B-G, B-C-G or C-A-G)

Symmetrical Faults: 

Symmetrical faults do not give rise to zero sequence or negative sequence components because they are perfectly balanced, symmetrical faults only have positive sequence values.

• Three Phase Line to Ground Fault: The 3-phase to ground faults are faults in which all the phases (A, B, and C) are shorted together and are grounded.
• Three Phase Line to Line Fault: The three-phase faults occur when you have A, B, and C phases are shorted together but the ground is not involved.

Concept:

The power rating of a resistor: The power rating of a resistor is the specification given with a resistor that serves to tell the maximum amount of power that the resistor can withstand. At this value, the current is maximum through the resistor.

\({{P}{rating}}=I{rated}^{2}R\)

Calculation:

Given, R = 2 Ω, I = 1 A

Prated = 1² × 2 = 2 W

Unilateral Circuits:

• Allows the current flow in one direction only
• Circuit properties changes by changing the direction of current or voltage
• Diode, transistors are the examples

Bilateral Circuits:

• Allows the current flow in both the directions
• Circuit properties will remain the same even by changing the direction of current or voltage
• Resistance, inductance and capacitance are the examples

Irreversible circuit:

An irreversible circuit is a circuit whose inputs cannot be reconstructed from its outputs.

Reversible circuit:

An reversible circuit is a circuit whose inputs can be reconstructed from its outputs.

Series connection:

• Increases output voltage.
• Capacity remains the same (measured in ampere-hours).

Parallel Connection:

In order to increase the ampere-hour rating of a battery, cells are connected in parallel.

This is explained with the help of the following diagram:

• The positive terminals of all batteries are connected together, or to a common conductor, and all negative terminals are connected in the same manner.
• The final voltage remains unchanged while the capacity of the assembly is the sum of the capacities of the individual batteries of this connection.

CONCEPT:

Elements of Earth's Magnetic Field:

• A compass is a device that uses a small, magnetized needle that aligns itself with the Earth's magnetic field.
• The needle points towards the Magnetic North Pole, which is the point where the Earth's magnetic field lines converge in the Northern Hemisphere.

EXPLANATION:

Earth has its own magnetic field whose poles are

• Magnetic North is in Geographical South pole.
• Magnetic South is in Geographical North Pole.
• A compass needle is a bar magnet with a pointed north pole. It is attracted to the magnetic south pole.
• Hence magnetic needle points in North Direction (Magnetic South). Therefore option 2 is correct.

• According to Planck's quantum theory : The radiant energy emitted or absorbed by a body not continuously but discontinuously in the form of small discrete packets of energy.
• These packets are called photons, also known as the quantum.
• The energy of a photon is given by:

E = h × f

where E = Photon energy

h = Planck's constant

f = Frequency

• The value of Planck's constant is 6.626 × 10^-34 J/Hz.

The correct answer is 0.7457 kW.

• Power is defined as the rate of doing work, it is the work done in unit time.
• SI unit of power is: watt (W)
• Another unit of power is horse-power (hp)
• 1 hp = 745.7 W
• 1 hp = 0.7457 kW
• Power is a scalar quantity.

Additional Information

• The term horsepower was adopted by Scottish inventor James Watt to measure the power output of his steam engine and to compare that power with that of horses.
• One watt is the power equal to the rate of energy of one joule per second in an electrical circuit.

Concept:

Force is a vector quantity.

If any two forces \(\vec A;& ;\vec B\) acts at an angle θ , then the resultant is given by $\vec R$

The magnitude of the resultant is given by-

$R = \sqrt {{A^2} + {B^2} + 2ABcos;\theta }$

The direction is given by α with respect to vector $\vec A$i.e.

 $tan\alpha = \frac{{Bsin;\theta }}{{A + Bcos;\theta }}$

Calculations:

Given:

A = B = P

θ = 90°

∴  $R = \sqrt {{P^2} + {P^2} + 2 \times P \times P \times \cos 90}$

⇒ $R = \sqrt {2}P$

$tan~\alpha = \frac{{P \times \sin 90}}{{P + (P \times \cos 90)}}$

⇒ tan α = 1

⇒ α = tan-1(1)

⇒ α = 45°

∴ To neutralize the resultant, the third force should act along the bisector in the opposite direction with magnitude $R = \sqrt {2P}$.

