BPSC AE Paper 4 (General Engineering Science) 10 Nov 2022 Official Paper (Previous Year Paper)

Explanation:

Maximum shear strain energy / Distortion energy theory / Mises – Henky theory: 

• It states that inelastic action at any point in body, under any combination of stress begging, when the strain energy of distortion per unit volume absorbed at the point is equal to the strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under the state of uniaxial stress as occurs in a simple tension/compression test.

• \(\frac{1}{2}\left[ {{{\left( {{{\rm{\sigma }}_1} - {{\rm{\sigma }}_2}} \right)}^2} + {{\left( {{{\rm{\sigma }}_2} - {{\rm{\sigma }}_3}} \right)}^2} + {{\left( {{{\rm{\sigma }}_3} - {{\rm{\sigma }}1}} \right)}^2}} \right] \le {\rm{\sigma }}{\rm{y}}^2\) for no failure

• \(\frac{1}{2}\left[ {{{\left( {{{\rm{\sigma }}_1} - {{\rm{\sigma }}_2}} \right)}^2} + {{\left( {{{\rm{\sigma }}_2} - {{\rm{\sigma }}_3}} \right)}^2} + {{\left( {{{\rm{\sigma }}_3} - {{\rm{\sigma }}1}} \right)}^2}} \right] \le {\left( {\frac{{{{\rm{\sigma }}{\rm{y}}}}}{{{\rm{FOS}}}}} \right)^2}\) For design

• It is the best suitable theory for ductile material.

• It cannot be applied to the material under hydrostatic pressure.​

Additional Information 

Maximum shear stress theory (Guest & Tresca’s Theory):

• According to this theory, failure of the specimen subjected to any combination of a load when the maximum shearing stress at any point reaches the failure value equal to that developed at the yielding in an axial tensile or compressive test of the same material.

Graphical Representation:

• \({{\rm{\tau }}{{\rm{max}}}} \le \frac{{{{\rm{\sigma }}{\rm{y}}}}}{2}\) For no failure

• \({{\rm{\sigma }}_1} - {{\rm{\sigma }}2} \le \left( {\frac{{{{\rm{\sigma }}{\rm{y}}}}}{{{\rm{FOS}}}}} \right)\) For design

• σ1 and σ2 are maximum and minimum principal stress respectively.

• Here, τmax = Maximum shear stress

• σy = permissible stress

• This theory is justified but a conservative theory for ductile materials. It is an uneconomical theory.

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Maximum principal stress theory (Rankine’s theory):

• According to this theory, the permanent set takes place under a state of complex stress, when the value of maximum principal stress is equal to that of yield point stress as found in a simple tensile test.
• For the design criterion, the maximum principal stress (σ1) must not exceed the working stress ‘σy’ for the material.
• \({{\rm{\sigma }}{1,2}} \le {{\rm{\sigma }}{\rm{y}}}\) for no failure
• ${{\rm{\sigma }}_{1,2}} \le \frac{{\rm{\sigma }}}{{{\rm{FOS}}}}$ for design
• Note: For no shear failure τ ≤ 0.57 σy

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Graphical representation:

• For brittle material, which does not fail by yielding but fail by brittle fracture, this theory gives a satisfactory result.
• The graph is always square even for different values of σ1 and σ2.

Maximum principal strain theory (St. Venant’s theory):

• According to this theory, a ductile material begins to yield when the maximum principal strain reaches the strain at which yielding occurs in simple tension.
• ${\epsilon_{1,2}} \le \frac{{{{\rm{\sigma }}_{\rm{y}}}}}{{{{\rm{E}}_1}}}$ For no failure in uniaxial loading.
• \(\frac{{{{\rm{\sigma }}_1}}}{{\rm{E}}} - {\rm{\mu }}\frac{{{{\rm{\sigma }}_2}}}{{\rm{E}}} - {\rm{\mu }}\frac{{{{\rm{\sigma }}3}}}{{\rm{E}}} \le \frac{{{{\rm{\sigma }}{\rm{y}}}}}{{\rm{E}}}\) For no failure in triaxial loading.
• \({{\rm{\sigma }}_1} - {\rm{\mu }}{{\rm{\sigma }}_2} - {\rm{\mu }}{{\rm{\sigma }}3} \le \left( {\frac{{{{\rm{\sigma }}{\rm{y}}}}}{{{\rm{FOS}}}}} \right)\) For design, Here, ϵ = Principal strain
• σ1, σ2, and σ3 = Principal stresses

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Graphical Representation:

This theory overestimates the elastic strength of ductile material.

Maximum strain energy theory (Haigh’s theory):

• According to this theory, a body complex stress fails when the total strain energy at the elastic limit in simple tension.
• Graphical Representation:
• \(\left{ {{\rm{\sigma }}_1^2 + {\rm{\sigma }}_2^2 + {\rm{\sigma }}_3^2 - 2{\rm{\mu }}\left( {{{\rm{\sigma }}_1}{{\rm{\sigma }}_2} + {{\rm{\sigma }}_2}{{\rm{\sigma }}_3} + {{\rm{\sigma }}_3}{{\rm{\sigma }}1}} \right)} \right} \le {\rm{\sigma }}{\rm{y}}^2\)  for no failure
• \(\left{ {{\rm{\sigma }}_1^2 + {\rm{\sigma }}_2^2 + {\rm{\sigma }}_3^2 - 2{\rm{\mu }}\left( {{{\rm{\sigma }}_1}{{\rm{\sigma }}_2} + {{\rm{\sigma }}_2}{{\rm{\sigma }}_3} + {{\rm{\sigma }}_3}{{\rm{\sigma }}1}} \right)} \right} \le {\left( {\frac{{{{\rm{\sigma }}{\rm{y}}}}}{{{\rm{FOS}}}}} \right)^2}\) for design
• This theory does not apply to brittle material for which elastic limit stress in tension and in compression are quite different.

Important Points

• For Brittle material:- Maximum  Principal Stress Theory (Rankine criteria) is used.
• Maximum Shear Stress Theory (Tresca theory), Total strain energy theory, Maximum Distortion Energy Theory (Von Mises) useful for a ductile material.
• Tresca's theory fails in the hydrostatic state of stresses.
• All theories will give the same results if loading is uniaxial.

Explanation:-

Critical or whirling speed of a shaft

• When the rotational speed of the system coincides with the natural frequency of lateral/transverse vibrations, the shaft tends to bow out with a large amplitude. This speed is termed as critical/whirling speed.
• Whirling speed or Critical speed of a shaft is defined as the speed at which a rotating shaft will tend to vibrate violently in the transverse direction if the shaft rotates in the horizontal direction.
• In other words, the whirling or critical speed is the speed at which resonance occurs.
• Hence we can say that whirling of the shaft occurs when the natural frequency of transverse vibration matches the frequency of a rotating shaft.
• It is the speed at which the shaft runs so that the additional deflection of the shaft from the axis of rotation becomes infinite is known as critical or whirling speed.

