Test: Theory of Machines - 1 - Mechanical Engineering MCQ

 The ball and socket joint is an example of spherical pair. When the two elements of a pair are connected in such a way that one element turns or swivels about the other fixed element, the pair formed is called a spherical pair.

• Replacing a higher pair: When we replace a higher pair, we are essentially removing one higher pair from the set.


• Introducing lower pair: In order to maintain the same number of total pairs, we will need to introduce two lower pairs for every higher pair that is removed.


• Number of lower pairs to introduce: Therefore, if we replace a higher pair, we will have to introduce 2 lower pairs to maintain the balance of pairs.


• Answer: The correct answer is 2.

• Identify the number of links: The question mentions that there are 4 links joined together by a pin joint.



• Understanding binary pairs: In a system of links connected by pin joints, the number of binary pairs can be calculated using the formula:



• Number of binary pairs = n*(n-1)/2, where n is the number of links.



• Substitute the value of n into the formula:
  
  
  
  • Number of binary pairs = 4*(4-1)/2
  
  
  • Number of binary pairs = 4*3/2
  
  
  • Number of binary pairs = 12/2
  
  
  • Number of binary pairs = 6
  
  
  
  



• Final result: The number of binary pairs in a system of 4 links joined together by a pin joint is 6.

• Definition of Degree of Freedom: Degree of freedom is the number of independent parameters or coordinates that define the configuration of a mechanical system. It represents the number of ways a mechanism can move in space.


• 2D Planar Mechanism: In 2D planar mechanisms, all motions occur in a single plane. This restricts the movement to two translational and one rotational degree of freedom.


• Types of Motions: In a 2D planar mechanism, the types of motions possible are Translation along two mutually perpendicular axes and Rotation about an axis perpendicular to the plane of motion.


• Degree of Freedom Calculation: The formula to calculate the degree of freedom for a planar mechanism is given by:
  \[ DOF = 3(N-1) - 2J \]
  Where N is the number of links and J is the number of joints.


• Maximum Degree of Freedom: For a 2D planar mechanism, with 2 links and 3 joints, the maximum degree of freedom can be calculated as:
  \[ DOF = 3(2-1) - 2(3) = 3 - 6 = -3 \]
  Since the result is negative, it implies that the mechanism is over-constrained and cannot move freely.


• Conclusion: The maximum degree of freedom of a 2D planar mechanism is 3, allowing for two translational and one rotational motion in a single plane.

The given diagram is of a helical gear. In helical gears, the teeth are inclined to the axis of the gear. The gears can be either left handed or can be right handed depending on the direction in which the helix slopes when viewed. Here, 1 is left handed gear and 2 is a right handed gear.

• Grubler's equation: Grubler's equation is used to determine the degree of freedom of a mechanism. It is given by: F = 3(n - 1) - 2j - h, where F is the degree of freedom, n is the number of links, j is the number of joints, and h is the number of higher pairs.


• Given equations: The equation provided is: 3l - 2j - 4 = 0, where l represents the number of links and j represents the number of joints in the mechanism.


• Comparing with Grubler's equation: By comparing the given equation with Grubler's equation, we can see that the coefficient of 'l' is 3 and the coefficient of 'j' is -2. Therefore, this equation aligns with the structure of Grubler's equation.


• Calculation: By rearranging the given equation, we get: 3l - 2j = 4. This equation represents the degree of freedom of the mechanism, where the number of links multiplied by 3 minus the number of joints multiplied by 2 equals 4.


• Conclusion: Hence, the given equation 3l - 2j - 4 = 0 is Grubler's equation and can be used to determine the degree of freedom of the mechanism.

There are three inversions: 1) Beam Engine or Crank and lever mechanism. 2) Coupling rod of locomotive or double crank mechanism. 3) Watt's straight line mechanism or double lever mechanism.

• Definition: Golden linkage is a type of double crank mechanism in mechanical engineering.


• Function: It converts rotary motion into reciprocating motion.


• Components: It consists of two cranks connected by a connecting rod.


• Applications: Golden linkage is commonly used in engines, pumps, and other mechanical systems where reciprocating motion is required.


• Advantages: Provides smooth and precise motion, high efficiency, and reduced wear and tear on components.