Explanation:

EQUILIBRIUM CONDITIONS FOR BODIES UNDER COPLANAR NON CONCURRENT FORCES

When a body is under the action of a coplanar non-concurrent force system it may rotate due to the resultant moment of the force system or it may set in a horizontal or vertical motion due to horizontal and vertical components of forces. The body thus can only be in equilibrium if the algebraic sum of all the external forces and their moments about any point in their plane is zero.

Mathematically, the conditions of equilibrium may be expressed as follows:

1. ΣH = 0 (ΣH means the sum of all the horizontal forces)
2. ΣV = 0 (ΣV means the sum of all the vertical forces)
3. ΣM = 0 (ΣM means the sum of all the moments)

When coplanar forces meet at a point the system is known as a coplanar concurrent force system. This system will be in equilibrium if it satisfies the conditions of equilibrium viz. ΣH = 0 and ΣV = 0.

When coplanar forces do not meet at a point the system is known as a coplanar nonconcurrent force system. This system will be in equilibrium if it satisfies all the three conditions of equilibrium viz. ΣH = 0, ΣV = 0, ΣM = 0.

The conditions ΣH = 0 and ΣV = 0 ensure that the system does not reduce to a single force and condition ΣM = 0 ensures that it does not reduce to a couple.

In the case of a coplanar non-concurrent force system, M may be equal to zero but the system may not still be called in equilibrium because the point where the moments are taken about may be lying on the line of action of the resultant. Hence, in this case, all the three conditions of equilibrium have to be fulfilled.

CONCEPT:

Law of Parallelogram of forces:

This law is used to determine the resultant of two coplanar forces acting at a point.

It states that “If two forces acting at a point are represented in magnitude and direction by two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes through that common point.”

Let two forces F1 and F2, acting at the point O be represented, in magnitude and direction, by the directed line OA and OB inclined at an angle θ with each other.

Then if the parallelogram OACB be completed, the resultant force R will be represented by the diagonal OC.

$R = \sqrt {F_1^2 + F_2^2 + 2{F_1}{F_2}cosθ }$

Calculation:

Given:

The resultant of two forces (P + Q) and (P - Q) is equal to $\sqrt {3{P^2} + {Q^2}}$

Let (P + Q) = A and (P - Q) = B

∵ Resultant, R = \(\sqrt{A^2~+B^2+~2AB\cos θ}\)

And R = $\sqrt {3{P^2} + {Q^2}}$

⇒ R² = 3P² + Q²

⇒ 3P2 + Q² = (P + Q)² + (P - Q)² + 2 × (P + Q) × (P - Q) × cosθ

⇒ 3P2 + Q2 = P² + Q² + 2PQ + P² + Q² - 2PQ + 2 × (P² - Q²) × Cos θ

⇒ 3P2 + Q2 = 2P² + 2Q² + 2 × (P² - Q²) × Cos θ

⇒ P² - Q² = 2 × (P² - Q²) × Cosθ

⇒ 2 × Cosθ = 1

⇒ Cosθ = $\frac{1}{2}$

⇒ θ = 60°

Explanation:

Earth has the maximum speed in the equatorial region. To take advantage the launch station should be in the equatorial region.

Satellite:

• A satellite is any body revolving around a large body under the influence of later. An artificial satellite or communication satellite of earth is one which appears stationary to an observer on the earth's surface.
• An artificial satellite is also known as a Geostationary satellite or geosynchronous satellite.

For geostationary satellites, the following conditions are necessary;

• The orbit of the satellite must be circular and in the equatorial plane of the earth.
• The period of rotation of the satellite must be equal to the period of rotation of the earth about its axis.
• The angular velocity of the satellite must be in the same direction as the angular velocity of the rotation of the earth.

CONCEPT:

• Equation of motion: The mathematical equations used to find the final velocity, displacements, time, etc of a moving object without considering force acting on it are called equations of motion.
• These equations are only valid when the acceleration of the body is constant and they move on a straight line.
• There are three equations of motion:

V = u + at

V2 = u2 + 2 a S

${\text{S}} = {\text{ut}} + \frac{1}{2}{\text{a}}{{\text{t}}^2}$

Where, V = final velocity, u = initial velocity, s = distance travelled by the body under motion, a = acceleration of body under motion, and t = time taken by the body under motion.