Explanation:

The three main chemical constituents of wood are cellulose, hemi-cellulose, and lignin. Their typical proportions are as follows:

1. Cellulose (40% - 50%):

• Cellulose is a polysaccharide consisting of a linear chain of glucose molecules. It provides strength and rigidity to the cell walls of wood. It is the primary structural component and the most abundant organic polymer in wood.

2. Hemi-cellulose (20% - 35%):

• Hemi-cellulose is a heterogeneous group of polysaccharides that surround the cellulose fibers. It acts as a matrix that holds cellulose fibers together. Hemi-cellulose has a lower degree of polymerization and is more amorphous than cellulose, making it more easily hydrolyzed.

3. Lignin (15% - 35%):

• Lignin is a complex organic polymer with a phenyl group structure that acts as a binder for cellulose fibers, providing rigidity and resistance to decay. It is hydrophobic, which makes it crucial for the water conduction properties of wood.

Option 1 provides the correct approximate percentage ranges of these three main constituents, making it the most accurate representation of the chemical composition of wood.

Explanation:

The typical composition of Portland cement includes the following major compounds:

1. Tricalcium Silicate (C₃S): About 45-60% – Responsible for the early strength and hardening of the cement. It hydrates quickly and contributes significantly to the initial setting time and early strength gain.
2. Dicalcium Silicate (C₂S): About 15-30% – Reacts more slowly than C₃S and contributes to the strength of the cement over a longer period, providing long-term strength and durability.
3. Tricalcium Aluminate (C₃A): About 8-12% – Reacts quickly with water, generating a significant amount of heat, which is why it is controlled in cement to reduce heat of hydration and prevent quick setting. It also affects sulfate resistance.
4. Tetracalcium Aluminoferrite (C₄AF): About 6-10% – Provides the grey color to the cement and has a minor role in the hydration process.
5. Other Bonding Agents: These usually include gypsum (about 3-5%) to control the setting time and minor impurities.

Option 4 matches the approximate composition of Portland cement most closely, making it the correct choice. Other options either have incorrect proportions or unlikely combinations that do not reflect the typical composition of standard Portland cement.

Explanation:

Cast iron:

• Cast iron is widely used in the manufacture of machine parts, since it has a good compressive strength and easy to make the castings.
• There are different problems in the welding of cast iron in comparison to mild steel, even though this is also in the group of ferrous metals.

Types of cast iron:

There are four basic types of cast iron available:

• Grey cast iron
• White cast iron
• Malleable cast iron
• Nodular cast iron (or) spheroidal graphite iron

Grey cast iron:

• Grey cast iron is soft and tougher than the white cast iron which is hard and brittle.
• Complex castings of gray cast iron can be done easily because it contains Good liquidity.
• The good mechanical properties of grey cast iron are due to the presence of particles of free state carbon or graphite, which separate out during slow cooling.
• Grey cast iron is of a weldable type.
• It contains 3 to 4% of carbon.
• The characteristic of gray cast iron is the graphic microstructure, which is able to cause fractures to the material and have a gray appearance.
• This one is the most commonly used type of cast iron and also commonly used cast material based on weight.

Explanation:

Silicon as a Crucial Alloying Element in Steel:

• Silicon acts as a Deoxidizing Agent during the steelmaking process. It helps in removing oxygen from the steel, reducing the formation of oxides, and improving the overall quality. The deoxidizing action ensures that the steel has fewer gas pockets and inclusions, leading to a more uniform and cleaner structure.
• Silicon improves the Surface Grain Structure of the steel. The presence of silicon enhances the formation of a finer grain structure on the surface, which contributes to better mechanical properties, including increased toughness and resistance to cracking. A finer grain structure also enhances the surface finish of the steel, making it more suitable for high-quality applications.
• Silicon contributes to an increase in Toughness. Toughness is the ability of the material to absorb energy and plastically deform without fracturing, which is crucial for applications where the steel is subjected to dynamic loads or impact forces. The increased toughness provided by silicon allows the steel to perform better in demanding environments.
• Silicon promotes Magnetic Properties. In electrical steels used in transformers and motors, silicon increases electrical resistivity and reduces energy losses due to eddy currents. The presence of silicon improves magnetic permeability and reduces hysteresis losses, making the steel more efficient for magnetic applications.

Conclusion

Based on the analysis of the effects of silicon in steel, the correct answer is:

4) Acts as a deoxidizing agent, improves the surface grain structure, increases toughness, and promotes magnetic properties.

This option comprehensively covers the key benefits of silicon as an alloying element in steel.

Explanation:

Steels Used for High-Speed Machining:

• Steels used for making tools that are designed to machine materials with heavy cuts at high speeds are typically chosen for their ability to maintain hardness and cutting efficiency at elevated temperatures. These tools need to withstand the high stresses and temperatures generated during high-speed machining.
• Plain-Carbon Steel: Although plain-carbon steel is used in some cutting tools, it is not suitable for high-speed machining where heavy cuts are involved, as it tends to lose its hardness quickly at elevated temperatures.
• Low-Carbon Steel: Low-carbon steels are generally softer and more ductile, making them unsuitable for cutting tools that require high hardness and strength, especially in high-speed machining applications.
• High-Speed Tool Steel: This steel is specifically designed for high-speed machining. It retains its hardness even at high temperatures, which is crucial for tools subjected to heavy cuts and high-speed conditions. High-speed tool steel contains alloying elements such as tungsten, molybdenum, vanadium, and cobalt, enhancing its performance in these demanding applications.
• Medium-Carbon Steel: Medium-carbon steels offer better hardness and strength than low-carbon steels, but they still do not perform as well as high-speed tool steels under high-speed machining conditions.

Conclusion

Based on the required properties for high-speed machining with heavy cuts, the correct answer is:

3) High-Speed Tool Steel

This type of steel is specifically engineered to perform well under the demanding conditions of high-speed and heavy-cut machining, making it the best choice among the options provided.

Explanation:

Delta Metal:

• Delta metal is a type of brass, which is a copper-zinc alloy with additional elements that improve its properties.
• The alloy is primarily composed of copper, zinc, lead, and iron. This specific composition results in a strong and corrosion-resistant material suitable for various industrial applications.
• Copper: Acts as the base metal in the alloy, providing ductility and conductivity.
• Zinc: Enhances strength and hardness, contributing to the alloy's overall mechanical properties.
• Lead: Improves machinability and provides lubrication, making it easier to process the metal.
• Iron: Increases strength and hardness, making the alloy suitable for heavy-duty applications.

Conclusion

Based on the composition and properties of Delta metal, the correct answer is:

2) Copper, zinc, lead, and iron

This option accurately reflects the elements that make up Delta metal, resulting in a durable and versatile alloy.

