Engineering Mechanics Statics

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. 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. Mass : Mass is a measure of resistance to acceleration. ( More generally, mass is a measure of resistance to change ) Mass is a scalar quantity associated with matter. When a system is composed of several objects it is the total mass that matters. The SI unit of mass is the kilogram [kg]. 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. Newton's second law of motion is more compactly written as the equation ∑F =

ma The concept implied in Newton's Second Law of Motion are found in many places, as shown below Cause of change

=

Resistance x to change

Rate of change of...

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 the, with a force whose magnitude is directly proportional to the product of masses and square of distance between them. F = G m1m2 / r2 Where G is universal gravitation constant. Force : 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. 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 1. 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 ),

2. 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 3. Magnetic forces ( E.g., two magnets can exert a magnetic pull on each other even when separated by a distance of a few centimeters ). 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. Parallel force : These are a set of forces, whose line of action is parallel to each other, then they are called as parallel forces. 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 : 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 3 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. 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, we have to specify the • •

Magnitude, Direction and

•

Point of application.

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. 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. Momement 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 gravity : is the point through which the resultant of the distributed gravitational forces, act irrespective of the orientation of the body. 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.

Friction : The friction is a force distribution at the surface of contact and acts tangential to the surface of contact. 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 tow moving bodies is called as kinetic or dynamic friction. Laws of dry friction : 1. The total frictional force developed is independent of the magnitude of area of contact. 2. 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. The area a is called as the forward length of deformation and is 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. 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 : 1. The joints are frictionless. 2. Loads are applied only in the joints. 3. The members are two force members with forces acting collinear to centre line of members. 4. The weight of members is negligible and 5. The truss is statically determinate. To determine the axial forces on the members, there are three methods. They are 1. Method of joints, 2. Method of sections and 3. 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 jo = ∫ r2dA jo = ∫ ( x2 + y2 ) dA = Ix + Iy 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 abut a axis normal to the lamina and passing through the point of intersection of Ox and OY axes then Iz = Ix + Iy Lastly updated on

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