STUDY4MECHANICAL for Mechanical Engineers

What is speed? Speed is the rate of change of displacement with respect to its surrounding. It is a scalar quantity as the speed of the body is irrespective of its direction. SI Unit: m/s  Symbol: S What is the velocity? The rate of change of displacement with respect to its surrounding is called velocity, in a particular direction. It is a vector quantity as the velocity is always expressed in a particular direction.  Formula: v = S / t  Symbol: v = Velocity, S = Speed, t = Time Unit: m/s What are the acceleration and retardation? The rate of change of velocity of the body is called acceleration. It is positive when the velocity of a body increases with time and negative when the velocity of a body decreases with time. The negative acceleration is called retardation.  Formula: a = v / t  Symbol: a =  Acceleration, v = velocity, t = time Unit: m/s 2

The theorem of Lami is related to the system of forces such as the magnitude of three coplanar, concurrent and non-colinear forces which keep a body in static equilibrium.  Lami's Theorem Lami's theorem statement: Lami's theorem states that if three concurrent, coplanar, or non-colinear forces acting at a point which is in equilibrium, then each force is proportional to the sine angle between  the other two forces.  Now we consider the three forces A, B and C acting on a rigid body or particle and all forces making respective angles  α,  β  and γ with each other  as we have shown in the above figure. Now we can represent the Lami's theorem in mathematically : A / sin α    = C / sinγ  = B / sinβ

Newton's laws of motion-related with the motion of an object to the force acting on it. There are three laws of newton are together laid the foundation for classical mechanics. They all describe the relationship between the body and the force acting upon the body. Let us check it out one by one below.  Newton's first law of motion:  The first law of Newton states that an object continues to remain in the state of rest or uniform motion in a straight line unless acted upon by an external force. It is because of inertia, that object will remain in their state of motion unless a force acts to change the motion.  Example: Wearing a seat belt while driving a car. If any accident occurs, or if brakes are applied to the car suddenly, the body will tend to continue its inertia and move forward.  Newton's second law of motion:  The second law of Newton states that the rate of change of momentum or acceleration of a system is directly proporti...

What is a free body diagram? In mechanics, a graphical, dematerialized, or symbolic representation used to visualize the applied forces, moments, and resulting reactions on a body in a given condition is called a free body diagram. A free body diagram used to draw a scaled drawing. It is a diagram that is modified as it solves the problem. There are flexibility and art to the process. The iconography of a free body diagram, not only how it is drawn but also how it is interpreted and depends upon how a body is modeled.  Features of free body diagram: The free-body diagram consists of the following. A coordinate system A simplified version of a body Force was shown as straight arrows pointing in the direction they act on the body The moment was shown as curved arrows pointing in the direction they act on the body In order to effectively use a free body diagram to analyze the motion of the body, you must know four skills.  Identify the force ...

The body is said to be in equilibrium if the resultant of all the force acting on a body is zero. There are mainly two types of equilibrium. One is translational and another one rotational equilibrium. Let us check out the equilibrium formulas for each type of force system following below.  Formulas of equilibrium of force system:  Concurrent force system:  ∑ F x = 0 ∑ F y = 0 Parallel force system: ∑ F = 0  ∑ M o = 0  Nonconcurrent nonparallel force system:  ∑ F x  = 0 ∑ F y  = 0 ∑ M o  = 0  Where  ∑ F x  is the sum of all forces in the x-direction ∑ F y  is the sum of all forces in the y-direction ∑ F is the sum of all forces ∑ M o  is the sum of the moment at any point O Important points on equilibrium force: Two forces are in equilibrium if they are equal and opposite direction.  Three coplanar forces in...

D-Alembert's principle is a statement of fundamental classical laws of motion. It is also known as Lagrange D-Alembert's principle. It is discovered by French physicist and mathematician Jean le Rond d'Alembert.  Principle of d'Alembert: D-Alembert's principle states that the resultant force acting on the body together with the inertia force or reversed effective force are in equilibrium.  Consider a rigid body that is acted upon by a system of force . The system may be reduced to a single resultant force on the body whose magnitude is given by the product of mass and linear acceleration of the body. According to newton's second law of motion,  F = ma   P - ma = 0  P + Fi = 0  Where Fi is called an inertia force. 

