THE CONCEPT OF MOMENT:

Moment of force (or moment) is the tendency of a force to twist or rotate an object. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the force and the moment arm. The moment arm is the perpendicular distance from the point of rotation, to the line of action of the force. The moment may be thought of as a measure of the tendency of the force to cause rotation about an imaginary axis through a point. (Note: In mechanical and civil engineering, "moment" and "torque" have different meanings, while in physics they are synonyms.)


The moment of a force can be calculated about any point and not just the points in which the line of action of the force is perpendicular. Image A shows the components, the force F, and the moment arm, x when they are perpendicular to one another. When the force is not perpendicular to the point of interest, such as Point O in Images B and C, the magnitude of the Moment, M of a vector F about the point O is

where

is the vector from point O to the position where quantity F is applied.

× represents the cross product of the vectors.

[In the figure a moment at Point O, when the components are perpendicular to the Point O. Image B and Image C illustrate the components of a Moment at Point O, when the components are not perpendicular to point O.]

In mechanical engineering (unlike physics), the terms "torque" and "moment" are not interchangeable. "Moment" is the general term for the tendency of one or more applied forces to rotate an object about an axis (the concept which in physics is called torque). "Torque" is a special case of this: If the applied force vectors add to zero (i.e., their "resultant" is zero), then the forces are called a "couple" and their moment is called a "torque".
For example, a rotational force down a shaft, such as a turning screw-driver, forms a couple, so the resulting moment is called a "torque". By contrast, a lateral force on a beam produces a moment (called a bending moment), but since the net force is nonzero, this bending moment is not called a "torque".

A particle is located at position r relative to its axis of rotation. When a force F is applied to the particle, only the perpendicular component F⊥ produces a torque. This torque τ = r × F has magnitude τ = |r|  |F⊥| = |r| |F| sinθ and is directed outward from the page.

A Couple is a system of forces with a resultant (a.k.a. net, or sum) moment but no resultant force. Another term for a couple is a pure moment. Its effect is to create rotation without translation, or more generally without any acceleration of the centre of mass.

The resultant moment of a couple is called a torque. This is not to be confused with the term torque as it is used in physics, where it is merely a synonym of moment. Instead, torque is a special case of moment. Torque has special properties that moment does not have, in particular the property of being independent of reference point.

Simple Couple:

The simplest kind of couple consists of two equal and opposite forces whose lines of action do not coincide. This is called a "simple couple". The forces have a turning effect or moment called a torque about an axis which is normal to the plane of the forces. The SI unit for the torque of the couple is newton metre.
If the two forces are F and −F, then the magnitude of the torque is given by the following formula:

where

τ is the torque

F is the magnitude of one of the forces

d is the perpendicular distance between the forces, sometimes called the arm of the couple

The magnitude of the torque is always equal to Fd, with the direction of the torque given by the unit vector , which is perpendicular to the plane containing the two forces. When d is taken as a vector between the points of action of the forces, then the couple is the cross product of d and F.

  
Independence of reference point:

The moment of a force is only defined with respect to a certain point P (it is said to be the "moment about P"), and in general when P is changed, the moment changes. However, the moment (torque) of a couple is independent of the reference point P: Any point will give the same moment.In other words, a torque vector, unlike any other moment vector, is a "free vector". (This fact is called Varignon's Second Moment Theorem.)









The proof of this claim is as follows: Suppose there are a set of force vectors F₁, F₂, etc. that form a couple, with position vectors (about some origin P) r₁, r₂, etc., respectively. The moment about P is

Now we pick a new reference point P' that differs from P by the vector r. The new moment is

Now the distributive property of the cross product implies

However, the definition of a force couple means that

Therefore,

This proves that the moment is independent of reference point, which is proof that a couple is a free vector.

2D FORCE ANALYSIS : HOW TO FIND REACTIONS IN A CASE OF CONCURRENT FORCE SYSTEM ACTING ON A BODY IS IN EQUILIBRIUM

DEFINITION : CONCURRENT FORCE SYSTEM

If the lines of actions of all the forces in a force system pass through a common point, then the force system is called as Concurrent Force System. The equilibrium conditions for a concurrent force system is

ΣF_x = 0 and   ΣF_y = 0

 

The steps to find out reactions when a coplanar concurrent force system acting on a body in equilibrium condition :

 

 

STEP 1 :

 

(i) Draw the diagram and identify all the contact points the body makes with other bodies including ground.

