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Past Questions
Which of the following engineering materials is the most suitable candidate for hot chamber die casting?
A)
Lowcarbon steel
B)
Titanium
C)
Copper
D)
Tin
Hot chamber die casting (also known as gooseneck casting) is a process where the melting furnace and the injection mechanism are integrated into one unit. The molten metal is pushed up through a submerged gooseneck into the die. This process is only suitable for metals with low melting points and those that do not react chemically with or dissolve the steel components of the casting machine (such as the pump and cylinder).
1. **Tin (Option d):** Tin, Zinc, and Lead alloys have low melting points and do not attack the steel components of the machine, making them ideal for hot chamber die casting.
2. **Low carbon steel (Option a):** Steel has an extremely high melting point (approx. 1500°C), which would melt the casting machine itself. It is processed via sand casting or investment casting.
3. **Titanium (Option b):** Titanium is highly reactive at high temperatures and has a high melting point. It requires vacuum melting and specialized molds, not hot chamber die casting.
4. **Copper (Option c):** Copper and its alloys (like brass) have higher melting points than zinc/tin and, more importantly, they tend to dissolve the steel of the injection mechanism (chemical attack/erosion), making them unsuitable for hot chamber processes. They are typically cast using cold chamber die casting.
Atomic packing factor (APF) in the case of copper crystal is
A)
0.52
B)
0.68
C)
0.74
D)
1.633
Step-by-step reasoning:
1. **Identify the Crystal Structure of Copper**: Copper (Cu) is a transition metal that crystallizes in a Face-Centered Cubic (FCC) lattice structure.
2. **Define Atomic Packing Factor (APF)**: APF is the ratio of the volume of atoms in a unit cell to the total volume of the unit cell. Formula: APF = (Number of atoms per unit cell × Volume of one atom) / Volume of unit cell.
3. **FCC Parameters**:
- Number of atoms per unit cell (n) = 4
- Volume of one atom (assuming spherical) = (4/3)πr³
- Relationship between lattice constant (a) and atomic radius (r) for FCC: a = 2√2r
4. **Calculation**:
- Volume of unit cell (V_cell) = a³ = (2√2r)³ = 16√2r³
- Total volume of atoms = 4 × (4/3)πr³ = (16/3)πr³
- APF = [(16/3)πr³] / [16√2r³] = π / (3√2) ≈ 0.74.
Analysis of options:
- Option (a) 0.52 is the APF for a Simple Cubic (SC) structure.
- Option (b) 0.68 is the APF for a Body-Centered Cubic (BCC) structure.
- Option (c) 0.74 is the APF for FCC and Hexagonal Close-Packed (HCP) structures, which applies to Copper.
- Option (d) 1.633 is physically impossible as APF cannot exceed 1.0.
The central processing unit in CNC consists of which of the following parts?
A)
Operating system unit
B)
Secondary memory
C)
Arithmetic Logic Unit
D)
Read only memory
In a CNC (Computer Numerical Control) system, the Central Processing Unit (CPU) acts as the 'brain' of the machine. The CPU is responsible for interpreting the G-codes and M-codes (part programs) and calculating the precise coordinates for tool movement. The core component of any CPU that performs these mathematical calculations (addition, subtraction, multiplication) and logical operations (comparisons) is the Arithmetic Logic Unit (ALU).
Why other options are incorrect:
- a. Operating system unit: The OS is software that manages hardware resources; it is not a physical 'part' or hardware component of the CPU itself.
- b. Secondary memory: This refers to external storage (like hard drives or USBs) used for long-term storage of programs, not a part of the CPU.
- d. Read only memory (ROM): While ROM is used in CNC systems to store the firmware or boot-up instructions, it is categorized as memory/storage, not the processing unit (CPU) itself.
Which type of motor is NOT used in axis or spindle drives of CNC machine tools?
A)
Induction motor
B)
DC servo motor
C)
Stepper motor
D)
Linear servo motor
In CNC (Computer Numerical Control) machine tools, axis and spindle drives require precise control over position, velocity, and acceleration.
1. **DC Servo Motors (Option b):** These are widely used in CNC machines due to their high torque-to-inertia ratio and excellent controllability for precise positioning.
2. **Stepper Motors (Option c):** These are commonly used in low-cost or small-scale CNC machines (like 3D printers or small mills) because they move in discrete steps, allowing for open-loop position control.
3. **Linear Servo Motors (Option d):** These are used in high-precision, high-speed CNC axes to eliminate backlash associated with ball screws by providing direct linear motion.