Explanation:

Sinking fund factor.

• The sinking fund factor is used to determine the regular (usually equal) payments needed to accumulate a specified future amount of money.
• It converts a future financial requirement into a series of equal annuity payments.

Other options like the capital recovery factor, present value factor, and unicost present value factor relate to different financial calculations and are not specifically for converting future sums into equal periodic payments.

Explanation:

• A Gantt chart is a horizontal bar chart that shows a project's planned schedule and its tasks or events between a start and finish date.
• Each bar in the Gantt chart represents a task, while the dates are laid out horizontally.
• ​The length of each bar is proportional to the duration of time planned for the corresponding activity.
• Henry Gantt (1861-1919), an American mechanical engineer, designed the Gantt chart.
• It helps in scheduling, managing, and monitoring specific tasks and resources in a project.
• It's the most widely used chart in project management.
• Gantt charts are mainly used to allocate resources to activities.
• The resources allocated to activities include staff, hardware, and software.
• Gantt charts are useful for resource planning.
• A Gantt chart is a special type of bar chart where each bar represents an activity.
• The bars are drawn along a timeline.

Explanation:

Crash cost is a term used in project management, particularly in the context of project scheduling and the Critical Path Method (CPM). Crashing refers to the process of accelerating a project activity to reduce the total project duration. This often involves allocating additional resources (like labor, equipment, or materials) or working overtime to complete an activity in less time than originally planned. The main points regarding crash cost are:

1. Crashing Activities: When you crash an activity, you aim to shorten its duration by increasing resources. This results in additional costs, known as crash costs. Crashing is done to meet project deadlines or to ensure that critical milestones are achieved sooner.
2. Crash Cost Implications: Crashing typically increases the direct costs of an activity due to the need for extra resources, overtime pay, or expedited materials. These additional expenses are referred to as crash costs. The decision to crash should weigh the cost increase against the benefit of reduced project time.
3. Trade-Offs: Project managers analyze the trade-off between time and cost to determine the most cost-effective way to shorten the project duration without significantly escalating the project budget.

Concept:

The PERT (Project Evaluation and Review Technique) technique is used, when activity time estimates are stochastic in nature. For each activity, three values of time (optimistic, most likely, pessimistic) are estimated.

• Optimistic time (to) estimate is the shortest possible time required for the completion of the activity
• Most likely time (tm) estimate is the time required for the completion of activity under normal circumstances
• Pessimistic time (tp) estimate is the longest possible time required for the completion of the activity

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In PERT expected time of an activity is determined by using the below-given formula:

${t_e} = \frac{{\left( {{t_o} + 4{t_m} + {t_p}} \right)}}{6}$

Variance, σ2 = ${\left[ {\frac{{{t_P} - {t_0}}}{6}} \right]^2}$

Calculation:

t_o = 2, t_p = 12, and t_m = 5

Expected time

\({{\text{t}}{\text{E}}}=\frac{{{\text{t}}{\text{o}}}+4{{\text{t}}{\text{m}}}+{{\text{t}}{\text{p}}}}{6}=\frac{2+4\left( 5 \right)+12}{6}=5.7\text{ }!!~!!\text{ days}\)

Variance, σ2 = ${\left[ {\frac{{{t_P} - {t_0}}}{6}} \right]^2}$ = ${\left[ {\frac{{{12} - {2}}}{6}} \right]^2}$= 2.8

Note: In official paper the given options are not match with correct values, so we updated the option 1.

Explanation

• Fulkerson’s rule is used for numbering events involved in the project scheduling network.
• Types of network diagram
• Event on the node ( EON )
• Activity on the node ( AON )

Important Points

• Hungarian  method – Assignment problems
• Johnson's  rule – scheduling jobs
• Simplex method -  linear programming

Concept:

When a jet of water discharges from a tap, it accelerates downward under the influence of gravity. As the jet falls, its velocity increases due to the gravitational pull. We can use the kinematic equations of motion to find the velocity of the jet at a specific point below the tap.

Calculation:

Given:

• Initial velocity ($u$) at the tap outlet = 2 m/s
• Vertical distance below the tap outlet ($h$) = 0.6 m
• Acceleration due to gravity ($g$) = 10 m/s²

Using the kinematic equation:

$v^2 = u^2 + 2gh$

Substitute the values:

$v^2 = (2 , \text{m/s})^2 + 2 \cdot (10 , \text{m/s}^2) \cdot (0.6 , \text{m})$

$v^2 = 4 + 12$

$v^2 = 16$

$v = \sqrt{16}$

$v = 4 , \text{m/s}$

Explanation:

Borda's mouthpiece is a short cylindrical tube projecting inward. The edge of the tube must be relatively thin and sharp to ensure perfect contraction, and its length, about ½ diameter, such that the jet will not touch the sides of the tube.

Pressure force at the entrance of the mouth piece

F = ρ × g × H × a , where a is the area of the mouthpiece.

H = distance of center of gravity of the mouthpiece from the free surface.

from Newton's second law of motion, the rate of change of momentum = net force 

ρ × a × vc × (vc - 0) = ρ × g × H × a 

ρ × ac × vc2 = ρ × g × H × a ..........(A) 

applying Bernoulli's equation

$\frac{{{P_1}}}{{\rho g}} + \frac{{V_1^2}}{{2g}} + {z_1}+ {h_f} = \frac{{{P}}}{{\rho g}} + \frac{{V^2}}{{2g}} + {z}$

$\frac{{v_c^2}}{{2g}} = H$ ⇒ $v_c = \sqrt {2gH}$ 

P1 = P2 = Patm ​= 0, hf  = 0 (since the fluid is frictionless)

from eqn. (A) 

ρ × ac × 2g × H = ρ × g × H × a

ac = $\frac{a}{2}$ ⇒ $C_c =\frac{a_c}{a} = 0.5$

Explanation:

In the case of incompressible flow around immersed bodies, Reynolds number is the primary dimensionless number that determines the similarity between the model and the prototype. The Reynolds number is defined as: 

R_e=ρ⋅V⋅L/μ​

Where:

• ρ\rho is the density of the fluid,
• VV is the velocity of the flow,
• LL is a characteristic length (e.g., diameter of the body),
• μ\mu is the dynamic viscosity of the fluid.

Importance of Reynolds Number:

1. Flow Similarity: For modeling the flow around immersed bodies, maintaining equal Reynolds numbers between the model and the prototype ensures that the flow patterns, including boundary layer behavior, separation points, and wake formation, will be similar. This allows accurate predictions and analyses of the flow characteristics of the prototype based on model experiments.
2. Incompressible Flow: In incompressible flow, the effects of compressibility (density changes) are negligible. The Reynolds number becomes the primary parameter because it encapsulates the relative effects of inertial forces to viscous forces in the flow, which dictate the flow behavior around immersed bodies.