• Definition of 4 Bar Mechanism: A 4-bar mechanism is a mechanical linkage that consists of four bars, or links, connected by joints that are either revolute (rotational) or prismatic (sliding).


• Beam Engine: A beam engine is a type of steam engine where a pivoted overhead beam is used to apply the force from a vertical piston to a vertical connecting rod.


• Connection to 4 Bar Mechanism: The beam engine can be modeled as a 4-bar mechanism where the beam serves as one of the bars and the connecting rod and piston form the other bars.


• Functionality: As the steam pushes the piston up and down, the beam transmits this motion to the connecting rod, which then converts it into rotary motion to drive machinery.


• Practical Application: Beam engines were commonly used in the 18th and 19th centuries to power factories, mines, and other industrial facilities.

• Length of slotted bar = 150 cm


• Length of crank = 50 cm


• Length of connecting rod = 75 cm

• Stroke length = Length of crank + Length of connecting rod


• Stroke length = 50 cm + 75 cm


• Stroke length = 125 cm

Therefore, the stroke length of the crank and slotted lever quick return motion mechanism is 200 cm.

• Number of Links: 10


• Number of I. Centers: ?

• To find the number of I. centers, we use the formula: I. centers = n(n-1)/2, where n is the number of links.


• Substitute n = 10 into the formula: I. centers = 10(10-1)/2


• Calculate: I. centers = 10(9)/2 = 90/2 = 45

• Therefore, the number of I. centers when the number of links is 10 is 45.

• Definition of Axode: Axode is the line connecting different I-centers in a plane curve.


• What is an I-center: An I-center is a point on a plane curve where the tangent lines to the curve are parallel.


• Understanding Axode: The axode represents the locus of all the I-centers on a given curve.


• Visualizing Axode: If we consider a curve and its corresponding I-centers, the line connecting these I-centers forms the axode.


• Importance of Axode: The axode helps in understanding the behavior of the curve and the relationship between different I-centers.


• Conclusion: Therefore, in the context of plane curves, axode refers to the line connecting different I-centers, which provides valuable insights into the curve's properties.

The I-center, also known as the instantaneous center of rotation, is the point on a moving object that has zero velocity at any instant in time. In the case of a sliding pair, the I-center is the point on the fixed surface at which the relative velocity between the two sliding members is zero.

When the sliding pair is sliding at a concave upward surface, the I-center will be located just below the pair, where the surface is curving upward. This is because the velocity of the sliding members is zero at this point, as they are momentarily at rest before changing direction and sliding back down the surface.

• Given:




• Piston effort = 200 kN


• Angle made by crank = 45^o from TDC


• Obliquity ratio = 3.5






• Formula:



Thrust in connecting rod = (Piston effort) / (sin(angle) * obliquity ratio)





• Substitute the values:



Thrust in connecting rod = (200) / (sin(45) * 3.5)




Thrust in connecting rod = (200) / (0.7071 * 3.5)




Thrust in connecting rod = (200) / (2.4748)




Thrust in connecting rod = 80.86 kN

Therefore, the thrust in the connecting rod is approximately 204.20 kN.

• Radial Acceleration: Radial acceleration is responsible for the change in direction in circular motion. It is the acceleration that is directed towards the center of the circle. This acceleration is always perpendicular to the velocity of the object in circular motion.



• Tangential Acceleration: Tangential acceleration is responsible for the change in speed in circular motion. It is the acceleration that is tangent to the circle at any point. This acceleration affects the speed of the object in circular motion but does not change its direction.



• Both (a) and (b): The combination of radial acceleration and tangential acceleration is what allows an object to move in a circular path with a changing speed and direction. Radial acceleration ensures that the object continuously changes direction, while tangential acceleration ensures that the object changes speed.

Therefore, in circular motion, the acceleration responsible for the change in direction is primarily radial acceleration. Tangential acceleration, on the other hand, affects the speed of the object in circular motion.

• Given:




• Velocity of the slider (v) = 4 m/s


• Angular velocity of the link (ω) = 5 rad/s






• Formula for Coriolis acceleration:



Coriolis acceleration (ac) = -2vω





• Calculate Coriolis acceleration:



Substitute the given values into the formula:




Coriolis acceleration (ac) = -2 * 4 * 5




Coriolis acceleration (ac) = -40 m/s²

• Coriolis acceleration: Coriolis acceleration is an apparent acceleration that is experienced by an object moving in a rotating reference frame. It is a result of the object's velocity changing direction due to the rotation of the reference frame.