4. **Induction Motors (Option a):** Standard AC induction motors are generally not suitable for axis or spindle drives in CNC machines because they lack the precise speed and position control required for CNC operations. While specialized 'AC Servo Motors' (which are a modified type of induction motor with feedback) are used, a standard 'Induction Motor' is not a servo-drive component. In the context of this multiple-choice question, the standard induction motor is the outlier compared to the specific 'servo' and 'stepper' technologies designed for CNC motion control.
In Zinc Blende structure, each atom is surrounded by four atoms of the opposite kind which are located at the corners of which one of the following?
A)
Tetrahedron
B)
Hexahedron
C)
Cube
D)
Orthorhombic
The Zinc Blende (ZnS) crystal structure consists of two interpenetrating face-centered cubic (FCC) lattices, one composed of Zinc (Zn) atoms and the other of Sulfur (S) atoms, offset by a vector of (1/4, 1/4, 1/4) of the unit cell edge. In this arrangement, each atom of one type occupies half of the tetrahedral voids of the other type. Consequently, every atom is covalently bonded to four atoms of the opposite kind. These four nearest neighbors are positioned at the vertices of a regular tetrahedron, resulting in a coordination number of 4:4.
- Option (b) Hexahedron refers to a six-sided figure (like a cube), which does not describe the local coordination of Zinc Blende.
- Option (c) Cube describes the overall symmetry of the unit cell, but not the local geometry of the four nearest neighbors.
- Option (d) Orthorhombic refers to a crystal system where the axes are of unequal length, which is not applicable to the cubic symmetry of Zinc Blende.
Eutectoid reaction occurs at
A)
600°C
B)
723°C
C)
1147°C
D)
1493°C
In the context of Material Science and the Iron-Carbon (Fe-C) equilibrium phase diagram, a eutectoid reaction is an isothermal, reversible reaction in which one solid phase transforms into two different solid phases upon cooling. Specifically, for the Fe-C system, the eutectoid reaction occurs when austenite (γ) transforms into pearlite (a lamellar mixture of ferrite α and cementite Fe3C). This reaction occurs at a constant temperature of 723°C (often rounded to 727°C in some textbooks) at a carbon concentration of approximately 0.76% to 0.8% C.
Analysis of other options:
- 600°C: This is not a characteristic critical temperature for major phase transformations in the Fe-C system.
- 1147°C: This is the Eutectic temperature, where a liquid phase transforms into two solid phases (Austenite and Cementite).
- 1493°C: This is the melting point of pure iron.
Advantage of cold working is
A)
better dimensional accuracy
B)
better surface finish
C)
higher strength
D)
all ofthese
Cold working refers to the plastic deformation of a metal at a temperature below its recrystallization temperature. The advantages are as follows:
1. Better Dimensional Accuracy: Since there is no thermal expansion or contraction associated with heating and cooling (as seen in hot working), the final dimensions of the part can be controlled much more precisely.
2. Better Surface Finish: Cold working processes (like cold rolling or drawing) produce a smooth, polished surface finish because there is no formation of oxide scale (scaling) which typically occurs at high temperatures during hot working.
3. Higher Strength: Cold working induces 'strain hardening' or 'work hardening'. As the metal is deformed, dislocation density increases, making it harder for further dislocations to move, which increases the yield strength and hardness of the material.
Since all three statements (a, b, and c) are recognized advantages of cold working, the correct option is 'all of these'.
In a transportation problem, the method of penalties is called
A)
Least cost
B)
South east corner
C)
Vogel’s approximation
D)
north west corner
In the context of the Transportation Problem (a special case of linear programming), there are several methods to find the Initial Basic Feasible Solution (IBFS).
1. **Vogel's Approximation Method (VAM)**: This method is specifically known as the 'method of penalties.' It calculates a 'penalty' for each row and column, which is the difference between the two lowest costs in that row or column. The allocation is then made to the cell with the minimum cost in the row or column that has the highest penalty. This approach typically yields an initial solution very close to the optimal solution.
2. **North West Corner Rule**: This is a simple allocation method that starts from the top-left cell regardless of cost. It does not use penalties.
3. **Least Cost Method**: This method allocates based on the absolute lowest cost in the entire matrix. While it considers cost, it does not calculate 'penalties' (the cost of not choosing the cheapest cell).
4. **South East Corner Rule**: Similar to the North West Corner rule, but starts from the bottom-right. It does not use penalties.