Concept:

To determine the rate of heat loss per unit area through a composite wall, we use the concept of thermal resistance in series.

Calculation:

Given:

• Thickness of silica brick, $t_1 = 119 , \text{mm} = 0.119 , \text{m}$
• Thickness of magnesite brick, $t_2 = 240 , \text{mm} = 0.24 , \text{m}$
• Temperature at the inside surface of silica brick, $T_1 = 780 , ^\circ \text{C}$
• Temperature at the outside surface of magnesite brick, $T_2 = 102 , ^\circ \text{C}$
• Thermal conductivity of silica brick, $k_1 = 1.7 , \text{W/m} \cdot ^\circ \text{C}$
• Thermal conductivity of magnesite brick, $k_2 = 6 , \text{W/m} \cdot ^\circ \text{C}$
• Contact thermal resistance, $R_c = 0.003 , ^\circ \text{C/W}$

Calculation:

First, calculate the thermal resistance of the silica brick layer:

$R_1 = \frac{t_1}{k_1} = \frac{0.119}{1.7} , \text{m}^2 \cdot ^\circ \text{C/W} = 0.07 , ^\circ \text{C/W}$

Next, calculate the thermal resistance of the magnesite brick layer:

$R_2 = \frac{t_2}{k_2} = \frac{0.24}{6} , \text{m}^2 \cdot ^\circ \text{C/W} = 0.04 , ^\circ \text{C/W}$

Then, calculate the total thermal resistance:

$R_{\text{total}} = R_1 + R_2 + R_c = 0.07 + 0.04 + 0.003 = 0.113 , ^\circ \text{C/W}$

Finally, calculate the rate of heat loss per unit area:

$q = \frac{\Delta T}{R_{\text{total}}} = \frac{T_1 - T_2}{R_{\text{total}}} = \frac{780 - 102}{0.113} , \text{W/m}^2 = \frac{678}{0.113} , \text{W/m}^2 \approx 6000 , \text{W/m}^2$

Therefore, the rate of heat loss per unit area of the wall is:

$\text{6000 W/m}^2$

Explanation:

In convection studies, it is common practice to nondimensional the governing equations and combines the variables which group together into dimensionless numbers in order to reduce the number of total variables.

Nusselt number is a parameter to nondimensional the heat transfer coefficient.

Non-dimensional groupings

• Nusselt No. Nu = hLc/k = (convection heat transfer strength)/ (conduction heat transfer strength)
• Prandtl No. Pr = n/a = (momentum diffusivity)/ (thermal diffusivity)
• Reynolds No. Re = Ux/ν = (inertia force)/ (viscous force)

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The dimensionless parameter represents the natural convection effects and is called the Grashof number.

Grashof number, Gr, as the ratio between the buoyancy force and the viscous force:

$Gr = \frac{{g\beta \left( {{T_s} - {T_\infty }} \right)L_c^3}}{{{\nu ^2}}}$

Nusselt number is a function of the Grashof number and the Prandtl number alone. Nu = f (Gr, Pr)

Concept:

When thermal radiation falls onto an object,

• The radiation will be absorbed by the surface of the object, causing its temperature to change
• The radiation will be reflected from the surface of the body, causing no temperature change
• The radiation will pass completely through the object, causing no temperature change

Absorptivity (α) is a measure of how much of the radiation is absorbed by the body

Reflectivity (ρ) is a measure of how much is radiation is reflected

Transmissivity (τ) is a measure of how much radiation passes through the object.

Each of these parameters is a number that ranges from 0 to 1.

For any given wavelength (λ), αλ + ρλ + τλ = 1. Objects may have different values of each of these parameters at different wavelengths.

Now,

For an opaque body, transmissivity is 0 since no wavelength is transmitted through it.

∴ For an opaque body sum of absorptivity and reflectivity is 1

Explanation:

Throttling process:

• A throttling process occurs when a gas or vapor flows through a minute aperture or restriction, leading to a drop in pressure.
• The process is characterized by constant enthalpy (no change in enthalpy) since there is no significant work done, and no heat transfer occurs.
• Example: Steam escaping through the safety valve of a pressure cooker.

Throttling processes are commonly encountered in engineering, particularly in valves, orifices, and similar devices where the fluid undergoes a reduction in pressure.

Key Characteristics of Throttling:

• Constant Enthalpy (h1 = h2): The enthalpy before and after the throttling process remains the same.
• Pressure Drop: The fluid experiences a significant reduction in pressure as it passes through the restriction.
• Temperature Change: Depending on the fluid and its initial conditions, the temperature may increase, decrease, or remain constant due to the Joule-Thomson effect.

Important Points

 In a throttling process, the most significant outcome is the pressure drop, while the enthalpy remains constant. This distinguishes throttling from other thermodynamic processes such as adiabatic or isothermal processes

Concept:

To determine the change in specific entropy of a system undergoing a reversible process, we use the relationship between heat transfer and temperature.

Calculation:

Given:

• Mass of fluid, $m = 2 , \text{kg}$
• Temperature, $T = 500 , \text{K}$
• Heat received in the first case, $Q_1 = 1 , \text{MJ} = 1000 , \text{kJ}$
• Heat received in the second case, $Q_2 = 2 \times Q_1 = 2000 , \text{kJ}$

The formula for the change in entropy is:

$\Delta S = \frac{Q_{\text{rev}}}{m \cdot T}$

Calculation:

For the first case, where the system receives 1000 kJ of heat:

$\Delta S_1 = \frac{1000 , \text{kJ}}{2 , \text{kg} \times 500 , \text{K}} = \frac{1000}{1000} = 1 , \text{kJ/kg·K}$

For the second case, where the heat supply is doubled to 2000 kJ:

$\Delta S_2 = \frac{2000 , \text{kJ}}{2 , \text{kg} \times 500 , \text{K}} = \frac{2000}{1000} = 2 , \text{kJ/kg·K}$

Therefore, the change in specific entropy for the two cases is:

$1 , \text{kJ/kg·K} \text{ and } 2 , \text{kJ/kg·K} \text{ respectively.}$

Concept:

Note that the Rankine cycle has a lower efficiency compared to the corresponding Carnot cycle 2’-3-4-1’ with the same maximum and minimum temperatures. The reason is that the average temperature at which heat is added in the Rankine cycle lies between T2 and T2' and is thus less than the constant temperature T2' at which heat is added to the Carnot cycle.

It is very difficult to build a pump that will handle a mixture of liquid and vapour at state 1' (refer T-s diagram; Carnot Vapor Cycle) and deliver saturated liquid at state 2’.

It is much easier to completely condense the vapour and handle only liquid in the pump (Rankine Cycle)

Additional Information

In the Carnot cycle, the compression is wet compression so the pump work requirement is more compare to the Rankine cycle where the pump compresses only the saturated liquid. Thus the specific work output for the Rankine cycle is more than the Carnot cycle for the same maximum and minimum temperature.

