Mathematics

Contents

Serial No. Topic Page No.

01 Introduction 02

02 Task 1 03

03 Task 2 09

04 Task 3 15

05 Task 4 19

06 Overview 23

07 References 24

Introduction

The first engineer known to history was the Egyptian legend Imhotep, circa 2550BC, who created one of the most astounding engineering marvels of all time, the Great Pyramids at Gizeh. Besides being a magnificent sight, the pyramids are a miracle of mathematics: the surface area of each face of the pyramid equals the square of its height; whether inside or outside, all its dimensions, orientations and angles are perfect; massive granite blocks are cut precise to one tenths of a millimetre; the slopes of the pyramid diverge in perfect proportion from the apex, in a magnificent convex that seems to embrace the sun. This is but one historical instance of millions, which shows engineering ingenuity through applied mathematics.

All engineering can be arguably said to rely heavily on mathematic fundamentals. Through the study of mathematical quantities and relations, an engineer can devise a system to exacting requirements and attain new heights in his practice.

This assignment aims to throw necessary light on how a compendium of many mathematical branches serves as necessary tools for engineering.

Task 1

A lathe is a machine tool which spins a block of material to perform various operations such as cutting, sanding, knurling, drilling, or deformation with tools that are applied to the work piece to create an object which has symmetry about an axis of rotation.

a) Here, the lathe tool has to accommodate work between two set extremities. For convenience and commonality of solution, we include these extremities as part of our calculations.

Bar diameters D1, D6 = 25mm, 300mm respectively.

Cutting speed S = 25mm min-1 = 25000mm min-1

No of spindle speeds n = 6

As , =318.31rpm

Similarly, =26.53 rpm

i) An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.

If a1 is the first term, an is the nth term and d is common difference, then

Here,

That is, = вЂ"58.36 rpm

The sequence of spindle speeds is hence:

318.31, 259.95, 201.59, 143.23, 84.87, 26.51 in rpm.

Bar diameters can be determined using as follows:

25, 30.61, 39.47, 55.56, 93.76, 300 in mm.

ii) A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number.

If a1 is the first term, an is the nth term and r is common ratio, then

Here, =0.6083

The sequence of spindle speeds is hence:

318.31, 193.63, 117.78, 71.64, 43.58, 26.51 in rpm.

Bar diameters can be determined using as follows:

25, 41.09, 67.56, 111.07, 182.6, 300 in mm.

b) AP graph

GP graph

c) AP graph

GP graph

Among spindle speeds in AP and GP, GP is preferred, as it gives a closer fit to ideal cutting speed over all possible sizes, although only six sizes can be cut at exactly 25m min-1.

d) Let the length of the first piece

x cm. By the definition in solution a) i),

the lengths of the other two pieces will be

xr cm and xr2 cm respectively, where r is the

common ratio.

By the question,

We also have

Hence

Or,

Or,

Or,

This is a quadratic equation which can be reduced to .

Sometimes, it is useful to know the characteristics of the equation by plotting its graph:

As r is a multiple of length, it cannot be negative; hence the second solution is rejected.

пÑ"Ñ› r = 2 пÑ"Ñ› =7cm.

Therefore, the lengths of each piece are 7, 14 (7п‚Ò'2) and 28 (7п‚Ò'22) cm.

Task 2

a) We can plot graphs with the help of comprehensive tables:

(Degrees)

(Units)

-360 0

-300 26

-240 26

-180 0

-120 -26