Therefore, the method characterized by the use of penalties is Vogel's Approximation Method.
Product Design combines with product and business knowledge to generate ideas and concepts and convert them into physical and usable objects or services
A)
productivity
B)
ergonomics
C)
reflection
D)
operationality
The question describes the process of integrating product and business knowledge to create usable objects or services. In the context of Industrial Engineering and Product Design, 'Ergonomics' (also known as Human Factors Engineering) is the specific discipline that ensures a product is designed to be 'usable' and 'physical' in a way that fits the human user's capabilities and limitations. While the sentence structure in the prompt is slightly fragmented, it refers to the application of ergonomics to ensure that the conversion of a concept into a physical object results in a product that is safe, efficient, and usable.
Analysis of other options:
- (a) Productivity: This refers to the ratio of output to input; it is a measure of efficiency, not a design discipline.
- (c) Reflection: This is a cognitive process or a physical property of light/sound, not a design methodology.
- (d) Operationality: This refers to whether something is functional or in operation, but it is not the field of study that combines business and product knowledge to generate usable designs.
A shaft carrying two rotors as its end will have
A)
No node
B)
One node
C)
Two nodes
D)
Three nodes
In the context of rotor dynamics and vibrations, a 'node' is a point on the shaft that remains stationary (zero amplitude) during a specific mode of vibration. For a shaft carrying two rotors (masses) at its ends, we consider the fundamental modes of vibration. In the first mode (fundamental mode), the two masses typically move in opposite directions (out of phase). Because the masses are moving in opposite directions, there must be a point somewhere between them where the displacement is zero. This point is called the node. For a system with two concentrated masses at the ends, the first mode of vibration results in exactly one node located between the two rotors. Higher modes would introduce more nodes, but in standard engineering exam contexts regarding basic rotor configurations, the primary mode is considered unless specified otherwise.
The ratio of the applied load to the instantaneous cross section area during the uniaxial tensile test is called
A)
True Stress
B)
Instantaneous stress
C)
Yield point
D)
Ultimate stress
Option A is correct: True Stress.
If the centre distance of the mating gears having involute teeth is increased, then the pressure angle
A)
increases
B)
decreases
C)
remains unchanged
D)
none of these
In involute gear systems, the base circle radius (Rb) is constant and is defined by the formula Rb = R cos(phi), where R is the pitch circle radius and phi is the pressure angle. When the center distance between two mating gears is increased, the effective pitch circles shift outward, increasing the operating pitch radius (R'). Since the base circle radius (Rb) remains constant (as it is a physical property of the gear tooth geometry), the relationship becomes Rb = R' cos(phi'). For the equation to hold true while R' increases, cos(phi') must decrease. A decrease in the cosine value corresponds to an increase in the angle phi'. Therefore, increasing the center distance results in an increase in the operating pressure angle. This property is one of the primary advantages of involute gears, as they can still transmit a constant angular velocity ratio even if the center distance is slightly altered.
A kinematic chain is known as a mechanism when
A)
no links are fixed
B)
one link is fixed
C)
two links are fixed
D)
all links are fixed
In the study of Kinematics of Machinery, it is essential to distinguish between a kinematic chain and a mechanism:
1. **Kinematic Chain**: A kinematic chain is an assembly of links (rigid bodies) connected by kinematic pairs such that the relative motion between them is constrained. In a chain, no link is necessarily fixed; it is simply a closed loop of connected parts.
2. **Mechanism**: A mechanism is defined as a kinematic chain in which one of the links is fixed. Fixing one link provides a reference frame, allowing the other links to move in a predictable, constrained manner to transmit or transform motion. For example, in a four-bar chain, if one link is bolted to the ground, it becomes a four-bar mechanism.
3. **Analysis of Options**:
- **Option (a)**: If no links are fixed, it remains a kinematic chain (or a structure if it lacks degrees of freedom), but it is not yet a mechanism because there is no reference for absolute motion.
- **Option (b)**: This is the correct definition. Fixing one link converts the chain into a mechanism.
- **Option (c)**: Fixing two or more links typically over-constrains the system, often turning it into a structure (where no relative motion is possible) rather than a mechanism.
- **Option (d)**: If all links are fixed, there is no motion, and it is simply a rigid structure.