(ii) Draw a tangent at each contact point with the object. These tangents are called Contact Surfaces.

(iii) Draw a perpendicular to the contact surface at each and every contact points. These perpendiculars will be the directions of reactions at each and every contact point.

(iv) Find the angles made by the reactions with respect to horizontal with the help of Geometry.

 

 

STEP 2 :

 

(i) Draw the Free Body Diagram (FBD) that consists of the external forces acting on the object. (applied forces, forces of gravity and reactions all are external forces)

(ii) Assign reactions by symbols like R₁, R₂ ....... and resolve all the external forces along X-axis and Y-axis.

(iii) Now add all the horizontal component forces as ΣF_x and put ΣF_x = 0 ---- eqn (1)

and add all the vertical component forces as ΣF_y and put ΣF_y = 0 --------eqn (2)

(iv) Solving these two equations we shall get values of  R₁, R_2.

 

FORCE: THE CAUSE OF ANY KIND OF CHANGE IN THE UNIVERSE

                           "When a student is introduced to the concept of force for the first time, the student would understand in a better way if we define force in formal way by explaining Mechanical Force which applied on an object produces or tends to produce certain kind of motion. Similarly, it can oppose a motion and thus it can a moving body to a halt."


                             In physics, a force is any influence that causes a free body to undergo an acceleration. Force can also be described by intuitive concepts such as a push or pull that can cause an object with mass to change its velocity (which includes to begin moving from a state of rest), i.e., to accelerate, or which can cause a flexible object to deform.

                            A force has both magnitude and direction, making it a vector quantity. Newton's second law, F=ma, can be formulated to state that an object with a constant mass will accelerate in proportion to the net force acting upon and in inversely  proportional to its mass, an approximation which breaks down near the speed of light.

                             Newton's original formulation is exact, and does not break down: this version states that the net force acting upon an object is equal to the rate at which its momentum changes.

                             Related concepts to accelerating forces include thrust, increasing the velocity of the object, drag, decreasing the velocity of any object, and torque, causing changes in rotational speed about an axis.

                            Forces which do not act uniformly on all parts of a body will also cause mechanical stresses, a technical term for influences which cause deformation of matter. While mechanical stress can remain embedded in a solid object, gradually deforming it, mechanical stress in a fluid determines changes in its pressure and volume.

                Newton's first law of Motion:

                    Newton's first law of motion states that objects continue to move in a state of constant velocity unless acted upon by an external net force or resultant force.

                   This law is an extension of Galileo's insight that constant velocity was associated with a lack of net force (see a more detailed description of this below). Newton proposed that every object with mass has an innate inertia that functions as the fundamental equilibrium "natural state" in place of the Aristotelian idea of the "natural state of rest". That is, the first law contradicts the intuitive Aristotelian belief that a net force is required to keep an object moving with constant velocity. By making rest physically indistinguishable from non-zero constant velocity, Newton's first law directly connects inertia with the concept of relative velocities. Specifically, in systems where objects are moving with different velocities, it is impossible to determine which object is "in motion" and which object is "at rest". In other words, to phrase matters more technically, the laws of physics are the same in every inertial frame of reference, that is, in all frames related by a Galilean transformation.



                         For example, while traveling in a moving vehicle at a constant velocity, the laws of physics do not change from being at rest. A person can throw a ball straight up in the air and catch it as it falls down without worrying about applying a force in the direction the vehicle is moving. This is true even though another person who is observing the moving vehicle pass by also observes the ball follow a curving parabolic path in the same direction as the motion of the vehicle. It is the inertia of the ball associated with its constant velocity in the direction of the vehicle's motion that ensures the ball continues to move forward even as it is thrown up and falls back down. From the perspective of the person in the car, the vehicle and every thing inside of it is at rest: It is the outside world that is moving with a constant speed in the opposite direction. Since there is no experiment that can distinguish whether it is the vehicle that is at rest or the outside world that is at rest, the two situations are considered to be physically indistinguishable. Inertia therefore applies equally well to constant velocity motion as it does to rest.



                        The concept of inertia can be further generalized to explain the tendency of objects to continue in many different forms of constant motion, even those that are not strictly constant velocity. The rotational inertia of planet Earth is what fixes the constancy of the length of a day and the length of a year.

