Best Mechanical engineering tutorials ENGINEERING MECHANICS

ENGINEERING MECHANICS - STATICS

            Engineering mechanics is the science that deals with the state of rest or motion of bodies under the action of forces. It is further divided into mechanics of rigid bodes, deformable bodes and fluids.
Rigid Bodies:
            Such bodies don't deform under the action of applied forces. However in many cases, it is negligible to affect the results. So it is assumed that bodies does not deform or the distance between two points on a body does not change because of external load. Mechanics of rigid bodies is further subdivided into statics and dynamics.
Statics : Study which deals with bodes in rest.
Particle : Refers to a object, whose mass is concentrated at a point. This assumption is made when the size of body is negligible. Many particles together constitute the particle.
Mass and Weight:
              Mass is defined as the measure of how much matter an object or body contains -- the total number of subatomic particles (electrons, protons and neutrons) in the object. So if your body weight is fluctuating, because of eating or exercising, it is actually the number of atoms that is changing.

Mass	Weight
It is quantity of matter contained in a body	It is the force with which the body is attracted towards the center of earth.
It is constant at all places	It is different at different places
It resists the motion in the body	It produces the motion in the body
It is a scalar quantity	It is a vector quantity
Measured with ordinary balance	Measured with spring balance
It is never zero	It is zero at the center of the earth.

The weight of a body of mass m should be measured in Newtons. W = mg = m (9.8 m/s²) = 9.8 N.
Newton's Law:
First law: Every body continues in a state of rest or uniform motion in a straight line, unless it is compelled by a external force to change the state.
Second Law: Change of momentum is proportional to impress force and takes place in the direction of the straight lines, in which the force acts. It states that

acceleration is directly proportional to net force when mass is constant, and
acceleration is inversely proportional to mass when net force is constant, and consequently
net force is directly proportional to mass when acceleration is constant.

This law enables to measure a force and establishes the fundamental equation of dynamics. Consider, a body moving along a straight line. Where 'm' is the mass of the body, 'u' is the initial velocity of body, 'v' the final velocity of body and 'a' acceleration of the body.
Initial momentum = m.u and final momentum = m.v. Thus the rate of change in momentum is m(v-u) / t = m.a Newton's second law of motionis more compactly written as the equation ∑F = ma

	Cause of change	Resistance to change	Rate of change of...
The concept implied in Newton's Second Law of Motion are found in many places, as shown below
Newton's second law	force	mass	velocity
rotational dynamics	torque	moment of inertia	angular velocity
Newtonian fluids	shearing stress	viscosity	shear
thermal conduction	temperature gradient	r-factor	heat
ohm's law	potential difference	electrical resistance	charge
faraday's law	potential difference	inductance	current

Third Law: to every action, there is a equal and opposite reaction. This goes to say, that the force of action and reaction are equal in magnitude by opposite in direction.
Law of Gravitation:
Two particles are attracted towards each other along the lines joining them, with a force whose magnitude is directly proportional to the product of masses and inversely proportional to the square of distance between them.

F = G m₁m₂ / r²

Where G is universal gravitation constant.

Scalar quantity: Some quantities like time, mass volume can be expressed in terms of magnitude alone and don't have any direction. They obey the law of algebra.

Vector quantity: Quantities like distance, velocity, acceleration and all are expressed in terms of both magnitude and direction. They obey the law of vectors. To define such a quantity Magnitude, Direction and Point of application has to be specified.

FORCE:

It is a derived unit. It is a force that imparts a acceleration of 1 m/s²on a body of mass one Kg. 1N = 1 Kg m/s²= It is a agency which changes or tends to change the state of rest or motion of a body. Force has the capacity to impart motion to a particle. Force can produce pull, push or twist. It is a vector quantity, hence to define force its point of application, its magnitude and its direction has to be specified. For simplicity sake, all forces (interactions) between objects can be placed into two broad categories.

Contact forces: Are types of forces in which the two interacting objects are physically contacting each other. Examples of contact forces include frictional forces, tensional forces, normal forces, air resistance forces, and applied forces.
Action-at-a-distance forces: are types of forces in which the two interacting objects are not in physical contact with each other, yet are able to exert a push or pull despite a physical separation. Examples

Gravitational forces ( E.g., the sun and planets exert a gravitational pull on each other despite their large spatial separation, even when our feet leave the earth and we are no longer in contact with the earth, there is a gravitational pull between us and the Earth ),

Electric forces ( E.g., the protons in the nucleus of an atom and the electrons outside the nucleus experience an electrical pull towards each other despite their small spatial separation ), and

Magnetic forces ( E.g., two magnets can exert a magnetic pull on each other even when separated by a distance of a few centimeters ).

