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Document Title
Fundamentals of Stress and Vibration 2. Engineering Mechanics Chapter
[A Practical guide for aspiring Designers / Analysts]

Situation 4: consider a thin disc of thickness ‘t’. Assuming that the disc is rotating about its
vertical diametral axis, as shown in [Fig 2.30]. Find its MMOI.

[Fig 2.30: thin disc rotation about the vertical diameter]

Solution 4: the mass of the elemental strip is given by:

dm = 2y ∗ dx ∗ t ∗ ρ , the elemental MMOI is given by: dI = dm ∗ x
2
= dI = 2y ∗ dx ∗ t ∗ ρ ∗ x
2
R
2
I total = 2y ∗ dx ∗ t ∗ ρ ∗ x - - - - (2.29)
−R

Since the integration is with respect to ‘dx’, let us represent ‘y’ in equation (2.29) in terms of ‘x’.

From [Fig 2.30], we have: y = R − x
2
2
Substituting the value of ‘y’ in equation (2.29), we get:

R
2
2
2
I total = 2 R − x ∗ t ∗ ρ ∗ x dx - - - - (2.30)
−R
Further, from [Fig 2.30], we have: x = R sinθ , upon differentiation, we get: dx = R cos θ dθ

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 41 age 41
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