Calculation:

Given:

A car moving with a speed u

The minimum distance over which the car can be stopped = x

Let a be the acceleration.

V2 = u2 + 2aS

⇒ \(a~=~\frac{v^2 ~-u^2}{2×~S}\)

∵ Final velocity, v = 0 and S = x

So, $a~=~\frac{-u^2}{2x}$

Now, initial velocity = n × u

Then the minimum distance over which the car can be stopped is:

d = \(\frac{v^2 -(nu)^2}{2~×a}=\frac{-n^2u^2}{2\times~\frac{(u^2)}{2x}}\)

∴ d = n²x

The minimum distance over which the car can be stopped = n2x

CONCEPT:

• Equation of motion: The mathematical equations used to find the final velocity, displacements, time, etc of a moving object without considering force acting on it are called equations of motion.
• These equations are only valid when the acceleration of the body is constant and they move on a straight line.
• There are three equations of motion:

V = u + at

V2 = u2 + 2 a S

${\text{S}} = {\text{ut}} + \frac{1}{2}{\text{a}}{{\text{t}}^2}$

Where, V = final velocity, u = initial velocity, s = distance traveled by the body under motion, a = acceleration of body under motion, and t = time taken by the body under motion.

Calculation:

Given:

A body fall from the top of a tower through half the height in 2 sec.

${\text{S}} = {\text{ut}} + \frac{1}{2}{\text{a}}{{\text{t}}^2}$

Here S is the distance travelled = $\frac{h}{2}$

Initial velocity, u = 0

Time taken to cover the distance $\frac{h}{2}$ , t = 2 seconds

So, $\frac{h}{2}$ = \(0~+\frac{1}{2}\timesg\times~t^2\)

⇒ $\frac{h}{2}$ = \(​​0~+\frac{1}{2}\times9.81\times~2^2\)

⇒ h = 39.24 m

Again using, ${\text{S}} = {\text{ut}} + \frac{1}{2}{\text{a}}{{\text{t}}^2}$

h = 39.24 = \(0~+\frac{1}{2}\times9.81\times~t^2\)

⇒ t² = 8

⇒ t = 2.828 seconds

The total time taken by the body to reach the ground is 2.828 seconds.

• The errors of instruments can be determined by a process known as calibration.
• Calibration is the comparison of measurement values delivered by a device under test with those of a calibration standard of known accuracy.
• If the difference between the measured value and the pre-defined calibrated value are outside the acceptable range, then the instrument is said to be producing error in its measurement.

Additional Information

• Slip gauges or gauge blocks are used as standards for precision length measurement.
• A optical projector is an optical device which is used to enlarge the image.

• Piezoelectric substance generates a voltage when they are subjected to mechanical forces or stress along specific planes.
• Thermoelectric materials are those materials that use temperature differences to generate electrical energy. They can therefore provide fully electric heating and cooling technology without any moving parts.
• Magnetoelectric materials are those whose magnetism can be affected by an external electric field or whose electric polarization is affected by a magnetic field.
• The materials which produce electrical energy when illuminated with light are known as photoelectric material.

Photoconductive cell:

• A photoelectric device in which the resistance of the metal is directly proportional to the light striking is known as a photo-conductive cell.
• Photoelectric transducers are used for the measurement of those parameters, which can be used to produce variation in light intensity.
• The photoconductive effect is a process in which the conductivity of a semiconductor material changes according to the wavelength and intensity of the radiation.

Photocell: 

• A photo cell is made up of a resistor that produces electrical energy depending upon the amount of light incident on photosensitive plates.
• As more light hits the plates, the resistance changes, turning the light on and off.

Photovoltaic cell:

• A photovoltaic cell, also known as a solar cell, is an electronic component that generates electricity when exposed to photons, or particles of light.

Photoemission cell:

• A photoemissive cell, commonly known as a phototube, makes use of the photoelectric effect, the phenomenon whereby light-sensitive surfaces give off electrons when struck by light.
• It consists of a photosensitive cathode(negative) from which electrons are emitted when hit by light. It also contains an anode(positive) that attracts these free electrons.

Concept:

The surface roughness on a drawing is represented by triangles.

Surface texture or roughness representation:

• The basic symbol consists of two legs of unequal length inclined at approximately 60° to the line representing the considered surface.