Apart from this force is also classified as internal and external force. Internal force are those that hold together the particles forming the rigid body. If the rigid body has several parts, the forces holding the component parts together are also called as internal force. External forces represent the action of other bodies on he rigid body under consideration. They will either cause it to move or assure that it remains at rest.

Types of forces:

Equal and Equivalent force:

Two forces of the same magnitude and direction but having a different point of application is called as equal force.

Two forces are said to be equivalent if they produce the same effect on a rigid body. Equivalent forces is based on some specific effect.

Coplanar forces:

When a number of forces lies in the same plane, then it is called as coplanar force. Other wise it is called as non coplanar forces.

Concurrent forces:

These forces are those in which the forces have the lines of action passing through common point. However, all of the individual vectors might not acutally be in contact with the common point.

Parallel force:

These are a set of forces, whose line of action is parallel to each other. Following are the types of parallel forces.

Like parallel force : When two parallel forces have the same direction but may or may not have the same magnitude.
Unlike unequal parallel force : when both the forces are unequal in magnitude and act in opposite directions.
Unlike equal parallel force : When two forces are opposite indirection and equal in magnitude.

Parallelogram Law for addition of forces:

If two forces acting on a point are represented in magnitude and direction, by two adjacent sides of a parallelogram, then the diagonal of parallelogram passing through the points of intersection, represents the resultant force in both magnitude and direction.

Triangle law of forces:

If two forces acting at a point are represented by two sides of a triangle taken in order, then their sum of resultant is the third side of triangle taken in opposite order.

Polygon law :

When a number of coplanar forces are acting at a point, such that they can be represented in magnitude and direction by the side of polygon taken in order, then the resultant can be represented both in magnitude and direction, by the closing side of polygon taken in opposite order.

Lami's Theorem:

When three forces acting at a point are in equilibrium, then each force will be proportional to the sine of the angle between the other two forces.

Principle of Transmissibility:

It states that condition of state of rest or motion of body does not change if the point of application of a force is transmitted to any other point, along its line of action. This principle is used to determine the external forces acting on the rigid body. But should not be used to determine the internal forces and deformation of the body.

Resultant of several force:

When a number of forces acting on a rigid body is replaced by a single force which has the same effect as all the forces on the rigid body, then that forces is called as resultant of several force.

Any concurrent set of forces, not in equilibrium, can be put into a state of equilibrium by a single force. This force is called the Equilibrant. It is equal in magnitude, opposite in sense and co-linear with the resultant. When this force is added to the force system, the sum of all of the forces is equal to zero.

Condition for equilibrium:

When the resultant of all the forces acting on a particle is zero, then the particle is said to be in a state of equilibrium.

Constraint, Action and Reaction:

A body is not always free to move in all directions. This restriction to the free motion of a body is called as constraint. A action of a constrained body on any support induces a equal and opposite reaction from the support.

Free body diagram:

To draw the free body diagram the supports are removed and replaced by the reactions the support exerts on the body.

Moment of force:

A force can produce a rotary motion. This measure of this turning effect produced by a force is called as moment of a force. The moment of a force about a point is equal to the product of the force and the perpendicular distance between the line of action of force and the point ( also called as Moment centre )

Varignon's Theorem:

The moment of a force about a axis is equal to the sum of the moments of components about the same axis.

Couple:

A system of two equal parallel forces acting in opposite directions can be replaced by a single force. In such a case a couple is produced, which has a tendency to rotate the body. The perpendicular distance between the line of action of two forces is called as arm of couple.

Moment of a couple:

The rotational tendency of a couple is measured by its moment. The moment of a couple is the product of magnitude of one of the forces and arm of the couple.

Central values:

Centre of mass: is the point through which the entire mass of the body is assumed to be concentrated. Both are different only when the gravitational field is not uniform and parallel, other wise it is the same.

Centroid: is the point where the entire area of the lamina is assumed to concentrated.