Also, the work ratio is defined as the ratio of net-work to the work done in the turbine.

$r_w=\frac{W_{net}}{W_T}=\frac{W_T-W_C}{W_T}$

Thus the work ratio is low for the Carnot Cycle.

Explanation:

In a Pelton turbine, which is a type of impulse turbine, the working principle involves converting the potential energy of water into kinetic energy before it strikes the runner buckets:

1. Conversion of Potential to Kinetic Energy: The high-pressure water from a reservoir is directed through a nozzle. The nozzle converts the high-pressure potential energy into a high-velocity jet of water (kinetic energy). This jet of water then strikes the buckets of the Pelton turbine runner, causing it to spin and generate mechanical energy, which is converted into electrical energy by a generator.
2. Impulse Turbine: Since the conversion of energy (from potential to kinetic) occurs outside the turbine (at the nozzle), the turbine itself works on the impulse principle. The runner operates in atmospheric pressure, and the pressure remains constant throughout the process, which characterizes the operation of impulse turbines.

Explanation:

In a single-acting reciprocating pump, the acceleration head is a phenomenon that occurs due to the change in velocity of the fluid inside the suction and delivery pipes as the piston moves back and forth. The acceleration of the fluid causes pressure variations, known as the acceleration head.

1. Length of the Pipe: The acceleration head is directly influenced by the length of the pipe. A longer pipe will have a greater mass of water that needs to be accelerated and decelerated with each stroke of the pump, resulting in a higher acceleration head. This is because the inertia of the fluid column is higher in longer pipes, leading to more significant pressure changes.

Additional Information

For a single acting reciprocating pump

$Q = \frac{{ALN}}{{60}}$

For a double-acting reciprocating pump

$Q = \frac{{2ALN}}{{60}}$

Thevenin’s theorem: Any two-terminal bilateral linear DC circuits can be replaced by an equivalent circuit consisting of a voltage source and a series resistor. 

It is applicable to both active and passive elements.

• Thevenin's theorem fails to simplify a complicated series and parallel linear two - terminal network.
• Thevenin's theorem states that a two - terminal electrical circuit (linear) can be replaced by an equivalent circuit consisting of a voltage source Vth in to series with a resistor known as equivalent resistance of entire network.
• For finding Thevenin equivalent impedance Zth, make the voltage source as a short circuit and the current source as an open circuit (i.e. voltage source and current source is replaced by their respective internal resistances).
• Thevenin's theorem is applicable to linear networks only.

Series Connection of Resistances:

Parallel Connection of Resistances:

• In a series configuration, resistors are connected end to end, allowing the same current to flow through each resistor.
• On the other hand, in a parallel configuration, each resistor is connected across the same voltage, but the current flowing through them is different.
• Parallel electrical circuits refer to a circuit with more than one path through which current flows.

Concept:

The RMS value of the voltage is given by:

$V_{rms}={V_m\over \sqrt{2}}$

The average value of the voltage is given by:

$V_{avg}={2V_m\over \pi}$

where, V_m is the maximum value of voltage.

Calculation

Given, V_m = 20 Volts

V_rms=$\frac{20}{\sqrt2}$

V_rms=14.14 Volts.

• In DC motor the classification is done according to the connection of the field winding to the armature.
• In a DC motor, the armature resistance is typically very low, usually less than 1 Ω, and sometimes even between 0.1 Ω and 1.5 Ω
• A DC generator is an electrical machine whose main function is to convert mechanical energy into electricity. When the conductor cuts magnetic flux, an emf will be generated based on the electromagnetic induction principle of Faraday’s Laws. This electromotive force can cause a flow of current when the conductor circuit is closed.
• The emf equation of the DC generator is given by the equation:

 

${E_b} = \frac{{PNϕ Z}}{{60A}}$

where

Z = Total number of armature conductor

P = The number of poles in a generator

A = The number of parallel paths

N = Speed of armature in r.p.m

ϕ = Flux per pole

Lap winding 

A = P

Wave winding

A = 2

• back e.m.f. depend on the number of poles
• back e.m.f. is is directly proportional to the speed of rotor

Explanation:

Greenhouse effect:

• The greenhouse effect is a natural process that warms the Earth's surface. It occurs when certain gases in the Earth's atmosphere trap heat, preventing it from escaping into space.
• These gases act like the glass of a greenhouse, allowing sunlight to enter but trapping some of the heat that radiates back from the Earth's surface.

Among the gases produced by the combustion of fossil fuels, the primary contributor to the greenhouse effect is carbon dioxide (CO2).

Carbon Dioxide (CO2):

• CO2 is the most significant greenhouse gas produced
• CO2 is the most significant greenhouse gas produced by the combustion of fossil fuels such as coal, oil, and natural gas.
• When fossil fuels are burned, carbon reacts with oxygen in the air to form CO2, which is then released into the atmosphere.
• CO2 has a long residence time in the atmosphere, meaning it accumulates and persists, contributing significantly to the warming of the Earth’s atmosphere.
• This gas traps heat within the atmosphere, leading to global warming and climate change.

Important Points

While other gases such as water vapor (H2O), methane (CH4), and nitrous oxide (N2O) also contribute to the greenhouse effect, CO2 is the primary gas emitted by human activities, particularly from the burning of fossil fuels.

Additional Information

Other greenhouse gases produced by fossil fuel combustion include:

(1) Methane (CH4):

• Methane is a potent greenhouse gas, though it is released in smaller quantities compared to CO2.
• It has a higher global warming potential (GWP) than CO2, making it much more effective at trapping heat in the atmosphere.

(2) Nitrous Oxide (N2O):

• N2O is another greenhouse gas produced during combustion, particularly in high-temperature processes like those in internal combustion engines.
• It has a significant global warming potential, even though its concentrations in the atmosphere are lower than CO2.

Conclusion:

The product of fossil fuel combustion that acts most like the glass in a greenhouse, trapping heat in the Earth's atmosphere and contributing to global warming, is CO2.

Correct Answer: 3) CO2

Explanation:

Oxides of nitrogen (NOx), primarily nitrogen dioxide (NO₂) and nitric oxide (NO), are pollutants formed during the combustion of fossil fuels at high temperatures, such as in vehicles, power plants, and industrial processes. These pollutants can have severe health impacts:

1. Formation of Acid in Lungs: When inhaled, nitrogen dioxide (NO₂) reacts with water in the lungs to form nitric acid (HNO₃). This acid formation can damage the sensitive tissues of the lungs, leading to inflammation and other respiratory issues. The acid essentially eats away at the lungs' spongy structure, impairing the ability to absorb oxygen.
2. Penetration Deep into the Lungs: NOx pollutants can penetrate deeply into the lung tissues, reaching the alveoli, where gas exchange occurs. This deep penetration exacerbates the impact on lung function.
3. Health Impacts: Exposure to NOx can lead to respiratory problems such as shortness of breath, bronchitis, asthma, and long-term exposure can increase the risk of lung infections. The damage caused by acid formation reduces the lungs' capacity to absorb oxygen, leading to serious health conditions and even chronic obstructive pulmonary disease (COPD).