• The symbol must be represented by a thin line.

• The value of roughness is added to the symbols.

1. Roughness ‘a’ obtained by any production process
2. Roughness ‘a’ obtained by machining
3. Roughness ‘a’ obtained without removal of material

If it is necessary to impose maximum and minimum limits of surface roughness, both values are indicated. a₁ = Maximum limit; a₂ = Minimum limit

Accuracy: It is the degree of closeness with which the reading approaches the true value of the quantity to be measured.

Precision: It is the measure of reproducibility i.e., given a fixed value of a quantity, precision is a measure of the degree of agreement within a group of measurements.

• The precision of an instrument does not guarantee accuracy
• An instrument with more significant figures has more precision

Important Points

Absolute Error: The deviation of the measured value from the true value (or) actual value is called error.

Absolute error (E) = Am – At

Am = Measured value

At = True value

Resolution: The smallest change in output to the change in input is known as resolution. Resolution is the smallest measurable input change.

Sensitivity: It is defined as the ratio of the changes in the output of an instrument to a change in the value of the quantity being measured. It denotes the smallest change in the measured variable to which the instrument responds.

Deflection factor or inverse sensitivity is the reciprocal of sensitivity.

The vernier bevel protractor is a precision instrument meant for measuring angles to an accuracy of 5 minutes (5').

Parts of a vernier bevel protractor

• Stock:
• This is one of the contacting surfaces during the measurement of an angle
• Preferably it should be kept in contact with the datum surface from which the angle is measured
• It is normally used as a reference base for measuring an angle
• Dial:
• The dial is an integrated part of the stock
• It is circular in shape, and the edge is graduated in degrees
• Blade:
• This is the other surface of the instrument that contacts the work during the measurement
• It is fixed to the dial with the help of the clamping lever
• A parallel groove is provided in the centre of the blade to enable it to be longitudinally positioned whenever necessary
• Locking screws:
• Two knurled locking screws are provided, one to lock the dial to the disc, and the other to lock the blade to the dial

Explanation:

Hermaphrodite caliper:

• A calliper is a measuring instrument with two adjustable legs or jaws used to measure the dimensions of material parts.
• A hermaphrodite calliper is a tool for laying out lines parallel to the edges of a workpiece. It can also be used to determine the centre of cylindrical workplaces.
• A hermaphrodite calliper has one bent leg inward and one straight leg that ends in a sharp point; this type of caliper is used for scribing lines at a specific distance from a flat or curved surface.
• Hermaphrodite calliper is also known as Odd leg calliper.

Sine bar:

A sine centre provides a convenient means of measuring angles of conical workpieces that are held between centres.

The procedure for setting the sine centres is similar to a sine bar.

For checking the included angle of the taper, the sine block is set to half the included angle of the gauge. The half included angle of the workpiece can be calculated as:

$\sin \theta = \frac{h}{L}\Rightarrow \theta = \sin ^{-1}( \frac{h}{L})$

The total included angle of the workpiece = 2θ 

∴ Sine centre is used for measurement of semi-angle of taper of a job

Active transducers:

• The thermocouple is not a source of power but it acts as an active transducer.
• While Photovoltaic cell, Solar cell, and Photoelectric cell acts as a source of power and also as an active transducer.
• Active transducers are those which do not require any power source for their operation.
• They work on the energy conversion principle.
• They produce an electrical signal proportional to the input (physical quantity).
• Piezoelectric, thermocouple, and photovoltaic cell transducers all are active transducers.

Transducers can be classified into the following types:

• Active or Passive Transducers
• Analog or Digital Transducers

Explanation:

Malleability:

Malleability is the property by virtue of which a material may be hammered or rolled into thin sheets without rupture. This property generally increases with the increase of temperature.

Malleability is the ability of a metal to exhibit large deformation or plastic response when being subjected to compressive force.

Lead, soft steel, wrought iron, copper and aluminium are some materials in order of diminishing malleability.

Ductility:

Ductility is the property of the material that enables it to be drawn out or elongated to an appreciable extent before rupture occurs.

The percentage elongation or percentage reduction in the area before rupture of a test specimen is the measure of ductility. Normally if percentage elongation exceeds 15% the material is ductile and if it is less than 5% the material is brittle.

Lead, copper, aluminium, mild steel are typical ductile materials.