Center of gravity of a Two-dimensional body:

Centre of gravity is defined as the point through which the resultant of the distributed gravitational forces, act irrespective of the orientation of the body.

For illustration, let us consider a flat horizontal plate is considered. We divide the plate into a small elements. The co-ordinates of the first element is denoted by (x1, y1) for the second element it is (x2, y2). Similarly the forces exerted by the earth on the elements on the plate will be denoted respectively by DW1, DW2 .... DWn. These forces or weights are directed towards the center of the earth, however for all practical purposes they are assumed to be parallel. The resultant W is a single force in the same direction. The magnitude W of this force is obtained by adding the magnitudes of the elementary weights.

W = DW1 + DW2 + ...... + DWn.

To obtain the co-ordinates of centroid (x, y) where the resultant W is applied, we write the moments of W about the x and y axes to be equal to the sum of the corresponding moments of the elementary weights.

xW = x1.DW1 + x2.DW2 + ......... + xn.DWn.
yW = y1.DW1 + y2.DW2 + ......... + yn.DWn.

Now the size of each element is decreased the number of elements is increased. We then obtain the limit of the following expressions.

W = ò dW xW = ò xdW yW = ò ydW

The magnitude of weight W is denoted by rgt DA. Substituting this value of W and W in the above equations and dividing it by rgt, we get

xA = x1.DA1 + x2.DA2 + ....... + xn.DAn
yA = y1.DA1 + y2.DA2 + ....... + yn.DAn, Similarly the values of integral also changes as shown.

xA = ò xdA yA = ò ydA

First moments of Areas and Lines:

The integral ò xdA in previous Para is known as the first moment of Area with respect to y axis and is denoted by Qy. Similarly the integral òydA defines the first moment of Area with respect to the x axis and is denoted by Qx. Mathematically we can also derive

Qx = yA Qy = xA

Beams:

It is a structural member designed to withstand loads at various points along the members. Usually the loads are applied perpendicular to the axis of the beam thus causing shear and bending in the beam. If the loads are not at right angles of the beam, then they will also produce axial forces in the beam. Beams are long, straight prismatic members designed to support loads applied at various points along the member.

In the design of a beam we have to consider the most effective cross section that will provide the most effective resistance the shear and bending moment produced by the applied loads. Hence the design of beam consists of two distinct parts. In the first part the shearing force and the bending moment produced by the loads are determined. In the second part there is a selection of cross section that best with stands the shear and bending moment determined in the first part.

Cables:

These are flexible members capable of withstanding only tension, designed to support either concentrated or distributed load.

Friction:

The friction is a force distribution at the surface of contact and acts tangential to the surface of contact. This force always develop when one surface attempts to move over the other. There are two types of friction. They are dry ( also called as coulomb friction ) and fluid friction.

Dry friction:

Is the one which exists between two dry surfaces. Such a friction is caused mainly because of minute projections present on the surface of body hindering relative motion. The friction between liquid surfaces is called as fluid friction.

Limiting friction:

When a body of mass m is there with a weight W a continuously increasing force P is applied on the body to move it. This force P is opposed and resisted by frictional force F. As P increases F also increases. The body also remains at rest. At a point F cannot increase, hence P > F and the body begins to move. The friction force at this instant is called as limiting friction.

Limiting friction is the maximum frictional force exerted at the time the body begins to move. The friction that exists between two moving bodies is called as kinetic or dynamic friction.

Laws of dry friction :

The total frictional force developed is independent of the magnitude of area of contact.
The total frictional force is directly proportional to the normal force acting at the surface of contact.

F = μN

Where F - Frictional force

μ - Coefficient of static friction and

N - Normal reaction.

Angle of Friction:

The normal reaction N and the frictional force F can be combined into a single resultant force R called resultant reaction. The angle which the resultant reaction R makes with the normal reaction N is called as angle of friction

Tan Ø = F / N = μN / N = μ

μ is called as coefficient of friction.

Angle of repose:

It is defined as the maximum angle of inclination at which the body remains in equilibrium at a inclined surface at the influence of friction alone, beyond which the body slides.

Rolling resistance:

A ball is present on the ground. They are in touch only at the point of contact. That a large amount of friction is eliminated. But then the when or ball starts rolling, the resistance increases. This is mainly due to deformation over which the ball creates on the surface. Thus there is no longer a point contact but a area contact. This area a is called as the forward length of deformation also called as coefficient of rolling resistance.