                        Albert Einstein extended the principle of inertia further when he explained that reference frames subject to constant acceleration, such as those free-falling toward a gravitating object, were physically equivalent to inertial reference frames. This is why, for example, astronauts experience weightlessness when in free-fall orbit around the Earth, and why Newton's Laws of Motion are more easily discernible in such environments.

                       If an astronaut places an object with mass in mid-air next to him/herself, it will remain stationary with respect to the astronaut due to its inertia. This is the same thing that would occur if the astronaut and the object were in intergalactic space with no net force of gravity acting on their shared reference frame. This principle of equivalence was one of the foundational underpinnings for the development of the general theory of relativity.


Newton's second law of Motion:

A modern statement of Newton's second law is a vector differential equation:
where p is the momentum of the system, and is the F net (vector sum) force. In equilibrium, there is zero net force by definition, but (balanced) forces may be present nevertheless. In contrast, the second law states an unbalanced force acting on an object will result in the object's momentum changing over time. 


                            By the definition of momentum,  p = mV ; where m is the mass and V is the velocity. In a system of constant mass, the use of the constant factor rule in differentiation allows the mass to move outside the derivative operator, and the equation becomes F = ma ; where m = mass of the body and a= acceleration of the body.

                           
The

SOLUTION OF PRESEMESTER EXAMINATION: ENGINEERING MECHANICS SOLUTION OF PRESEMESTER EXAMINATION: ENGINEERING MECHANICS©subhankar_karmakar

GROUP-A Q.1 Answer the following questions as per the instructions 2x20=20 Choose the correct answer of the following questions: (i) A truss hinged at one end, supported on rollers at the other, is subjected to horizontals load only. Its reaction at the hinged end will be (a) Horizontal;                   (b) Vertical (c) Both horizontal & vertical (d) None of the above. Ans: (c) both horizontal & vertical (ii) The moment of inertia of a circular section of diameter D about its centroidal axis is given by the expression (a) π(D)4/16           (b) π(D)4/32 (c) π(D)4/64           (d) π(D)4/4 Ans: (c) π(D)4/64 Fill in the blanks in the following questions: (iii)The distance of the centroid of an equilateral triangle with each side(a) is …………. from any of the three sides. Ans a /(2√3) (iv)Poisson’s ratio is defined as the ratio between ……. and ………… . Ans: Lateral Strain, Longitudinal Strain (v)If two forces of equal magnitudes P having an angle 2Ө between them, then their resultant force will be equal to ________ . Ans: 2P CosӨ Choose the correct word/s. (vi) Two equal and opposite force acting at different points of a rigid body is termed as (Bending Moment/ Torque/ Couple). Ans: Couple

Choose correct answer for the following parts: (vii) Statement 1: In stress strain graph of a ductile material, yield point starts at the end of the elastic limit. Statement 2: At yielding point, the deformation becomes plastic by nature. (a) Statement 1 is true, Statement 2 is true. (b) Statement 1 is true, Statement 2 is true and they are unrelated with each other (c) Statement 1 is true, statement 2 is false. (d) Statement 1 is false, Statement 2 is false. Ans: (a) Statement 1 is true, Statement 2 is true. (viii) Statement 1: It is easier to pull a body on a rough surface than to push the body on the same surface. Statement 2: Frictional force always depends upon the magnitude of the normal force. (a) Statement 1 is true, Statement 2 is true. (b) Statement 1 is true, Statement 2 is true and they are unrelated (c) Statement 1 is true, statement 2 is false. (d) Statement 1 is false, Statement 2 is false. Ans: (a) Statement 1 is true, Statement 2 is true. (ix) In a cantilever bending moment is maximum at (a) free end            (b) fixed end (c) at the mid span (d) none of these Ans: (b) fixed end (x) The relationship between linear velocity and angular velocity of a cycle (a) exists under all conditions (b) does not exist under all conditions (c) exists only when it moves on horizontal plane. (d) none of these Ans: (a) exists under all conditions

NEXT ARTICLE

SOLUTION OF PRESEMESTER EXAMINATION: ENGINEERING MECHANICS PART-II ©subhankar_karmakar

QUESTION PAPER: ENGINEERING MECHANICS

Vivekanand Institute of Technology & Science; Ghaziabad
PRE-SEMESTER EXAMINATION (odd SEMESTER 2009-10)
B.Tech…first Semester

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Sub Name: Engineering Mechanics Max. Marks: 100
Sub Code: EME-102 Max. Time: 3: 00 Hr

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(i) This paper is in three sections, section A carries 20 marks, section B carries 30 marks and section C carries 50 marks.
(ii) Attempt all the questions. Marks are indicated against each question
(iii) Assume missing data suitably if any.