Plasticity:

This is the ability of a material to deform without any rupture by a non-returnable way. After removing the load there are staying permanent deformations.

Concept:

Simply supported beam of length (L) loaded with uniform loading intensity (w = W/L):

Bending Moment (Mx) at a section x-x is given by:

When plotted against x, bending moment gives rise to a parabolic curve and the shear force and bending moment can be drawn in the following way will appear as follows:

The maximum bending moment for a simply supported beam with a uniformly distributed load  

The bending moment at the mid-span $\frac{{W{l^2}}}{8}$

Concept:

Maximum Bending moment will occur at a fixed end. Mmax = -WL

A cantilever beam subjected to point load on free end will have a maximum bending moment at the fixed end and constant shear force throughout the length.

So maximum stress will be at the fixed end (σ=My/I) and failure will occur at that point.

∴ A beam is loaded as a cantilever. If the load at the end is increased, the failure will occur at the support.

Explanation:

• The strain is defined as the ratio of change in dimension to the original dimension.
• If ‘δl’ changes in the length and ‘l’ is the original length, then strain = δl/l.

Hence the unit of Strain is mm/mm.

Therefore Strain has no unit. Hence it is a dimensionless term.

Important Points

• The ratio of change in length to the original length is called the longitudinal strain.
• The ratio of change in diameter/breadth to the original diameter/breadth is called the lateral strain.
• The ratio of lateral strain to longitudinal strain is called Poisson's ratio.
• The volumetric strain of a deformed body is defined as the ratio of the change in volume of the body to the deformation to its original volume.
• Shear strain is defined as the strain accompanying a shearing action. It is the angle in radian measure through which the body gets distorted when subjected to external shearing action.

Explanation:

The design value of the effective length factor for various combinations is given below:

∴ The equivalent length of a column of length L having one fixed free and the other free is 2L.

Concept:

Torsion Equation of a shaft is given by,

$\frac{T}{J} = \frac{τ }{r} = \frac{{Gθ }}{L}$

Where, T = Torque, J = Polar moment of inertia, fs or τ = Shear stress, r = Radius of shaft, G = Shear modulus, θ = Angle of twist and L = Length of shaft

Polar moment of inertia for a circular shaft of diameter d is $\frac{{\pi {d^4}}}{{32}}$

Polar modulus is defined as the ratio of the polar moment of inertia to the radius of the shaft. It is also called as torsional section modulus. It is denoted by Zp.

Polar section modulus of shaft is given by,

${Z_p} = \frac{J}{r} = \frac{{\frac{{\pi {D^4}}}{{32}}}}{{\frac{D}{2}}} = \frac{{\pi {D^3}}}{{16}}$

The maximum shear stress developed on the surface of the shaft due to twisting moment T:

$τ_{max} = \frac{{T}}{{Z_p}} = \frac{{16T}}{{\pi {D^3}}}$

∴ τmax ∝ 1/D3

Maximum and minimum values of normal stresses occur on planes of zero shearing stress. The maximum and minimum normal stresses are called the principal stresses, and the planes on which they act are called the principal plane.

${\sigma _{max,;min}} = \frac{{{\sigma _{xx}} + {\sigma _{yy}}}}{2} \pm \sqrt {{{\left( {\frac{{{\sigma _{xx}} - {\sigma _{yy}}}}{2}} \right)}^2} + \tau _{xy}^2}$

But the maximum shear stress planes may or may not contain normal stresses as the case may be.

${\tau _{max}} = \frac{{{\sigma _{max}} - {\sigma _{min}}}}{2} = \sqrt {{{\left( {\frac{{{\sigma _{xx}} - {\sigma _{yy}}}}{2}} \right)}^2} + \tau _{xy}^2}$

Explanation:

$\frac{{dM}}{{dx}} = F = 0$

This implies that the slope of the bending moment distribution is equal to the shear force at any position along the beam. It also means that mathematical maximum and minimum values of bending moment, which are located where the slope of the bending moment distribution is zero, occur at points where the shear force is zero.

In the case of the simply supported beam, the bending moment will be zero at supports. And it will be maximum (positive or negative) where shear force is zero.

All metals used in engineering work can be classified into two categories.

