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Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics
Let us now find the shortest distance, using the distance formula, as follows:

x − x + y − y + z − z
2
2
2
1
2
1
2
2
1
= 1357 − 1479.8 + 6385 − 6408 + 1507 − 1278 = 260.2 kms
2
2
2
Now let us find the distance between Bangalore and Coimbatore in the great circle as shown in
[Fig 1.74].

[Fig 1.74: Distance between the cities on the great circle]

Š‡ ƒ” Ž‡‰–Š ȋ Ȍ ‘‡ –‹‰ ƒ‰ƒŽ‘”‡ ƒ† ‘‹„ƒ–‘”‡ ‹• ‡“—ƒŽ –‘ ȋ ɅȌǡ ™Š‡”‡‹ǡ ǮɅǯ ‹• –Š‡ ˜ƒŽ—‡
–Šƒ– ‡‡†• –‘ „‡ ‘’—–‡†Ǥ ‡– —• ‘’—–‡ ǮɅǯ „› • ƒŽƒ” ’”‘†— –Ǥ

ȋ ‘• ɅȌ ‹• ‰‹˜‡ „›ǡ •— ‘ˆ ’”‘†— – ‘ˆ ”‡•’‡ –‹˜‡ ‘’‘‡–• ‘ˆ –Š‡ –™‘ ˜‡ –‘”•ǡ †‹˜‹†‡† „› –Š‡
’”‘†— – ‘ˆ ƒ‰‹–—†‡ ‘ˆ –Š‡ ˜‡ –‘”•Ǥ ‡”‡ǡ –Š‡ –™‘ ˜‡ –‘”• ƒ”‡ ȋ Ȍ ƒ† ȋ Ȍǡ ™Š‘•‡ ƒ‰‹–—†‡ ‹•
–Š‡ ”ƒ†‹—• ȋ Ȍ ‘ˆ –Š‡ ‡ƒ”–ŠǤ

ƒ–Š‡ƒ–‹ ƒŽŽ›ǡ ™‡ Šƒ˜‡ǣ

x x + y y + z z 1357 ∗ 1479.8 + 6385 ∗ 6408 + 1507 ∗ 1278
1 2
1 2
1 2
cos θ = = = 0.9992

R R 6700 2
Therefore, we have: θ = cos −1 0.9992 = 2.29 degrees = 0.0407 radians
And we have, Rθ = 6700 ∗ 0.0407 = 272.69 kms

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
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