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Group A

Q.1 Answer the following questions as per the instructions 2x20=20
Choose the correct answer of the following questions:

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(i) The magnitudes of the force of friction between two bodies, one lying above the another depends upon the roughness of the
(a) Upper body;                 (b) Lower body
(c) Both the bodies               (d) The body having more roughness

(ii)The moment of inertia of a circular section of diameter D about its centroidal axis is given by the expression
(a) π(D)⁴/16               (b) π(D)⁴/32
(c) π(D)⁴/64               (d) π(D)⁴/4

Fill in the blanks in the following questions:

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(iii)The distance of the centroid of an equilateral triangle with each side(a) is …………. From any of the three sides.
(iv)Poisson’s ratio is defined as the ratio between ……………………. and
………………………… .
(v)If two forces of equal magnitudes P having an angle 2Ө between them,
then their resultant force will be equal to ________ .

Match the following columns for the following two parts:

(vi) Match the column I to an entry from the column II:

COLUMN – I	COLUMN - II
(i) BMD of an UDL	(a) stored strain energy per unit volume
(ii) Resilience is	(b) brittle materials
(iii) Bulk Modulus	(c) parabolic in nature
(iv) Yield Point	(d) volumetric stress & strain
	(e) Ductile materials
	(f) Shear stress

(vii) Match the Following columns:

COLUMN – I	COLUMN - II
(i) Square of side (b)	(p) π b4 / 64
(ii) Equilateral Triangle of side (b)	(q) b4 / 12
(iii) Circle of diameter (b)	(r) b4/ 36
(iv) Isosceles right angle triangle of base (b)	(s) b4/(32√3)
	(e) Ductile materials
	(f) Shear stress

Column II gives the value of Moment of Inertia Ixx about a centroidal axis.

Choose correct answer for the following parts:

(viii) Statement 1:
In stress strain graph of a ductile material, yield point starts at the end of
the elastic limit.

Statement 2:
At yielding point, the deformation becomes plastic by nature.

(i) Statement 1 is true, Statement 2 is true.
(ii) Statement 1 is true, Statement 2 is true and they are unrelated with
each other
(iii) Statement 1 is true, statement 2 is false.
(iv) Statement 1 is false, Statement 2 is false.

(ix) Statement 1:
It is easier to pull a body on a rough surface than to push the body on
the same surface.

Statement 2:
Frictional force always depends upon the magnitude of the normal force.

(i) Statement 1 is true, Statement 2 is true.
(ii) Statement 1 is true, Statement 2 is true and they are unrelated with
each other
(iii) Statement 1 is true, statement 2 is false.
(iv) Statement 1 is false, Statement 2 is false.

Choose the correct word/s.
(x) Two equal and opposite force acting at different points of a rigid body
is termed as (Bending Moment/ Torque/ Couple).

SECTION-B

Q.2: Answer any three parts of the followings: 10X3=30

(a)	Find the shear force and moment equation for the beam as shown in the figure. Also sketch SFD (shear force diagram) and BMD (bending moment diagram)
(b) Explain and prove “the parallel axis theorem of moment of inertia.” Also find the centroid of the following composite area.
(c) Find the centroidal Moment of Inertia of the following shaded area.
	(d) Two cylinders P and Q rest in a channel as shown in fig. below. The cylinder P has a diameter of 100 mm and weighs 200 kN, where as the cylinder Q has a diameter of 180 mm and weighs 500 kN. Find the support reactions at all the point of contact.
(e)	Two blocks A and B of weights 1 kN and 2 kN respectively are in equilibrium as Shown in the figure. If the co-efficient of friction everywhere is 0.3, find the force P required to move the block B.

SECTION C:

(3) Answer any two parts of the following 5X2=10

(a) A simply supported beam of 2 cm wide and 4 cm high having a length 2 m long
in concentrated load of 3 kN (acting perpendicular to the axis of beam) at a point
0.5 m from one of the supports. Determine
(i) the maximum fiber stress (σb max); (ii) the stress in a fiber located at a
distance of 1 cm from the top of the beam at mid-span.

(b) Explain and justify the assumptions taken during analysis of a perfect truss.

(c)A flywheel is making 180 rpm and after 20 second it is running at 120 rpm.
How many revolutions will it make and what time will elapse before it stops, if the retardation is constant.


