First slide

Moment of intertia

Question

A metallic sheet of mass 2 kg has the shape of an isosceles triangle OAB as shown in figure. Then its moment of inertia about an axis passing through O and perpendicular to its plane is

Difficult

A

$\frac{4}{3} {Kg-m}^{2}$
B

$\frac{2}{3} {Kg-m}^{2}$
C

$\frac{8}{3} {Kg-m}^{2}$
D

$\frac{1}{4} {Kg-m}^{2}$

Solution

If four such triangles are assembled a square ABCD is formed as shown in figure. Whose side $l = 2 \times 2 \cos 45^{o} m = 2 \sqrt{2} m$ .
$m^{l} =$ Mass of the square $=4×2 Kg=8 Kg$
Moment of inertia of the square about an axis passing through O and perpendicular to its plane is given by
$I_{0}^{1} = \frac{1}{12} m^{1} l^{2} + \frac{1}{12} m^{1} l^{2} = \frac{1}{6} \times 8 \times {(2 \sqrt{2})}^{2} K g - m^{2} = \frac{32}{3} K g - m^{2}$
$∴$ Moment of inertia of the triangle ABC, $I_{0} = \frac{I_{0}^{1}}{4} = \frac{8}{3} K g - m^{2}$

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