In transverse fillet welded joint, the size of weld is equal to
A)
0.5 x Throat of weld
B)
Throat of weld
C)
V2 x Throat of weld
D)
2 Throat of weld
In a fillet weld, the 'size of weld' (s) is the length of the leg of the weld triangle. The 'throat of weld' (t) is the minimum distance from the root of the weld to its face, which represents the weakest section of the weld. For a standard fillet weld with a 45-degree angle (isosceles right triangle), the relationship between the size (s) and the throat (t) is given by the formula: t = s * sin(45°) = s / √2. Rearranging this formula to solve for the size of the weld (s), we get: s = t * √2. Therefore, the size of the weld is √2 times the throat of the weld. Option (c) 'V2 x Throat of weld' is a typographical representation of √2 * Throat of weld.
In the disc clutch, the flywheel of the engine plays the role of-
A)
driven plate
B)
driving plate
C)
pressure plate
D)
idle plate
In a disc clutch system, the purpose is to transmit torque from the engine to the transmission system. The flywheel is rigidly attached to the crankshaft of the engine; therefore, it is the component that provides the initial rotational energy (the input). In mechanical terms, the component that provides the power to be transmitted is called the 'driving member' or 'driving plate'. The clutch plate (friction disc) acts as the driven member, which then transmits that power to the gearbox.
Breakdown of options:
- a. Driven plate: This is typically the friction disc that receives power from the flywheel.
- b. Driving plate: Correct. The flywheel acts as the driving member because it is connected to the engine's power source.
- c. Pressure plate: This is the component that clamps the friction disc against the flywheel to ensure torque transmission; it is not the source of power.
- d. Idle plate: This is not a standard term used to describe the primary functional components of a disc clutch.
Which of the following cotter joint is used to connect strap end of a connecting rod?
A)
Socket and spigot cotter joint
B)
Sleeve and cotter joint
C)
Gib and cotter joint
D)
None of these
A cotter joint is a temporary fastening used to connect two co-axial rods subjected to axial tensile or compressive loads.
1. **Socket and Spigot Cotter Joint**: This joint consists of a spigot (a male end) and a socket (a female end). It is specifically designed for applications where one rod is inserted into another. In an internal combustion engine, the strap end (small end) of the connecting rod is often connected to the gudgeon pin using a socket and spigot arrangement to ensure a secure, axial connection that can be disassembled for maintenance.
2. **Sleeve and Cotter Joint**: This is typically used to connect two coaxial rods where neither rod is designed to fit inside the other; instead, both fit into a separate sleeve. This is not the standard configuration for a connecting rod strap end.
3. **Gib and Cotter Joint**: This joint is used when the connection needs to be adjustable or when the load is applied in a way that requires a wedge-like action (e.g., in lathe tool posts or guide rails). It is not used for the axial connection of a connecting rod.
Therefore, the socket and spigot cotter joint is the correct choice for connecting the strap end of a connecting rod.
When engine torque is more than mean resisting torque, then the flywheel has-
A)
zero velocity
B)
uniform velocity
C)
acceleration
D)
retardation
The dynamics of a flywheel are governed by the principle of conservation of energy and Newton's Second Law for rotation. The fundamental equation for the torque acting on a flywheel is:
Net Torque (T_net) = Engine Torque (T_e) - Resisting Torque (T_r)
According to the rotational analogue of Newton's Second Law:
T_net = I * α
where 'I' is the moment of inertia of the flywheel and 'α' is the angular acceleration.
1. If Engine Torque (T_e) > Mean Resisting Torque (T_r), the net torque is positive (T_net > 0). This results in a positive angular acceleration (α > 0), meaning the flywheel speeds up. Therefore, the flywheel has acceleration.
2. If Engine Torque (T_e) < Mean Resisting Torque (T_r), the net torque is negative, leading to angular retardation (slowing down).
3. If Engine Torque (T_e) = Mean Resisting Torque (T_r), the net torque is zero, and the flywheel maintains a uniform velocity (constant speed).
Since the question specifies that engine torque is more than mean resisting torque, the flywheel must accelerate.
If a composite bar is cooled, then the nature of stress in the part with high coefficient of thermal expansion will be
A)
tensile
B)
zero
C)
compressive
D)
none of these
In a composite bar consisting of two different materials bonded together, the materials are forced to contract or expand by the same amount due to the bond.
1. **Thermal Expansion/Contraction**: The change in length due to temperature change is given by $\Delta L = L \alpha \Delta T$, where $\alpha$ is the coefficient of thermal expansion.
2. **Cooling Process**: When the composite bar is cooled ($\Delta T$ is negative), both materials want to contract.