(4) Answer any one part of the following: 1X10=10

(a) Explain the terms angular momentum and radius of gyration.
A wheel rotates for 5 seconds with a constant angular acceleration and
describes during this time 100 rotations. It then rotates with a constant angular
velocity and during the next five seconds describes 80 radians. Find the initial
angular velocity and the angular acceleration.

(b) Analyse the following truss:



(5) Answer any three question;                           3X10=30

(a) Compare the stress – strain diagrams of a ductile material to that of a brittle Material. Also explain the term bulk modulus and modulus of elasticity.

(b) Explain and prove the “Bending Equation” (M/I)=(б/y)=(E/ρ). What is
called as Section Modulus?

(c) A cylinder of radius R, length L and total mass M is suspended vertically
from the floor, if the modulus of elasticity of the cylinder be E, find the total
deflection and maximum stress induced in the cylinder.

(d) A cylinder of 200 mm diameter is subjected to a twisting moment of
250 kN-m, the length of the cylinders is 1 m, if the modulus of rigidity of the
cylinder material be 150 GPa, find the maximum shear stress induced in the
cylinder. Also find the total angular deformation.

MULTIPLE CHOICES : ENGG MECHANICS Sub: Engineering Mechanics, Sub Code: EME-202, Semester: 2nd Sem, Course: B.Tech,

MULTIPLE CHOICES:
sub: ENGINEERING MECHANICS

Sub Code: EME - 102 /201
B.Tech First Semester (all branch)
University: UPTU (GBTU), Lucknow, Uttar Pradesh
OBJECTIVE TYPE QUESTIONS

PROBLEM SET - 1

Q1) If the coefficients of friction of an inclined plane be (1/√3), then the angle of repose of the plane will be
a) 90° b) 60° c) 45° d)30°

Q2) Three forces of equal magnitude acts along the side of an equilateral triangle, then the body will be in
a) static equilibrium                                 b) dynamic equilibrium
c) translational motion                             d) rotational motion.

Q3) (1/2).E.e² is called
a) total strain energy                                  b) resilience
c) dynamic loading energy                          d) none of this

Q4) The ratio of shear stress and shear strain is known as,
a) modulus of elasticity                                      b) poissons' ratio
c) modulus of rigidity                                         d) bulk modulus

Q5) The bending moment curves generated by UVL is,
a) parabolic                                          b) straight line
c) cubic                                                d) constant

Q6) The centroidal moment of inertia of a quarter circle lamina is
a) 0.11r⁴                                              b) 0.05r⁴
c) 0.4242r⁴                                          d) 0.5868r⁴

Q7) The Point of Contraflexure is a point in the beam where
a) shear force is zero                        
b) bending moment is zero
c) bending moment is zero and it changes sign
d) none of these

Q8) If a body of mass M is moving with an acceleration (a) then the Inertial Force on the body is equals to
a) -Ma          b) Ma        c) |Ma|     d) none of these

Q9) A truss is made of seven linkages and five joints, then the truss is
a) deficient                                              b) redundant
c) perfect                                                d) none of these

Q10) A beam has 3 no.s of supports is known as
a) cantilever beam
b) continuous beam
c) overhanging beam
d) simply supported beam


PROBLEM SET - 2:

Q1) The maximum value of frictional force that comes into play when a body tends to move on a surface called:
a) sliding friction,                                   b) limiting friction,
c) milling friction,                                   d) none of these.

2) The ratio of static friction to dynamic friction is:
a) less than 1,                                      b) equal to 1,
c) greater than 1,                                 d) none of these.

3) The angle of friction is equal to the:
a) ratio of frictional force to the normal reaction.
b) angle of inclined plane when a body tends to slide down.
c) angle of an inclined plane when a body is sliding.
d) none of these.