Definition of Torsion:  The action of twisting or the state of being a twist of the one end of an object related to another end. The torsion is the twisting of an object due to an applied torque.  Torsion Formula:  For shaft,  T = J T /  τ  × r =  J T  / l  ×  G ψ   Where, T = Applied torque τ =  Tau J T  = Torsional Constant r = Perpendicular distance between rotational axis and point in the section l = Length of an object ψ = Angle of twist  G = Shear modulus ( Modulus of rigidity) J T  G = Torsional rigidity  In noncircular cross-sections, twisting is applied by a distortion called warping, in which transverse sections don't remain plane. Torsion Units: SI Unit: N/m2 = Pascal (Pa) Other Unit: Pounds per square inch (psi) 

Definition of Torque:   The twisting force that causes motion is known as torque refers to the turning effect. The point of the rotation of the object is called the axis of rotation. The concept of torque originated with the studies by Archimedes when ideas of the usage of the levers. Force is what causes a body or object to accelerate in linear kinematics, likewise, torque is what causes an angular acceleration. Thus torque can be defined as the rotational equivalent of linear force.  Types of torque:  Torque can be classified into two types following below.  Static Torque The torque that does not produce an angular acceleration is called static torque. Example: A person pushing a closed-door is applying a static torque because the door is not rotated despite the force applied.  Dynamic Torque The torque that does produce an angular acceleration is called dynamic torque.  Example: The drive shaft in a racing car accele...

Definition of angle of repose:  The angle of repose is also called a critical angle of repose. The angle of repose is the steepest angle of dip or descent relative to the horizontal plane to which is possible to pile material without slumping.  The angle of repose can be range from 0 0 to 90 0 . Angle of repose The angle of inclination  α of the plane is equal to the limiting angle of the friction   ϕ. It is the angle of the plane to the horizontal, at which the body just begins to move down the plane.  The formula for confirming that is the following below.  W sin  α = F =  µ × R N = µ  W cos  α  tan  α =  µ = tan  ϕ  α =   ϕ  Uses of the angle of repose: The angle of repose is used in the design of equipment for the processing of particulate solids.  

Definition of limiting angle of friction:  The angle between resultant reaction R and normal reaction R N is known as limiting angle of friction. Let us check out the formula of the limiting angle of friction to know more about it.  Limiting Angle of friction Let us consider an object of weight W resisting on a horizontal plane B. A horizontal force P is applied to the body, no relative motion takes place until the force P applied is equal to the force of friction F, acting opposite to the normal direction. The magnitude of the force of friction is given by the formula below.  F =  µ W =  µ  R N   Where  R N  = Normal reaction  When the body just begins to move, it is in equilibrium under the action of a force which is following below, in the limiting case.  Weight of the body W A horizontal force applied P Reaction R between the body and the plane  The resultant reactio...

Definition of friction:  A force that is acting in the opposite direction to the motion of the body is called friction. Friction is resistance to the motion of one object moving relative to another object.  Frictional force f = µ × N Where, f = frictional force µ = coefficient of friction  N = Normal force Types of friction:  Friction mainly classified into two types.  Static friction  Dynamic friction  The friction experienced by a body, when an object is at rest is known as static friction.  The friction experienced by a body, when the body is in motion is known as dynamic friction. It is also called kinetic friction. Kinetic friction also divided into two types which are the following below.  Sliding friction Rolling friction The friction experienced by a body, when the one body slides over another body is known as sliding friction.  The friction...

Definition of a moment of inertia:  The moment of inertia also called a mass moment of inertia, angular mass or rotational inertia. The moment of inertia is a quantity expressing a tendency of the body to resist angular acceleration about a rotational axis. It is similar to how mass determines the force needed for the desired acceleration.  Mathematically, the moment of inertia is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation.  Units of a moment of inertia:  SI Unit: kg m 2 Other Unit: lbf ft s 2 Dimension: M L 2 Moment of inertia usually denoted by I. 