3. **Differential Contraction**: The material with the higher coefficient of thermal expansion ($\alpha_{high}$) wants to contract more than the material with the lower coefficient ($\alpha_{low}$).
4. **Constraint**: Because they are bonded, the material with the lower $\alpha$ prevents the material with the higher $\alpha$ from contracting fully.
5. **Resulting Stress**: Since the material with the higher $\alpha$ is being 'pulled' or prevented from shrinking to its natural cooled length, it is placed under tension. Conversely, the material with the lower $\alpha$ is 'pushed' or compressed by the other material, placing it under compressive stress.
Therefore, the part with the high coefficient of thermal expansion will experience tensile stress.
T1 = tension on tight-side, T2 = tension on slack side, @= angle of lap of the belt around pulley, f= coefficient of friction between belt and pulley. Then for flat belt drive
A)
T1/T2 = e^(f@)
B)
T1/T2 = e^-f@
C)
T1/T2 = e^(f/ @)
D)
T1/T2 = e^( @/f)
In the analysis of a flat belt drive, the relationship between the tensions on the tight side (T1) and the slack side (T2) is governed by the belt friction equation. This is derived by considering a small element of the belt and balancing the radial forces (centripetal/normal) and tangential forces (friction and tension). The fundamental formula for a flat belt is given by: T1/T2 = e^(fθ), where 'e' is the base of the natural logarithm, 'f' is the coefficient of friction, and 'θ' (represented as @ in the question) is the angle of lap or wrap in radians.
Analysis of options:
- Option (a) T1/T2 = e^(f@) is the standard derived formula for belt drives.
- Option (b) T1/T2 = e^-f@ would imply that the slack side tension is greater than the tight side tension, which is physically impossible for a driving pulley.
- Options (c) and (d) incorrectly place the friction coefficient or angle in the denominator of the exponent, which does not align with the derivation of the Capstan equation/Belt friction law.
The effect of the mass of the spring is accounted for by adding X times its value to the mass for calculating the natural frequency of the spring-mass system-
A)
1/2
B)
1/3
C)
2/3
D)
3/4
In a basic spring-mass system, the spring is often assumed to be massless. However, in real-world applications, the spring itself has a mass (m_s). Because the parts of the spring move at different velocities (the end attached to the mass moves at maximum velocity, while the end attached to the fixed support does not move), the entire mass of the spring does not contribute equally to the kinetic energy of the system.
Using the energy method (Rayleigh's Method), the kinetic energy of the spring is integrated along its length. For a linear spring, the effective mass contribution of the spring to the system's total mass is found to be 1/3 of its actual mass.
Therefore, the equivalent mass (m_eq) used to calculate the natural frequency (ω_n = √(k/m_eq)) is:
m_eq = M + (1/3)m_s
where M is the mass of the attached load and m_s is the mass of the spring.
Comparing this to the question, X = 1/3.
A solid shaft transmits 100 kW at 150 rpm. What will be the diameter of the required shaft to transmit the power at the torque exceeding 20% of the mean torque in each revolution? (Given tau = 60 MPa)
A)
85mm
B)
86mm
C)
85.6mm
D)
86.6mm
Step-by-step solution:
1. **Identify Given Data:**
- Power (P) = 100 kW = 100,000 W
- Speed (N) = 150 rpm
- Allowable Shear Stress (τ) = 60 MPa = 60 N/mm²
- Maximum Torque (T_max) = Mean Torque (T_mean) + 20% of T_mean = 1.2 * T_mean
2. **Calculate Mean Torque (T_mean):**
The formula for power is P = (2πNT) / 60
T_mean = (P * 60) / (2 * π * N)
T_mean = (100,000 * 60) / (2 * π * 150)
T_mean = 6,000,000 / 942.477
T_mean ≈ 6366.2 N-m = 6,366,200 N-mm
3. **Calculate Maximum Torque (T_max):**
The problem states the torque exceeds the mean torque by 20%.
T_max = 1.2 * T_mean
T_max = 1.2 * 6,366,200
T_max = 7,639,440 N-mm
4. **Calculate Shaft Diameter (d):**
For a solid shaft, the torsion formula is T = (π/16) * τ * d³
d³ = (16 * T_max) / (π * τ)
d³ = (16 * 7,639,440) / (π * 60)
d³ = 122,231,040 / 188.495
d³ ≈ 648,455.5
d = (648,455.5)^(1/3)
d ≈ 86.55 mm
Rounding to the nearest option provided, the diameter is approximately 86.6 mm.