Q.4) A particle is moving along a circle with constant speed. The acceleration of the particle is:
a. along the circumference                      b. along the tangent
c. along the radius                                 d. zero

Q.5) The area moment of inertia of a quarter circle of radius (r) about the centroidal
axes is
a. (0.055r²)                                        b. (0.11r²)/3
c. (πr²)/4                                           d. π.r²

Q.6) The centroid of a semi circular area of radius r is
a. (r, 3r/4π )                                      b. ( r, 2r/π)
c. (r/π, r)                                           d. (r, r/2π)

Q.7) In the truss analysis, if the no. of linkages is 12 and the no. of joints is 7, then
a. perfect truss                            b. redundant truss
c. deficient truss                          d. indeterminate truss

Q.8) When shear force is zero at a point in a beam, bending moment at a certain point is,
a. zero                                                 b. maximum
c. minimum                                          d. increasing

Q.9) The mass moment of inertia of a solid sphere of mass M and radius R about an diameter of the sphere will be
a. (1/5).M.R²              b. (2/5).M.R⁴           c. (2/5).M.R²       d. (1/5).M.R⁴

Q.10) In a simply supported beam of length L, a concentrated load W acts at the mid span, the maximum bending moment would be
a. W.L/4                                                   b. W.L²
c. W.L/2                                                   d. 0

PROBLEM SET - 3

Q1) A couple can be balanced by
a) a direct Force                                    b) a moment
c) a torque                                            d) an equal and opposite couple.

Q2) Opening a Cold drinks bottle, one has to apply
a) a moment                                            b) a torque
c) parallel forces                                      d) a couple

Q3) A truss is said to be plane truss if
a) all the members lies in one plane
b) any two members lies in one plane
c) all the members are perpendicular to one plane
d) none of these

Q4) The ratio of frictional force normal reaction is called
a) angle of repose
b) angle of friction
c) limiting friction
d) coefficient of friction

Q5) Yielding means
a) elastic deformation
b) plastic deformation
c) fatigue failure
d) shear failure

Q6) The force required to move a body up an inclined plane will be least when the angle of inclination is:
a) equal to friction angle,
b) greater than friction angle,
c) less than friction angle,
d) none of these.

Q7) In a cantilever beam loaded with a point load at the free end, has maximum bending moment,

(i) free end,                                            (ii) fixed end,
(iii) at the mid span,                                (iv) none of the above.

Q8) An idealized truss can only be loaded at
a) only at the mid point of the linkages
b) only at the joints
c) both at the mid points and joints
d) none of the above

Q9) UTM can be used to find
a) Modulus of Elasticity
b) compressive breaking strength
c) ultimate tensile strength
d) all of the above

||| the question paper is made by subhankar karmakar, 2011 |||

Q10) Bulk Modulus is defined as the ratio between
(i) linear stress and strain,                 (ii) shear stress and strain,
(iii) volumetric stress and strain,        (iv) none of the above.

PROBLEM SET - 4:

Q.1 Answer the following questions as per the instructions 2x20=20

Choose the correct answer of the following Questions:

(i) The magnitudes of the force of friction between two bodies, one lying above the another depends upon the roughness of the
(a) Upper body                               (b) Lower body
(c) Both the body                           (d) The body having more roughness

(ii) The moment of inertia of a circular section of diameter D about its centroidal axis is given by the expression
(a) π(D)⁴/16
(b) π(D)⁴/32
(c) π(D)⁴/64
(d) π(D)⁴/4

Fill in the blanks in the following questions:

(iii) The distance of the centroid of an equilateral triangle with each side (a) is …………. From any of the three sides.

(iv) Poisson’s ratio is defined as the ratio between ……………. and…………………………

(v) If two forces of equal magnitudes P having an angle 2Ө between them, then their resultant force will be equal to ________

Match the following columns for the following two parts:

(vi) Match the column I to an entry from the column II:

COLUMN I----                            ---COLUMN II

(i) BMD of an UDL                      (a) stored strain energy per unit volume
(ii)Resilience is                            (b) brittle materials
(iii) Bulk Modulus                        (c) parabolic in nature
(iv) YieldPoint                             (d) volumetric stress and strain
                                                  (e) Ductile materials
                                                  (f) Shear stress

(vii) Match the Following columns:

COLUMN I  ------------------       COLUMN II

(i) Square of side b                                              (p) π b⁴ / 64

(ii) Equilateral Triangle of side b                           (q) b⁴ / 12

(iii) Circle of diameter b                                       (r) b⁴/ 36

(iv) Isosceles right angle triangle of base b            (s) b⁴/(32√3)

Column II gives the value of Moment of Inertia I_xx about a centroidal axis.

Choose correct answer for the following parts:

(viii) Statement 1:
In stress strain graph of a ductile material, yield point starts at the end of the elastic limit.
Statement 2:
At yielding point, the deformation becomes plastic by nature.