Explanation:

Flue gas desulfurization (FGD) is a common method for removing sulfur oxides (SOx) from flue gases generated by burning fossil fuels like coal and oil. One of the most effective ways to achieve this is by using lime (calcium oxide) or limestone (calcium carbonate). The process works as follows:

1. Lime or Limestone Reaction: The sulfur oxides (SOx) in the flue gas react with lime or limestone to form calcium sulfite or calcium sulfate (gypsum). This reaction can effectively reduce the concentration of sulfur oxides in the flue gases:

SO_2​+CaCO_3​→CaSO_3​+CO_2​

SO_2​+CaO+1/2​O_2​→CaSO_4​

2. Removal of By-products: The resulting compounds, such as calcium sulfate (gypsum), are collected and can be used in other industries (e.g., construction materials) or disposed of safely.

Explanation:

Ecological Impact:

• Dams and their associated lakes (reservoirs) significantly alter the local ecological conditions. They create new water bodies where none existed, leading to changes in the natural habitats of plants and animals. Aquatic and terrestrial ecosystems are affected by the changes in water availability, temperature, and flow patterns. This can lead to the displacement of species that were previously adapted to the original environment, and in some cases, the introduction of invasive species that thrive in the new conditions.

Pressure on Land:

• The accumulation of large volumes of water in a reservoir increases the pressure on the underlying land. This can cause geological changes such as land subsidence, seismic activity, or even induced earthquakes in some cases. The added weight from the water also changes the stress distribution within the Earth's crust.

Groundwater Levels:

• Dams affect groundwater levels in several ways. The reservoir can recharge aquifers, raising the groundwater table. Conversely, changes in water flow downstream can reduce natural groundwater recharge, leading to a drop in groundwater levels. These changes can impact water availability for plants, animals, and human consumption.

Explanation:

Sedimentation:

• It is a physical method that allows heavier particles to settle at the bottom, separating them from the water. It is commonly used as a primary treatment step in wastewater treatment.

Oxidizing Agents:

• Like chlorine and ozone are chemical methods used to disinfect water, effectively killing bacteria, viruses, and other pathogens, making the water potable.

Additional Information

• Filtration and sedimentation are physical methods, not chemical methods.
• Screening and skimming are physical methods, not biological methods.
• Aerobic and anaerobic processes are biological methods, not physical methods.

• Synchronous generator or alternator is an electrical machine that converts the mechanical power from a prime mover into an AC electrical power at a particular voltage and frequency.
• The synchronous machine always runs at a constant speed called synchronous speed.
• To excite the field winding of the rotor of the synchronous machine, a direct current is required.
• Thus, an alternator is a polyphase synchronous machine in which excitation is provided by rotor winding connected to dc supply.
• With D.C excitation, there exists magnetic locking between stator poles and rotor pole. So that alternator rotates at synchronous speed.
• In a three-phase load, three different impedances are connected together in a star or delta fashion.
• The delta in a three-phase system is formed by connecting one end of the winding to the starting end of other winding and the connections are continued to form a closed loop.
• The star in the three-phase system is formed by connecting one end of all three impedances are connected together.

 

          The relationship between line and phase voltages and currents in star connected and delta connected networks are: 

Concept:

To find the magnitude of the resultant force and the angle it makes with the horizontal, we use vector addition. Forces acting along the sides of the square can be considered as vectors, and we can resolve these forces into x and y components to find the resultant.

Calculation:

Given:

• Force $P$ acts along side $AB$ (positive x-direction)
• Force $2P$ acts along side $BC$ (positive y-direction)
• Force $3P$ acts along side $CD$ (negative x-direction)
• Force $4P$ acts along side $DA$ (negative y-direction)

1. Resolving Forces into Components:

• Net force in the x-direction ($R_x$):
• $R_x = P - 3P = -2P$
• Net force in the y-direction ($R_y$):
• $R_y = 2P - 4P = -2P$

2. Calculating the Magnitude of the Resultant Force ($R$):

$R = \sqrt{R_x^2 + R_y^2}$

$R = \sqrt{(-2P)^2 + (-2P)^2}$

$R = \sqrt{4P^2 + 4P^2}$

$R = \sqrt{8P^2}$

$R = 2\sqrt{2}P$

3. Calculating the Angle with the Horizontal ($\theta$):

The angle $\theta$ that the resultant makes with the horizontal is given by:

$\tan \theta = \frac{R_y}{R_x}$

$\tan \theta = \frac{-2P}{-2P} = 1$

$\theta = 45^\circ$

Concept:

To solve this problem, we can use the work-energy principle and Newton's second law. The system consists of two blocks connected by a rope over an ideal pulley, with one block moving upward and the other downward. The change in kinetic energy will be related to the work done by gravity and tension.

Calculation:

Given:

• Weight of Block 1 ($W_1$) = 200 N (upward displacement)
• Weight of Block 2 ($W_2$) = 300 N (downward displacement)
• Initial velocity ($v_1$) = 3 m/s
• Final velocity ($v_2$) = 5 m/s
• Acceleration due to gravity ($g$) = 10 m/s²

1. Finding the Acceleration:

The change in velocity is from 3 m/s to 5 m/s. We need to find the acceleration ($a$):

$a = \frac{v_2^2 - v_1^2}{2h} = \frac{25 - 9}{2h} = \frac{16}{2h} = \frac{8}{h}$

2. Equations of Motion for Each Block:

Let $T$ be the tension in the rope:

• For Block 1 (200 N, moving upward):
• $T - W_1 = \frac{200}{10} \cdot a$
• $T - 200 = 20a \quad ...(1)$
• For Block 2 (300 N, moving downward):
• $W_2 - T = \frac{300}{10} \cdot a$
• $300 - T = 30a \quad ...(2)$

3. Solving the Equations:

Adding equations (1) and (2) to eliminate $T$:

$(T - 200) + (300 - T) = 20a + 30a$

$100 = 50a$

$a = 2 , \text{m/s}^2$

4. Finding the Tension $T$:

Substitute $a = 2$ m/s² into equation (1):

$T - 200 = 20 \cdot 2$

$T = 240 , \text{N}$

The tension is the same for both blocks.

5. Calculating the Displacement $h$:

Using the kinematic equation:

$v_2^2 = v_1^2 + 2ah$

$25 = 9 + 4h$

$4h = 16$

$h = 4 , \text{m}$

Concept:

To solve this problem, we use Newton's second law of motion. When a balloon descends or ascends, it experiences forces due to gravity and buoyancy. The weight removed from the balloon changes its net acceleration.