• Ferrous material: Ferrous metals are those in which chief constituent is iron, although other constituent like carbon, manganese, phosphorus and sulphur etc. Example steel and cast iron
• Non-ferrous metals are those which do not contain iron. Metals like copper, aluminium, lead, tin etc and their alloys.

Steel is manufactured from iron ore. In the blast furnace, pig iron is produced by reducing iron ore. Thus, raw material for iron and steel product is pig iron.

Explanation:

Cast iron

• Cast iron is an alloy of iron, carbon and silicon and it is hard and brittle.
• Carbon content in CI may be within 2% to 4% and carbon may be present as free carbon or iron carbide Fe3C.
• In general, the types of cast iron are (a) grey cast iron and (b) white cast iron (c) malleable cast iron etc

Additional Information

Wrought iro n

• Wrought iron is a very pure iron where the iron content is of the order of 99.5%.
• It usually contains less than 0.1% carbon and 1 or 2% slag.
• It is produced by re-melting pig iron and some small amount of silicon, sulphur, or phosphorus may be present.
• It is tough, malleable and ductile and can easily be forged or welded.
• It cannot however take sudden shock. Chains, crane hooks, railway couplings and such other components may be made of this iron.

Pig iron 

• Pig iron is an intermediate product of the iron industry.
• Pig iron has a very high carbon content, typically 3.8–4.7%, along with silica and other constituents of dross, which makes it very brittle, and not useful directly as a material except for limited applications.

Steel 

• Steel is basically an alloy of iron and carbon in which the carbon content can be less than 1.7% and carbon is present in the form of iron carbide to impart hardness and strength.
• Two main categories of steel are (a) Plain carbon steel and (b) alloy steel.
• Thus, wrought iron is an iron alloy with a very low carbon (less than 0.08%) content in contrast to cast iron.

Explanation:

Cast iron is a ferrous metal, an alloy of iron with 2 to 4.5% carbon and 3.5% silicon. 

• Compared to tensile strength cast iron is strong in compressive strength.
• This is because cast iron is having 2 to 4% carbon in it which makes it brittle in nature and all brittle materials are comparatively stronger in compression than in tension.

Important Points

Ductile materials are stronger in tension and brittle material are stronger in compression.

Additional Information

Properties of Cast iron

• Cast iron is a hard and brittle material
• It has excellent vibration damping property
• Cast iron has a good compressive strength
• Graphite is present in cast iron in free form and it adds self-lubricating properties to cast iron
• The corrosion resistance of cast iron is improved via the addition of minor elements such as silicon, nickel, chromium, molybdenum, and copper

Explanation:

Chromium is one of the important metal which is used to improve the following characteristics of a material.

• To increase the abrasion resistance
• To provide the better hardness
• To improve the corrosion and the oxidation resistance
• To improve the temperature resistance
• With high C resists wear and abrasion
• Manganese is used to improve the hardness and strength of the material. It also acts as a desulfurizing agent.
• Vanadium is used to refine grain size. Steels containing vanadium have a much finer grain structure than steels of similar compositions without vanadium. Vanadium is used to improve the mechanical properties of the materials such as hardness and toughness.
• Cobalt improves strength at high temperatures and magnetic permeability. It contributes to red hardness by hardening ferrite. It is an alloying element is certain high-speed steels. Cobalt is used in martensitic steels, where it increases hardness and tempering resistance, especially at higher temperatures.

Explanation: 

• Bronze is the mixture of the metals tin and copper.
• The bronze alloy was made as early as the Harappan age.
• An alloy is considered a homogeneous mixture because, by definition, a homogeneous mixture has the same uniform appearance and composition throughout.
• Since an alloy is a mixture of two or more metals or a mixture of a metal with some other element, and these mixtures are uniform throughout, they fit the definition of being homogeneous.

Explanation:

Thermoplastics:

• Thermoplastics are defined as polymers that can be melted and recast almost indefinitely.
• They are molten when heated and harden upon cooling.
• When frozen, however, thermoplastic becomes glass-like and subject to fracture.
• These characteristics, which lend the material its name, are reversible, so the material can be reheated, reshaped, and frozen repeatedly. As a result, thermoplastics are mechanically recyclable.
• Some of the most common types of thermoplastic are polypropylene, polyethene, polyvinylchloride, polystyrene, Polyethylenetheraphthalate (PET), Teflon, Nylon, and polycarbonate.

