MECHANICS OF SOLIDS Abishek yadav

Types of Quantities Scalar Quantities: The Quantities which possess magnitude only are called scalar quantities Eg: Length, Area, Mass etc. Vector Quantities: The Quantities which possess magnitude as well as direction are called vector quantities Eg: Force, Velocity, Acceleration

Resolution of vectors The process of determining the magnitude of a vector is known as vector resolution The two methods of vector resolution that we will examine are: 1.Parallelogram Method 2.Trigonometric Method

Units of Measurement Unit : The known amount used as a reference in the measurement of a physical quantity called an Unit Basic Units : The units which are used for measurement of basic or fundamental quantities (Mass, Length, Time) Derived Units: All units which are used for measurement of physical quantities other than fundamental quantities(Area, Volume,Speed)

System of Units There are four system of units in use : 1.Foot Pound Second system i.e, FPS system The system of units is a scheme for measuring dimensional and material quantities. The fundamental units are the foot for length, the pound for weight, and the second for time 2.Centrimetre Gram Second system i.e, CGS system It is a variant of the metric system of physical units based on centimetre as the unit of length, grams as a unit of mass, and second as a unit of time 3.Metre, Kilogram Second system i.e, MKS system It is a variant of the metric system of physical units based on metres as the unit of length, kilograms as a unit of mass, and second as a unit of time 4.System of International i.e, SI system It is a variant of the metric system of physical units based on ampere as the unit of electric current, Kelvin as a unit of temperature, candela for luminous intensity in addition to fundamental units of length, mass and time

S.NoQuantityUnits 1.Lengthm 2.Aream2 m2 3.DensityKg/m 3 4.Velocitym/s 5.ForceN 6.Accelerationm/s 2 7.WorkNm(Joule) 8.PowerNm/sec – (Watt) 9.EnergyNm(Joule) 10.PressureN/m 2 11.MassKg 12.WeightN 13.Times Important S.I Units:

Definitions: Space: The dimensions of height, depth, and width within which all things exist and move. Time: The indefinite continued progress of existence and events in the past, present, and future regarded as “Time”. Particle: An extremely small piece of matter. Rigid body: Non flexible body / fixed body / stiff body

Characteristics of force : A Force is characterised by 1.Magnitude 2.Lines of action 3.Direction

Magnitude: Magnitude of a force is denoted by certain number of units. In SI system of units magnitude of a force is in Newton(N). Sometimes, magnitude of a force is also denoted by the multiples of Newton as Kilo Newton (KN), Mega Newton (MN),Giga Newton (GN). 1KN = 10 3 N 1MN = 10 6 N 1GN = 10 9 N

Line of Action : An infinite straight line along which the force acts is called line of action of force. Direction : Line of action of force is denoted by an angle with some fixed axis. This angle with fixed axis and sense of force represents direction of force. Sense of force (arrow head) indicates whether the force acts outwards from a particle or towards a particle Note : Outwards - Tensile Towards - Compressive

System of Forces : A body with two or more forces acting simultaneously on it constitute a system of forces. The force system is classified is classified into subdivisions. 1.Coplanar forces 2.Non – coplanar forces 3.Collinear forces 4.Concurrent forces 5.Parallel forces 6.Like collinear coplanar forces 7.Unlike collinear coplanar forces 8.Coplanar concurrent forces 9.Coplanar non-concurrent forces 10.Non-coplanar concurrent forces 11.Non coplanar non-concurrent forces

Coplanar Forces : In coplanar force system, all the forces act in one plane. This system is also called as “forces in plane”. Non – coplanar Forces: In Non – coplanar force system, the forces do not act in one plane. This system is also called as “forces in space”. Collinear Forces: The forces which acts on a common line of action are called collinear forces. If they act in same direction, they are called “Like collinear” and if they act in opposite direction, they are called “Unlike collinear” Concurrent Forces : In concurrent force system, forces intersects at an common point.

Parallel Forces : In parallel force system,the line of action of forces are parallel to each other parallel forces acting in same direction are called “Like parallel forces” and parallel forces acting in opposite direction are called “Unlike parallel forces”. Like collinear coplanar forces : Forces acting in same direction, lies on a common line of action and acts in a single plane. Unlike collinear coplanar forces : Forces acting in opposite direction, lies on a common line of action and acts in a single plane. Coplanar concurrent forces : Forces intersects at a common point and lies in a single plane.

Coplanar non – concurrent forces : Forces which do not intersect at a common point, but acts in one plane. They might be either parallel or non-parallel. Non – Coplanar concurrent forces : Forces intersect at one point,but their lines of action do not lie on the same plane. Non – Coplanar non – concurrent forces : Forces do not intersects at one point and also their lines of action do not lie on the same plane.

Parallelogram law of forces : It states that, “If two forces acting simultaneously at a point, be represented in magnitude and direction by the two adjacent sides of a parallelogram, then the resultant of these two forces is represented in magnitude and direction by the diagonal of that parallelogram originating from that point”

Now to find out the magnitude of the resultant force, we require to draw a perpendicular from C on OA produced at D. As OB & AC are parallel. Therefore BOA = CAD = (Corresponding Angles). As OCD is a right angled triangle therefore by Pythagoras Theorem, Replacing the sides by their corresponding forces, [OB = AC = P]

[By Pythagoras Theorem] Let us take a right angled triangle CDA, and let the hypotenuse CA makes an angle(θ) with the horizontal i.e.AD. If CA is known, the other two sides CD & DA can be expressed in terms of CA. DA = CA cos(θ) & CD = CA sin(θ) Note: The side which makes an angle (θ) with the hypotenuse will be the hypotenuse multiplied by cos(θ) and the other side will be hypotenuse multiplied by sin(θ).

Therefore the Resultant of the two Coplanar Concurrent Forces is R, where Now we will find the value of (α) i.e. the resultant of the two Coplanar Concurrent Forces makes an angle(α)with the force P. Now we know that,

Triangular law of forces It states that “If two forces acting at a point area represented by two sides of a triangle, taken in order, then their resultant force is represented by the third side taken in opposite order “

Polygon law of forces : It states that “If number of coplanar concurrent forces are represented in magnitude and direction by the sides of a polygon taken in an order, then their resultant force is represented by closing side of polygon taken in opposite order “

Polygon laws of forces (Con.) Consider the forces F 1, F 2, F 3 and F 4 are acting at a point O as shown in Figure. Starting from the point O, the vector OA represents the force F 1 in magnitude (using suitable scales) and direction. From the tip A, draw vector AB representing the force F 2. Similarly, vector BC represents the force F 3 and vector CD represents force F 4. Join the starting point O to the end point D giving a vector OD in opposite order. Vector OD represents the resultant force R = F 1 + F 2 + F 3 + F 4 in magnitude and direction as shown in Figure From the triangle law of forces