Definition of center of gravity: The center of gravity is the point that shows the average location of the weight of an object. The point through which the whole mass of body acts, irrespective of the position of the body is known as a center of gravity OR centroid.  The plain geometrical figures such as rectangle, triangle, circle all have only areas not mass. The center of an area of all such figures is known as a center of gravity of the area of the body. Every object has one and only one centroid.  Types of figure and its center of gravity: The center of gravity of a uniform rod is at a middle point of the rod.  The center of gravity of a rectangle or parallelogram lies at a point where its diagonal intersects.  The center of gravity of a triangle lies at a point where the three medians of the triangle intersect.   For a general shaped object, there is a simple mechanical way to determine the center of gravity which is the follow...

Definition of a couple in Mechanics: In mechanics, a couple refers to two equal and opposite forces, whose line of action is different from a couple. The couple also called a force couple or pure moment.  Moment of couple The perpendicular distance between L and M of the two equal and opposite forces is known as the arm of the couple. The magnitude of the couple is the product of one of the forces and the arm of the couple.  Mathematically, Moment of couple = LM × F A little analysis can show that a couple produces no translational motion but a couple produces a body rotation motion that acts on it.

Definition of parallel forces: A parallel force is a situation in which two forces of equal magnitude act whose line of action is parallel to each other within the same plane are said to be a parallel force.  Example -  A see-saw Types of parallel forces: Like parallel forces  If the parallel forces act in the same direction then these forces are known as, like parallel forces.  Unlike parallel forces  If the parallel forces act in the opposite direction then these forces are known as, unlike parallel forces. 

Varignon's theorem also called the principle of moments.  Varignon's Principle: Varignon's principle states that if a number of concurrent forces acting on a particle are in equilibrium, then the algebraic sum the moments that each force creates about a single point will be equal to the moment of their resultant force about the same point.   Varignon's Theorem From the above figure, F1 and F2 are two concurrent forces and their resultant is R. The moment produced by R with respect to a moment center O is R × d. The moment produced by the force F1 and F2 is respectively F1 × d1 and F2 × d2.  As per varignon's therorem R d = F1 d1 + F2 d2 Notes: This theorem initially stated for two concurrent forces, but, it is true for any system of forces and any number of concurrent or coplanar forces.  One of the practical applications of this theorem is to find the unknown reactions when a system is known to be in equilibrium under...

The turning effect produced by a force applied to a rotational system or on the body at a distance from the axis of rotation is called the moment of force. The moment of a force is the product of the force and the perpendicular distance of the point, about which the moment is required and the line of action of the force.  Moment of force Mathematically, the moment of force F1 about point 2 = F1 × d  The unit of moment of force depends on the units of force and perpendicular distance. If the unit of force is in newtons and the perpendicular distance in meters, then the unit of the moment will be Newton-meter (N-m). 

When two or more than two forces of different magnitude and different directions act on the body they form a system of forces. Depending upon plane and direction the different types of system of forces categorize are following below.  System of forces: Coplanar forces If all the forces in the system lie in a single plane or whose line of action lies on the same plane are known as coplanar forces.  Concurrent forces  The forces that are meet at one point or if the line of action of all the forces in a system passes through a single point are known as concurrent forces.  Coplanar concurrent forces The forces that are meet at one point and their line of action also lie on the same plane are known as coplanar concurrent forces.  Coplanar parallel forces  The forces are parallel to each other and lie in the single plane are known as coplanar parallel forces.   Coplanar like parallel forces The forces are ...

Definition of resultant force: The resultant force is a single force that produces the same effect as produced by all the given force acting on the body. A resultant force also called the net force. The resultant force may be determined by the laws which are following below.  Parallelogram law of forces Triangle law of forces  Polygon law of forces  Parallelogram law of forces: The law states that, if two forces acting simultaneously on a particle can be represented in magnitude and direction by two adjacent sides of a parallelogram. Their resultant maybe represent in magnitude and direction by the diagonal of a parallelogram which passes through their point of intersection.  Parallelogram law of forces Now, let us consider two forces A and B acting at angle p. The resultant is given by the following formula.  R = √ A 2 + B 2 + 2ABCosp  If the resultant force R makes an angle q with the force B, then...