Additional Information

Thermosetting plastic.

• The plastic which keeps its shape and does not soften i.e the plastic that when moulded once, can not be softened by heating is called Thermosetting plastic.
• Thermosetting plastics are synthesized by condensation polymerization and have primary bonds between molecular chains and held together by strong cross-links.
• Thermosetting plastics have a high melting point and tensile strength as compared to thermoplastics.
• one of the most common types of Thermosetting  are Bakelite, Melamine, Polyesters etc.

Concept:

• Nichrome is an alloy of nickel, chromium, manganese, and a little percentage of iron.
• It is used for making heating elements of electrical appliances such as toasters, water heaters, etc.
• It is used to make the coils used in heaters.
• It has high resistivity and melting point.
• It does not undergo oxidation or burns easily even at high temperatures.

Nichrome wire is used as a heating element because-

• it is very stable at high temperatures and unlike other metals, it will not melt when generating a large amount of heat.
• It has high resistivity and high resistance which can produce a lot of heat on passing the current.
• It does not oxidize or burn easily at high temperatures and can be kept red-hot without burning or breaking in the air.

Additional Information

• An alloy is a substance made by combining two or more metallic elements, especially to give greater strength or resistance to corrosion.

Explanation:

• Break-even analysis is a method used to determine the sales volume required for a company to “break even”, or experience neither a profit nor a loss on the sale of its product.
• The break – even represents the number of units that must be made and sold for income from sales to equal the cost of producing the product.
• A breakeven analysis is used to determine how much sales volume your business needs to start making a profit, based on your fixed costs, variable costs, and selling price.
• Break-even point usually means the business volume that balances total costs with total gains. At break-even volume, in other words, total cash inflows equal total cash outflows. At Break-Even, in other words, net cash flow equals zero.

Break Even Point:

• It is the volume of production where total cost equal to total sale and an organisation neither earn profit nor suffer from loss
• It is also known as No Profit – No Loss point
  

${\rm{Break;Even;Point}} = \frac{{{\rm{Total;Fixed;cost}}}}{{{\rm{Selling;cost;per;unit}} - {\rm{Variable;cost;per;unit}}}}$

${\rm{BEP}} = \frac{{\rm{F}}}{{{\rm{s}} - {\rm{v}}}}$

• When sales revenue > total cost ⇒ There will be profit
• When sales revenue < total cost ⇒ There will be loss

Explanation:

A project may be defined as a combination of interrelated activities which must be executed in a certain order before the entire task can be completed.

The aim of planning is to develop a sequence of activities of the project so that the project completion time and cost are properly balanced.

To meet the objective of systematic planning, the management has evolved several techniques applying network strategy.

PERT (Programme Evaluation and Review Technique) and CPM (Critical Path Method) are network techniques which have been widely used for planning, scheduling and controlling the large and complex projects.

• PERT (Project Evaluation and Review Technique) approach takes account of the uncertainties. In this approach, 3-time values are associated which each activity. So it is probabilistic.

• CPM (Critical Path Method) involves the critical path which is the largest path in the network from starting to ending event and defines the minimum time required to complete the project.So it is deterministic.

Difference between PERT and CPM (Critical Path Method)

Concept:

PERT stands for "Project Evaluation and Review Technique". This network model is used for project scheduling.

Difference between PERT and CPM (Critical Path Method)

Explanation:

Project Appraisal:

• Project appraisal is a process of exploration, review and evaluation taken on by the decision maker as the alternative options for development are defined within the project planning process.
• It can also be expressed in terms of a number of mathematical techniques that simplify the comparison of project options on the basis of an agreed criterion or set of criteria.
• These techniques provide a rational and significant approach to evaluating diverse aspects of different alternatives and the ability of these alternatives to achieve a set objective.
• The primary objective is to aid in the process of making informed and rational choices regarding the most effective use of available scarce resources.
• Students of engineering must develop an awareness of the relevant economic, financial, social and political factors of significance to engineering development projects, along with an understanding of risk analysis and quality systems.
• This knowledge is a vital building block in an engineering student's education, given that the ability to analyse and solve engineering problems must include a capacity to make choices on the basis of environmental/commercial as well as engineering/technical constraints. The ultimate objective of project appraisal is to secure the greatest benefit from the available scarce resources or present value of the money.