First slide

Moment of intertia

Question

Seven identical disc are arranged in a hexagonal, planar pattern so as to touch each neighbor, as shown in the figure. Each disc has mass m and radius r. What is the moment of inertia of the system of seven discs about an axis passing through the centre of central disk and normal to plane of all discs?

Moderate

A

$\frac{7}{2} {mr}^{2}$
B

$\frac{13}{2} {mr}^{2}$
C

$\frac{29}{2} {mr}^{2}$
D

$\frac{55}{2} {mr}^{2}$

Solution

$I = I_{0} + 6 I^{'}$
$I_{0}$ is the moment of inertia of central disc and $I^{'}$ is moment of inertia of rest of the each disc about specified axis.
By parallel axes theorem
$I = \frac{{mr}^{2}}{2} + 6 (I_{cm} + {md}^{2}) = \frac{{mr}^{2}}{2} + 6 [\frac{{mr}^{2}}{2} + m {(2 r)}^{2}]$
= $\frac{{mr}^{2}}{2} + \frac{6 {mr}^{2}}{2} + 24 {mr}^{2} = \frac{55 {mr}^{2}}{2}$

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