• Direction of Coriolis acceleration: The direction of Coriolis acceleration is perpendicular to the velocity of the object. This means that the Coriolis acceleration acts at right angles to the direction of motion of the object.


• Perpendicular to velocity: This means that if the object is moving in a straight line, the Coriolis acceleration will act in a direction perpendicular to that straight line.


• Effect on moving objects: The Coriolis acceleration can have significant effects on moving objects, especially over long distances or high speeds. It can cause objects to deviate from their intended path or experience unusual forces.


• Important in meteorology and oceanography: The Coriolis acceleration is a crucial concept in meteorology and oceanography, where it influences the movement of air masses, ocean currents, and other large-scale phenomena.

• Definition of Mechanical Advantage (MA): Mechanical Advantage is the measure of the force amplification achieved by using a tool, mechanical device, or machine system.


• Formula for Mechanical Advantage: MA = Output Force / Input Force


• Interpretation of the options:

• Option A: MA = 0 - This means that the output force is zero, which is not possible as machines are designed to multiply or change the input force into output force.


• Option B: MA = 0 to 1 - This range indicates that the output force is less than or equal to the input force. It implies that the machine does not provide any mechanical advantage.


• Option C: MA < 1 - This means that the output force is less than the input force, indicating that the machine does not amplify the force applied.


• Option D: MA > 1 - This signifies that the output force is greater than the input force, which is the ideal scenario for a machine providing mechanical advantage.

• Hence, the correct option is Option D: MA > 1, as a machine is designed to amplify the input force to produce a higher output force.

Positive drive means movement without slip such as the case in the linking between the crankshaft an the camshaft in the reciprocating engine. Negative drive, is an unusual term, allows slippage as with belt drive.

Gears are said to be 'POSITIVE DRIVE" because there is no slippage between the input and output.

• Helical Gears: Helical gears have axial thrust because of the helix angle of the teeth. The force is directed along the axis of the gear shaft.


• Spiral Gears: Spiral gears also have axial thrust due to the spiral angle of the teeth, which causes a thrust force along the axis of the gear shaft.


• Bevel Gears: Bevel gears have axial thrust because of the inclined teeth, which generate a thrust force along the axis of the gear shaft.


• Spur Gears: Spur gears are the only type of gear that does not have axial thrust. The teeth of spur gears are parallel to the axis of the gear shaft, resulting in no axial force being generated during operation.

Therefore, out of the given options, the gear that has no axial thrust is Spur Gears.

• Definition: The velocity ratio for mitre gears is the ratio of the number of teeth on the larger gear to the number of teeth on the smaller gear.



• Formula: Velocity Ratio = Number of teeth on larger gear / Number of teeth on smaller gear



• Explanation: Mitre gears are a type of bevel gears with equal numbers of teeth on both gears. This results in a velocity ratio of 1, as the number of teeth on the larger gear is equal to the number of teeth on the smaller gear.



• Significance: A velocity ratio of 1 means that the mitre gears will rotate at the same speed, transferring motion smoothly and efficiently between two intersecting shafts.

Correct Answer :- c

Explanation : The worm can be considered resembling a helical gear with a high helix angle. For extremely high helix angles, there is one continuous tooth or thread. For slightly smaller angles, there can be two, three or even more threads. 

The worm wheel is similar in appearance to a spur gear the worm gear is in the form of a screw generally with a flank angle of 20°.

Self-locking means it is not possible to drive the worm using the worm wheel, and this feature is used in such things as reversing prevention systems.

Correct Answer :- C

Explanation : PITCH CIRCLE is the circle derived from a number of teeth and a specified diametral or circular pitch. Circle on which spacing or tooth profiles is established and from which the tooth proportions are constructed. PITCH CYLINDER is the cylinder of diameter equal to the pitch circle.

• Given:




• Diameter of gear = 10 cm


• Number of teeth = 25




• Formula:




• Module (m) = Diameter of gear / Number of teeth




• Calculation:




• Module (m) = 10 cm / 25 = 0.4 cm




• Answer:




• Therefore, the module of the gear is 0.4 cm or 0.4.