(i) Statement 1 is true, Statement 2 is true.
(ii) Statement 1 is true, Statement 2 is true and they are unrelated with each other
(iii) Statement 1 is true, statement 2 is false.
(iv) Statement 1 is false, Statement 2 is false.

(ix) Statement 1:
It is easier to pull a body on a rough surface than to push the body on the same surface.
Statement 2:
Frictional force always depends upon the magnitude of the normal force.
(i)Statement 1 is true, Statement 2 is true.
(ii)Statement 1 is true, Statement 2 is true and they are unrelated with each other
(iii) Statement 1 is true, statement 2 is false.
(iv) Statement 1 is false, Statement 2 is false.

Choose the correct word/s.
(x) Two equal and opposite force acting at different points of a rigid body
is termed as (Bending Moment/ Torque/ Couple).

PROBLEM SET - 5

Q.1) The example of Statically indeterminate structures are,

a. continuous beam,
b. cantilever beam,
c. over-hanging beam,
d. both cantilever and fixed beam.

Q.2) A redundant truss is defined by the truss satisfying the equation,
a. m = 2j - 3,
b. m < 2j + 3,
c. m > 2j - 3,
d. m > 2j + 3

Q.3) The property of a material to withstand a sudden impact or shock is called,
a. hardness                                           b. ductility,
c. toughness,                                        d. elasticity  of the material

Q.4) The ratio of the stress generated by a dynamic loading to the stress developed by the gradually applying the same load is
a) 1           b) 2               c) 3                    d) none of the above

Q.5) The ratio between the volumetric stress to the volumetric strain is called as
a. young's modulus
b. modulus of elasticity
c. rigidity modulus,
d. bulk modulus

Q.6) In a Cantilever beam, the maximum bending moment is induced at
a. at the free end
b. at the fixed end
c. at the mid span of the beam
d. none of the above

Q.7) The forces which meet at a point are called
a. collinear forces
b. concurrent forces
c. coplanar forces
d. parallel forces

Q.8) The coefficients of friction depends upon
a. nature of the surface
b. shape of the surface
c. area of the contact surface
d. weight of the body

Q.9) The variation of shear force due to a triangular load on simply supported beam is
a. uniform                                                       b. linear
c. parabolic                                                     d. cubic

Q.10) A body is on the point of sliding down an inclined plane under its own weight. If the inclination of the plane is 30 degree, then the coefficient of friction between the planes will be
a. (1/√3)        b. √3            c. 1            d. 0

PROBLEM SET - 6:

Q.1) Moment of Inertia of a body means
a) The resistance of the body against any movement or the tendency of movement of the body
b) The resistance of the body against any rotation of the body about a certain point
c) The force multiplied by the distance of the force from the body
d) None of these

Q.2) A truss is a
a) load bearing structure
b) framed mechanical structure
c) structure made of linkages and joints
d) all of the above

Q.3) The type of joints that can resist a moment is called as
a. roller joint                                            b. pin joint
c. hinged joint                                          d. fixed joint

Q.4) The slope of the normal stress-strain graph is equal to the
a. modulus of rigidity
b. bulk modulus
c. modulus of elasticity
d. stiffness constant.

Q.5) The shear stress in a circular shaft under torsion varies
a. linearly                                                  b. parabolically
c. hyperbolically                                        d. uniformly

Q.6) If for a material poissons ratio is 0.2 and modulus of elasticity is 200 GPa, then the value of modulus of rigidity would be
a) 100.33 GPa                                          b) 93.33 GPa
c) 83.33 GPa                                            d) 100 GPa

Q.7) If the section modulus of a beam decreases, then bending stress will
a. decrease                                   b. increase
c. remain same                             d. bending stress is independent of section modulus

Q.8) Which one of the following statements is correct?
a. Energy and work are scalars
b. Force and work are vectors
c. Energy, momentum and velocity are vectors
d. Force, momentum and velocity are scalars

Q.9) Stress may be defined as:
a. load per unit area
b. external force per unit area
c. internal resistance per unit area
d. same as pressure

Q.10) Hooke's law is valid up to the
a. yield point                                           b. elastic limit
c. proportional limit                                 d. ultimate point.

