From a triangular lamina of side a and mass m , a small triangular Lamina made by joining midpoints of sides of the triangular lamina is cut. The Moment of Inertia of complete Lamina about COM is Io . If the Moment of Inertia of remaining Lamina about the centroid is (p/q)I0 , then find the value of (p + q) . Where p and q are the least positive integers.

Question

From a triangular lamina of side a and mass m , a small triangular Lamina made by joining midpoints of sides of the triangular lamina is cut. The Moment of Inertia of complete Lamina about COM is Io . If the Moment of Inertia of remaining Lamina about the centroid is (p/q)I0 , then find the value of (p + q) . Where p and q are the least positive integers. <img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-fnk231siwlsb.png" /> (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Dimension analysis Io=KMa2 Now for small lamina Iâ²=K(M/4)((9/2))2=(k m a2/16) Iâ²=(Io/16) So moment of Inertia of remaining part IL=Io-Iâ² =Io-(Io/16) IL=(15 Io/16)

Dimension analysis Io​=KMa2 Now for small lamina I′=K4M​(29​)2=16kma2​ I′=16Io​​ So moment of Inertia of remaining part IL​=Io​−I′ =Io​−16Io​​ IL​=1615Io​​

Dimension analysis
$I_{o} = K M a^{2}$
Now for small lamina
$I^{'} = K \frac{M}{4} (\frac{9}{2})^{2} = \frac{km a ^{2}}{16}$
$I^{'} = \frac{I _{o}}{16}$
So moment of Inertia of remaining part
$I_{L} = I_{o} - I^{'}$
$= I_{o} - \frac{I _{o}}{16}$
$I_{L} = \frac{15 I _{o}}{16}$