Calculation:

Let:

• $W$ = Weight of the balloon (initial)
• $w$ = Weight removed
• $a$ = Acceleration (upward and downward)
• $g$ = Acceleration due to gravity

1. When the balloon descends with acceleration $a$:

The net downward force is:

$W - F_b = \frac{W}{g} \cdot a$

Rearranging, we get:

$F_b = W - \frac{W \cdot a}{g}$    ...(1)

2. When weight $w$ is removed, the balloon accelerates upwards with acceleration $a$:

The net upward force is:

$F_b - (W - w) = \frac{W - w}{g} \cdot a$

Using equation (1) for $F_b$:

$W - \frac{W \cdot a}{g} - (W - w) = \frac{(W - w) \cdot a}{g}$

Simplifying this:

$w = \frac{W \cdot a}{g} + \frac{(W - w) \cdot a}{g}$

3. Solving for $w$:

Combine terms and solve:

$w(1 + \frac{a}{g}) = \frac{2W \cdot a}{g}$

$w = \frac{2W \cdot a}{a + g}$

The weight $w$ that should be removed is:

$w = \frac{2aW}{a + g}$

Concept:

When a vehicle comes to rest after applying brakes, it decelerates due to the frictional force between the tyres and the road. The deceleration can be calculated using kinematic equations, and the coefficient of kinetic friction is determined from the relationship between frictional force and normal force.

Kinematic Equation:

The kinematic equation for uniformly accelerated (or decelerated) motion is given by:

$v^2 = u^2 + 2as$

Where:

• v = final velocity (m/s)
• u = initial velocity (m/s)
• a = acceleration (m/s²)
• s = distance (m)

Calculation:

Given:

• Initial velocity (u) = 25 m/s
• Final velocity (v) = 0 m/s (since the car comes to rest)
• Distance (s) = 100 m
• Acceleration due to gravity (g) = 10 m/s²

Using the kinematic equation to find the deceleration (a):

$0 = 25^2 + 2 \cdot a \cdot 100$

$0 = 625 + 200a$

$a = -3.125 , \text{m/s}^2$

The negative sign indicates deceleration.

To find the time (t) required for the car to stop:

$v = u + at$

$0 = 25 + (-3.125)t$

$t = \frac{25}{3.125} = 8 , \text{s}$

Now, calculate the coefficient of kinetic friction (μ):

Using the formula: $a = \mu g$

$3.125 = \mu \times 10$

$\mu = \frac{3.125}{10} = 0.3125$

Hence, the time required for the car to stop is 8 seconds, and the coefficient of kinetic friction is 0.3125.

Concept:

Non-Conservative Force:

• A non-conservative force is a force for which the work done depends on the path taken by the object, not just on the starting and ending points. Common examples of non-conservative forces include friction, air resistance, and applied forces.
• In contrast to conservative forces (like gravitational force and electrostatic force), non-conservative forces dissipate energy, often converting it into heat or other forms of energy.

Characteristics of Work Done by Non-Conservative Forces:

• Path Dependence: The work done by a non-conservative force is dependent on the path taken by the particle. If the path changes, the work done will also change, making non-conservative forces path-dependent.
• Energy Dissipation: Non-conservative forces usually result in energy dissipation, meaning they convert mechanical energy into other forms like heat, sound, etc. This can lead to a decrease in the mechanical energy of the system.
• Non-Zero Net Work in Closed Paths: When a particle moves in a closed path under the action of a non-conservative force, the net work done by that force is not zero. This is because energy is lost due to dissipation (e.g., friction).

Driving system of Energy meter:

• In this system there are two silicon steel laminated electromagnet.
• The upper magnet called shunt magnet it carries a voltage coil consisting many turns of thin wire which connected to supply voltage so it's behave like a potential coil. Voltage coil is connected to the supply voltage.
• The coil of shunt electro - magnet is called pressure coil
• The lower magnet called series magnet it carries the two current coils consisting of few turns of thick wire and behave like a current coil. Current coils are connected in series with the circuit and load current passes through it.

Key Points

Energy meter:

• Energy meter or Watt-hour meter is used to measure the energy in kWh.
• It is an integrating type instrument.
• Its working principle is similar to the transformer.
• There are three essential mechanisms required in the energy meter named Driving torque, Braking torque, and registered mechanism.

 

Driving torque:

• This torque is required to revolve the disc or rotate the disc.
• It is obtained by the electromagnetic induction effect.

 

Braking torque:

• It is required to rotate the disc at a constant speed.
• It is obtained by using a permanent magnet placed inside the energy meter near the Aluminum disc.
• Eddy currents have induced in the magnet due to the movement of the rotating disc through the magnetic field. This eddy current reacts with the flux and exerts a braking torque which opposes the motion of the disk. The speed of the disk can be controlled by changing flux.
• Breaking torque of induction type single-phase energy meter is:

${T_b} = k\frac{{{\phi ^2}}}{{{R_e}}}N \times R$

K = constant

ϕ = flux

N = speed in rpm

R = radius of the disc

R_e = resistance in path of current (i.e. disc)

The braking torque of induction type single-phase energy meter is directly proportional to the square of the flux.

Registered mechanism:

• It registers the no. of rotations or revolutions of the disc which is proportional to the energy consumed in kWh.
• Meter constant = (No. of revolutions / kWh)

 Important Points

Creeping:

• Sometimes the disc of the energy meter makes the slow but continuous rotation at no load i.e. when the potential coil is excited but with no current flowing in the load called creeping error
• This error may be caused due to overcompensation for friction, excessive supply voltage, vibrations, stray magnetic fields, etc
• It can be reduced by making two opposite holes on the disc.

Concept:

Planimeter:

• A planimeter is a precision instrument used to measure the area of an irregularly shaped region on a map, plan, or photograph. It provides an accurate area measurement by tracing the boundary of the region in question.
• The planimeter consists of a tracing arm and a measuring wheel that moves along the boundary of the area. As the tracing arm moves around the perimeter, the measuring wheel calculates the area enclosed by the traced path.

Features of a Planimeter:

• Direct Area Measurement: Planimeters allow for direct measurement of the area by tracing the contour of the region. This method is straightforward and does not require complex calculations or approximations.
• Versatile Application: Planimeters can be used on maps, blueprints, photographs, and any other surface where area measurement is needed. They are widely used in fields like cartography, surveying, and engineering.
• Accuracy: Planimeters provide high accuracy for area measurements, making them suitable for professional use in surveying and engineering tasks where precise area computations are critical.

Additional Information 

• Rotameter: A rotameter is a device used to measure the flow rate of liquids and gases, not for area measurement.
• Tachometer: A tachometer measures the rotational speed of an object, typically in revolutions per minute (RPM), and is not used for area calculation.
• Odometer: An odometer measures the distance traveled by a vehicle, not the area of a region on a map or photograph.