Engineering structures:

Any system of interconnected members builds to support or transfer force acting on them and to safely withstand these forces are called as engineering structure. Following are the types.

Truss : It is a system of members which are joined together at the ends, by riveting or welding at the ends. All members are two force members. Load is applied only at joints.

Frame : Here one or more members are subject to more than two forces.

Assumptions Made:

The joints are frictionless.
Loads are applied only in the joints.
The members are two force members with forces acting collinear to centre line of members.
The weight of members is negligible and
The truss is statically determinate.

To determine the axial forces on the members, there are three methods. They are

Method of joints,
Method of sections and
Graphical method.

Moment of Inertia:

By analogy the role played by the moment of inertial in the rotary motion is similar the role played by mass in translatory motion. The moment of Inertia of area is called as the area moment of inertia. The moment of Inertia of mass is called as the mass moment of inertia.

dA is a element at a distance ( x, y ) from the axes.

The moment of area with respect to X axis is = Ix = ∫ y²dA

The moment of area with respect to Y axis is = Iy = ∫ x²dA

Polar moment of Inertia :

The moment of inertia of a area of plane figure with respect to the axis that is perpendicular to x-y plane and passing through O is called polar moment of Inertia it is denoted by

j_o = ∫ r²dA

j_o = ∫ ( x² + y² ) dA = Ix + Iy

Theory of Papus-Guidinus:

It states that the surface of revolution is a surface which may be generated by rotating a plane curve about a fixed axis. The surface of sphere may be obtained by rotating a semicircular arc about its diameter, the surface of a cone by rotating a straight line inclined about its axis.

A body of revolution is a body which may be generated by rotating a plane area about a fixed axis. A solid sphere may be generated by rotating a semi circular area, a cone by rotating a triangular area and a torus by rotating a full circular area.

Parallel axis theorem:

The moment of Inertia of a lamina about any axis in the plane is equal to the sum of the moment of inertia abut a parallel centroidal axis in the plane of the lamina and the product of the area and square of distance between two axes.

Perpendicular Axis theorem:

If Ix and Iy are the moment of inertia about two mutually perpendicular axis OX and OY. Iz be the moment of inertia of lamina about a axis normal to the lamina and passing through the point of intersection of Ox and OY axes then

Iz = Ix + Iy

Quantity	Unit	Symbol
Acceleration	meter / sec²	m/s²
Angle	Radian	Rad
Angular velocity	Radian/second	Rad/s
Angular acceleration	Radian/second²	Rad/s²
Area	metre²	m²
Density	Kilogram / meter³	Kg/m³
Energy	Joule	J = Nm
Force	Newton	N = Kg m/s²
Frequency	Hertz	Hz
Length	meter	m
Mass	Kilogram	Kg
Moment of force	Newton-metre	Nm
Power	Watts	W = J/s
Pressure	Pascal	Pa = N/m²
Stress	Pascal	Pa = N/m²
Torque	Newton-metre	Nm
Velocty	metre/second	m/s
Volume	metre³	m³
Work	Joule	J = Nm

NUMERICAL PROBLEMS

Problem 1: For the system of forces shown, calculate the resultant force and its angle of inclination.

Solution: Each force shown is resolved into the x and y components. To get the x components, the magnitude is multiplied with the Cosine of the angle of inclination ( F Cos a ). For y components the magnitude of force is multiplied with Sine of the angle of inclination ( F Sin a ).

Force	Magnitude	x component	y component
F1	150	129.9	75
F2	80	-27.4	75.2
F3	110	0	-110
F4	100	96.6	-25.9

Sum of x component of force = 199.1 N
Sum of y component of force = 14.3 N

Resultant = Ö (199.1)² + (14.3)² = 199.6 N

Angle of inclination = Tan ^-1 ( 14.3 / 199.1 ) = 4.1^o

Problem 2: For the plane shown determine (a) the first moments and the location of centroid.

Solution: To proceed further the plane is considered to be a combination of Rectangle + Triangle + Semi circle - Circle.

Component	Area (a)	Centroid X	Centroid Y	X.a	Y.a
Rectangle	9600	60	40	576000	384000
Triangle	3600	40	-20	144000	-72000
Semi-circle	5655	60	105.5	339300	596602.5
Circle	-5026	60	80	-301560	-402080
Total	13829			757740	506522.5