PROBLEM SET - 7:

Q.1) The first law of motion provides the definition of
a. momentum                                         b. force
c. energy                                               d. acceleration

Q.2) A man in a lift will weigh more when the lift is
a. accelerated upwards
b. accelerated downwards
c. descends freely
d. lift going up is slowing down

Q.3) The motion of a bicycle wheel is
a. linear                                                   b. rotary
c. translatory                                           d. rotary as well as translatory

Q.4) A particle is moving along a circle with constant speed. The acceleration of the particle is :
a. along the circumference
b. along the tangent
c. along the radius
d. zero

Q.5) The mass moment of inertia of a thin disc of mass (m) and radius (r) about the centroidal axes is
a. (m.r²)/2                                                  b. (m.r²)/3
c. (m.r²)/4                                                  d. m.r²

Q.6) The centroid of a semi circular arc of radius r is
a. (3r)/π                                                     b. 2r/π
c. r/π                                                        d. r/(2π)

Q.7) In the method of sections for trusses, the section must be passed so as to cut not more than
a. two members
b. three members
c. four members
d. five members

Q.8) When bending moment at a certain point is maximum, shear force is
a. zero b. maximum
c. minimum d. increasing

Q.9) The mass moment of inertia of a solid sphere of mass M and radius R about an diameter of the sphere will be
a. (1/5).M.R²
b. (2/5).M.R⁴
c. (2/5).M.R²
d. (1/5).M.R⁴

Q.10) In a cantilever beam of length L, a concentrated load W acts at the free end, the bending moment at the free end would be
a. W.L                              b. W.L²
c. (W.L)/2                         d. 0

Multiple Choice type Question:

TOPIC : FRICTION

1) The maximum value of frictional force that comes into play when a body tends to move on a surface called :
a) sliding friction,
b) limiting friction,
c) milling friction,
d) none of these.

2) The ratio of static friction to dynamic friction is :
a) less than 1,
b) equal to 1,
c) greater than 1,
d) none of these.

3) The angle of friction is equal to the :
a) ratio of frictional force to the normal reaction.
b) angle of inclined plane when a body tends to slide down.
c) angle of an inclined plane when a body is sliding.
d) none of these.

4) The coefficient of friction depends upon :
a) area of contact,
b) shape of the body,
c) nature of contact surfaces,
d) none of these.

5) Kinetic friction is :
a) limiting friction,
b) friction when a body is moving,
c) friction when a body is stationary,
d) none of these.

©subhankar_karmakar

PROBLEM SET - 8

Q1) The enclosed area in the shear force diagram for a beam is the magnitude of
a) shear force
b) bending moment
c) applied load
d) bending stress.

Q2) If a particle is moving in a circular path at constant velocity on a plane, then the direction of the angular velocity will be
a) along the tangent to the circular path
b) along the radial direction
c) normal to the plane
d) none of this.

There will be 2 statements, which may be right or may be wrong, read those sentences and then choose the right choice

Q3) statement 1: Two forces act on a particle are collinear forces.
statement 2: The ratio of longitudinal strain and lateral strain is the value of poisson's ratio.

a) both of them are true
b) statement 1 is true, but statement 2 is false
c) statemen 2 is true while stateme 1 is false
d) both of them are false.

ENGINEERING MECHANICS

CONCEPTS OF FORCE AND FORCE SYSTEM                                 ©subhankar_karmakar

CONCEPTS OF FORCE AND FORCE SYSTEM

               To understand force one has to understand Energy. Energy is the transferable physical quantity of an object which has the capacity to do work. Energy can be transferred from one place to another place, or from one object to another object.



               Force is the physical quantity which when acts on a body, may do some work on the body by displacing from a certain point in the space to another point in the space. When a body gets displaced due to the application of force, some work has been done on the object. So work done on a body is basically nothing but putting some more energy on the body.



               Any kind of displacement of an object is called a change in position and it is a universal truth that to go through any kind of change in position, an object needs some time interval, the process of changing position involves time, hence we can say an amount of ∆x change in position it needs ∆t amount of time. Hence, we can get a rate of change of position with respect to time.



               If in (∆t) time the body travels a distance (∆x) than,
in unit time the body travels a distance (∆x/∆t) and it is known as average velocity.



               When the time interval ∆t is very small, then the ratio of (∆x/∆t) is represented by differential operator and then it is called instantaneous velocity, V=dx/dt



               An heavier object needs more energy to attain the same velocity of that of a lighter object. It has been observed that body at a higher motion needs more energy to change velocity incrementally. Suppose we have two identical object A and B, while the velocity of A is (u), the velocity of B is larger at (v), now if we try to increase the velocity of both the body by a amount ∆v, then the body with the higher motion would need more energy per unit change in velocity.