Unconstrained rigid link in a plane has _
A)
one degree of freedom
B)
two degrees of freedom
C)
three degrees of freedom
D)
zero degree of freedom
Degree of Freedom (DOF) refers to the number of independent coordinates required to completely specify the position and orientation of a body in space. For a rigid body moving in a 2D plane (planar motion), the body can undergo three independent types of motion: 1) Translation along the X-axis, 2) Translation along the Y-axis, and 3) Rotation about an axis perpendicular to the plane (Z-axis). Since the link is 'unconstrained', there are no joints or supports restricting its movement. Therefore, it requires three independent parameters (e.g., the coordinates (x, y) of one point on the link and the angle θ of rotation) to define its position. Thus, an unconstrained rigid link in a plane has 3 degrees of freedom.
The endurance or fatigue limit is defined as the maximum value of the stress which a polished standard specimen can withstand without failure, for infinite number of cycles, when subjected to
A)
static load
B)
dynamic load
C)
static as well as dynamic load
D)
completely reversed load
The endurance limit (or fatigue limit) is a fundamental concept in machine design and materials science. It is defined as the maximum stress level below which a material can withstand an infinite number of loading cycles without failure. By standard definition (such as in the ASTM or ISO standards for fatigue testing), this limit is determined using a polished specimen subjected to a 'completely reversed load'. A completely reversed load is one where the maximum stress is equal in magnitude to the minimum stress but opposite in sign (i.e., the stress ratio R = -1, and the mean stress is zero).
Why other options are incorrect:
- (a) Static load: Static loads are used to determine the yield strength or ultimate tensile strength, not the fatigue limit.
- (b) Dynamic load: While fatigue is caused by dynamic loading, 'dynamic load' is a broad term. The specific standard test for the endurance limit requires a symmetric, completely reversed cycle.
- (c) Static as well as dynamic load: This describes combined loading or fatigue with a mean stress (fluctuating stress), which is analyzed using modified diagrams (like Goodman or Soderberg), but the baseline endurance limit itself is defined based on the reversed load case.
For circular arc cam with flat footed follower in contact with the flank, the displacement of follower is where R = flank radius, r = base circle radius, 9 = angle of rotation of cam.
A)
x=(R+nr) (1 - cos)
B)
x=(R-1) (1 - cos@)
C)
x=(R+1r) (1 + cos)
D)
x=(R-r) (1+ cos@)
In the context of a circular arc cam with a flat-footed follower, the displacement (lift) of the follower is determined by the geometry of the arc and the base circle. Let R be the radius of the arc (flank radius) and r be the radius of the base circle. When the cam rotates through an angle θ, the distance from the center of the cam to the contact point changes. For a flat-footed follower, the displacement x is the difference between the current distance from the center to the follower face and the base circle radius. Using the law of cosines or geometric projection for a circular arc profile, the displacement is typically expressed as x = (R + r) (1 - cosθ) or a variation depending on the specific offset. Looking at the provided options, option (a) 'x=(R+nr) (1 - cos)' contains a typographical error where 'n' likely represents a coefficient or is a misprint for the base radius relation, but it follows the standard mathematical form for circular arc displacement (1 - cosθ). Options (b), (c), and (d) are mathematically incorrect for a standard lift profile: (b) uses (R-1), (c) uses (1 + cosθ) which would imply the follower starts at maximum lift and moves down, and (d) also uses (1 + cosθ). Therefore, based on the standard kinematic equation for circular arc cams, option (a) is the only one that correctly represents the 'lift' behavior (starting at 0 when θ=0).
The graphical representation of maximum principal strain theory has a shape of —
A)
Circle
B)
Square
C)
Rhombus
D)
Rectangle
The Maximum Principal Strain Theory (also known as Saint-Venant's Theory) states that failure occurs when the maximum principal strain reaches the strain at the yield point in a simple tension test. Mathematically, for a biaxial stress state, the condition for failure is: ε₁ = ε₀ (where ε₀ is the strain at yield). In terms of stresses, this is expressed as: σ₁ - μ(σ₂ + σ₃) = σ_y. When plotted on a 2D graph of principal stresses (σ₁ vs σ₂), the boundary for failure forms a square shape. This is because the limits for the principal stresses are independent and bounded by the yield strength of the material. In contrast, the Maximum Shear Stress Theory (Tresca) results in a hexagon, and the Maximum Distortion Energy Theory (von Mises) results in an ellipse/circle.