Principles of Dairy Machine Design

Module 1. Statics and dynamics

Lesson 1
BASIC CONCEPTS IN STATICS AND DYNAMICS, FORCE SYSTEMS, EQUILIBRIUM CONDITIONS

1.1 Introduction

Statics and dynamics are important terms in designing of any machine parts or equipments. The science of dynamics is based on the natural laws governing the motion of a particle.

1.1.1 Mechanics

It is a branch of science which deals with study of forces and their effects on structure, machines etc.

The branch of mechanics is further divided in to Engineering mechanics, mechanics of solids and mechanics of fluids. (Fig 1.1)

Fig. 1.1 Mechanics

1.1.2 Engineering mechanics

It deals with mechanics of rigid body and study of external forces and their effects on rigid body.

1.1.3 Mechanics of solids

Study of internal resisting forces developed in a deformable body under the action of external forces.

1.1.4 Mechanics of fluids

It deals with mechanics of compressive forces on the fluid & in compressive forces on fluid particles. Mechanics is an extensive field of operation which can be subdivided in various ways. A subdivision addressed in the given description of mechanics is based on the perspective of rest and movement.

1.1.5 Statics

Branch of a mechanics which deals with the study of forces on a body which is at rest.

1.1.6 Dynamics

It deals with study of motion and forces on a body which is in motion. Subcomponent of dynamics is kinematics.

1.1.7 Kinematics

The study of forces and the displacement of bodies, without addressing the cause of movement.

1.1.8 Newton’s Laws

The basic laws for the displacement of a particle were first formulated by Newton. Newton’s three laws are as follows.

1.1.9 First law (or) law of inertia

First Law states that when a body is in motion (or) is at rest unless until it is forced to change that state by forces imposed on it. The property with which a particle resists a change in its state of rest (or) movement is called its inertia. Newton’s first law is therefore also known as the law of inertia.

1.1.10 Second law of motion

It states that the rate of change in momentum of a particle is proportional to the force applied to it, and takes place in the direction of that force. Second law is defined by the following formula.

F = K.M.a

Where,

F= force in N

M= mass in kg

a = Acceleration in m/s²

k= Proportionality constant

1.1.11 Third law of motion

It states that for every action there is an equal and opposite reaction. If a body A exerts a force on particle (or) body B, it will exert an equal and opposite force on body A.

1.2 Force Systems

1.2.1 Force

It is defined as agent which produces (or) tends to produce destroys (or) tends to destroy motion. It may be noted that the force may have either of the two functions i.e. produces (or) tends to produce motion.

1.2.2 Resultant force

If a number of forces P, Q, R etc are acting simultaneously on a particle. It is possible to find out a single force, which would replace them, i.e. which would produce the same effect as produced by all the given forces. This force is called the resultant force and the forces P, Q, R etc are called component forces.

1.3 Analytical Method to Find out the Resultant Force

The resultant forces, of a given system of forces, may be found out analytically by methods as discussed below.

1.3.1 By parallelogram law of forces

It states that if two forces, acting simultaneously on a particle are represented in magnitude and direction by the two adjacent sides of a parallelogram; their resultant may be Fig.1.2 Parallelogram law of forces represented in magnitude and direction by the diagonal of the parallelogram which passes through their point of intersection:

Fig. 1.2 Parallelogram law of forces

Let the two forces P and Q, acting at ‘O’, be represented by the straight line OA and OC in magnitude and direction If parallelogram OABC be completed, with OA and OC as adjacent sides, the resultant ‘R’ of the forces OA and OC may be represented by the diagonal OB.

Let ‘θ’ be the angle between the forces P and Q. Extend the OA and draw perpendicular BM from B on this line. Let these lines meet at the point ‘M’ as shown in fig. no 1.2 from the geometry of the figure, we known that

In right angle triangle OBM, we know that

1.4 Methods for the Determination of Resultant Force

The magnitude of resultant force may be found out by the following methods

1. Analytical method

2. Graphical method

1.4.1 Analytical method

It is also known as the method of resolution of forces. The magnitude and direction of the resultant force of a given system of forces may be found out by the analytical method as discussed below.

Resolve all the forces vertically and find the algebraic sum of all the vertical components {i.e. ∑V.}

Resolve all the forces Horizontally and find the algebraic sum of all the horizontal components {i.e. ∑H.}

The resultant R of the given forces will be given by the each

R=√[ ∑V²+∑H²]

The resultant forces will be inclined at an angle θ, with Horizontal such that.

Tan θ= ∑V / ∑H

1.4.2 Graphical method for the resultant forces

The resultant forces of a given system of forces may be found out graphically by the follows methods as discussed below.

Triangle Law of Forces

It states that “if two forces acting simultaneously on a particle, be represented in magnitude and direction by the two sides of a triangle, taken in order, their resultant may be ,represented in magnitude and direction by the third side of the triangle, taken in opposite order”.

The algebraic sum of the resolved parts of a number of forces, in a given direction is equal to the resolved part of their resultant in the same direction.

Fig. 1.3 Triangle Law of Forces

Polygon Law of Forces

It states that “if more than two forces acting simultaneously on a particle, be represented in magnitude and direction by the sides of a polygon, taken in order, their resultant may be ,represented in magnitude and direction by the side closing of the polygon, taken in opposite order”.

Fig. 1.4 Polygon Law of forces

1. A, B, C, D, E are five forces.2. R is the resultant forces i.e. the closing side of the polygon taken in opposite order.

1.5 Method of Vectors to find out the Resultant Force

This method also used to find out the resultant of given forces. First of all draw the space diagram for the given forces. Now select some suitable points and go on adding the forces vertically. Then the closing side taken in opposite direction, will represent the resultant.

1.5.1 Equilibrium of forces

If the resultant of a number of forces, acting on a particle is zero, the particle will be in equilibrium. Such a set of forces, whose resultant is zero are called equilibrium forces.

1.5.2 Coplanar forces

The forces, whose lines of action lie on the same plane, are known as coplanar forces.

1.5.3 Concurrent forces

The forces, which meet at one point, are known as concurrent forces.

1.5.4 Lames theorem

It states that three coplanar forces acting on a point be in equilibrium, then ratio of force to the sine of the angle between the other two, forces are always equal to the ratio of other force to the sine of the angle between the remaining forces.(Fig.1.5)

Fig. 1.5 Lames Theorem

Mathematically,

P/Sinα =θ/Sinβ= R/Sinγ

Where P, Q, R are three forces and α,β,γ are the angles.

1.6 Graphical Method of Studying the Equilibrium of Forces

The equilibrium of such forces may also be studied, graphically, by drawing the vector diagram.

1.6.1 Converse of the law of triangle of forces

If three forces acting at a point be represented in magnitude and direction by the three sides of a triangle taken in order, the forces shall be in equilibrium.

1.6.2 Converse of the law of polygon of forces

If any number of forces acting at a point be represented in magnitude and direction by the sides of a closed polygon, taken in order, the forces shall be in equilibrium.