Question

find the moment of inertia of a uniform square lamina of mass m and side a about its diagonal.

Answer 1

Moment of inertia equals:

$$I = \int r ^ {2} d m$$

For one side:

$$I _ {1} = \int r ^ {2} d m,$$

where $r = x\sin 45 = \frac{x}{\sqrt{2}}$

and $dm = \frac{dx}{a} * \frac{m}{4}$ - lamina is uniform and one side has mass $\frac{m}{4}$

$$I _ {1} = \frac {m}{4 a} \int_ {0} ^ {a} \frac {x ^ {2}}{2} d x = \frac {m}{4 a} \frac {1}{2} \frac {a ^ {3}}{3} = \frac {m}{4} * \frac {1}{6} a ^ {2}$$

Total moment of inertia equals:

$$I = 4 I _ {1} = \frac {1}{6} m a ^ {2}$$

Answer: $I = \frac{1}{6} ma^2$