Concept:

Theodolite:

• A theodolite is a precise optical instrument used in surveying and engineering to measure horizontal and vertical angles between visible points. It is a crucial tool for accurate angular measurements in the field.
• The theodolite consists of a telescope mounted on a rotating base, allowing it to rotate in both the horizontal and vertical planes. This dual rotation capability enables surveyors to accurately measure both horizontal and vertical angles.

Features of a Theodolite:

• Horizontal and Vertical Rotation: The telescope of a theodolite can rotate 360 degrees both horizontally and vertically. This allows surveyors to measure angles in all directions, making it versatile for various surveying applications.
• Precision Measurements: Theodolites are known for their high precision in measuring angles. They are often equipped with vernier scales or digital displays to read angles accurately, usually to a fraction of a degree.
• Leveling Mechanism: Theodolites come with leveling screws and a leveling bubble to ensure the instrument is perfectly level before taking measurements. This is essential for maintaining accuracy and reliability in angular readings.

Concept:

Dumpy Level:

• A dumpy level is an optical instrument used in surveying to establish horizontal lines of sight. It is mainly used for measuring height differences and transferring, measuring, and setting horizontal levels.
• It consists of a telescope firmly fixed to its support, allowing only horizontal rotation. This design ensures accuracy and stability, which is why dumpy levels are widely used in construction and civil engineering projects.

Features of a Dumpy Level:

• Horizontal Rotation: The telescope of a dumpy level can rotate 360 degrees in the horizontal plane. This allows surveyors to sight and measure multiple points around the instrument while keeping the line of sight level.
• No Vertical Rotation: The telescope is fixed vertically, meaning it cannot rotate up or down. This ensures that all measurements taken are along the same horizontal plane.
• Leveling Bubble: A bubble level is used to ensure the instrument is level before taking measurements. The bubble helps maintain accuracy by confirming the instrument's stability and leveling.

Explanation:

Micrometer:

• A micrometer is a precision instrument used to measure a job, generally within an accuracy of 0.01 mm.
• Micrometers used to take the outside measurements are known as outside micrometers.
• The ratchet stop ensures uniform pressure between the measuring surfaces.

Important Points 

The parts of a micrometer are listed here.

• Ratchet stop: The ratchet stop ensures uniform pressure between the measuring surfaces.
• Frame: The frame is made of drop-forged steel or malleable cast iron. All other parts of the micrometer are attached to this.
• Barrel/Sleeve: The barrel or sleeve is fixed to the frame. The datum line and graduations are marked on this.
• Thimble: On the beveled surface of the thimble also, the graduation is marked. The spindle is attached to this.
• Spindle: One end of the spindle is the measuring face. The other end is threaded and passes through a nut. The threaded mechanism allows for the forward and backward movement of the spindle.
• Anvil: The anvil is one of the measuring faces which is fitted on the micrometer frame. It is made of alloy steel and finished to a perfectly flat surface.
• Spindle lock nut: The spindle lock nut is used to lock the spindle at the desired position.

Permanent Magnet Moving Coil (PMMC)

• Permanent Magnet Moving Coil (PMMC) instruments, also known as D'Arsonval meters or galvanometers, are used to measure direct current (DC) quantities, such as voltage and current.

• The instrument which uses the permanent magnet for creating the stationary magnetic field between which the coil moves is known as the permanent magnet moving coil or PMMC instrument.

• It operates on the principle that the torque is exerted on the moving coil placed in the field of the permanent magnet. The PMMC instrument gives accurate result for DC measurement.

 

Construction of PMMC Instrument:

The main parts of PMMC instruments are:

1.) Moving coil

• The coil is the current-carrying part of the instruments which is freely moved between the stationary field of the permanent magnet.
• The current passes through the coil and deflects it due to which the magnitude of the current or voltage is determined.

 

2.) Magnet system

• The PMMC instrument uses a permanent magnet for creating stationary magnets.
• The Alnico material is used for creating the permanent magnet because this magnet has a high coercive force.

 

3.) Controlling system

• In the PMMC instrument, the controlling torque is because of the springs. The springs are made up of phosphorous bronze and placed between the two jewel bearings.
• The current passes through control springs to the moving coil. These coil springs are two in number so if one of them is broken then the current path to the moving coil is disrupted. Hence, the meter would read zero.

 

4.) Damping system

• The damping torque is used for keeping the movement of the coil at rest.
• This damping torque is induced because of the movement of the aluminum core which is moving between the poles of the permanent magnet.

Concept:

Three-Dimensional State of Stress:

In a three-dimensional stress state, the stress at a point is represented by a stress tensor, which is a 3 × 3 matrix. This matrix includes normal and shear stress components.

The general form of the stress tensor is:

Where:

• σ_xx, σ_yy, σ_zz are the normal stress components.
• τ_xy, τ_yz, τ_zx are the shear stress components.

Due to the symmetry of the stress tensor, shear stress components on mutually perpendicular planes are equal (i.e., τ_xy = τ_yx, τ_yz = τ_zy, τ_zx = τ_xz).

Calculation:

The number of independent stress components in a three-dimensional state of stress is calculated as follows:

• 3 normal stress components: σ_xx, σ_yy, σ_zz
• 3 shear stress components: τ_xy, τ_yz, τ_zx (since τ_xy = τ_yx, τ_yz = τ_zy, τ_zx = τ_xz)

Total number of independent stress components = 3 (normal) + 3 (shear) = 6

Explanation:

The strain energy of a cylindrical specimen subjected to a gradually applied tensile load can be calculated using the formula for strain energy in elastic materials. Strain energy (UU U U) is the energy stored in a body due to deformation under loading.

For a cylindrical specimen of length L, cross-sectional area A, extension δ, and under a gradually applied tensile load P, the strain energy is given by:

U=1/2​×P×δ

Additional InformationThe elastic strain energy stored in a member of length (it may be curved or straight) due to axial force, bending moment, shear force and torsion are summarized below:

Explanation:

• Bending Stress:
• Varies linearly across the depth of the beam.
• Zero at the neutral axis, maximum at the top and bottom fibers.
• Shear Stress:
• Parabolically distributed over the cross-section.
• Maximum at the neutral axis.
• Zero at the top and bottom fibers.

Concept:

Euler buckling load for a column is given by,

\({{\rm{P}}{\rm{E}}}{\rm{;}} = {\rm{;}}\frac{{{{\rm{\pi }}^2}{\rm{EI}}}}{{{\rm{L}}{\rm{e}}^2}}\)

Where, Le is the effective length of the column that depends on the end support conditions, and EI is the flexural rigidity of the column.

Now, effective length (Le) for different end conditions in terms of actual length (L) are listed in the following table:

The crippling load for a direction - fixed strut is four times that of a pin - ended strut for the same length.