Figure shows triangular lamina which can rotate about different axes. The moment of inertia is maximum about the axis
Show Hint
To maximize the moment of inertia of any planar lamina, choose the axis that is located as far away from the center of mass as possible, or an edge that is directly opposite the most distant vertex. The further the mass stretches from the line of rotation, the larger the $r^2$ term becomes.
Step 1: Understanding the Question:
We are given a flat triangular lamina that can rotate about four distinct axes lying in or perpendicular to its plane ($PR$, $QS$, $QR$, and $PQ$). We need to identify the specific axis about which the lamina exhibits its maximum moment of inertia.
Step 2: Key Formula or Approach:
The moment of inertia $I$ for a collection of continuous mass points is defined by:
$$I = \sum m_i r_i^2$$
where $m_i$ represents elemental masses and $r_i$ represents their perpendicular distances from the chosen axis of rotation. Consequently, the moment of inertia increases when the bulk mass distribution is located farther away from the operational axis.
Step 3: Detailed Explanation:
Let's analyze the geometry of the triangular lamina relative to each option:
An axis running through the bulk body, like the altitude $QS$ or the diagonal segments, keeps the surrounding mass relatively close to the line of rotation, leading to a smaller distribution radius ($r$).
Now consider the baseline edges $PQ$ and $QR$. In a standard triangular layout where $QR$ serves as the primary base opposite to the furthest vertex $P$, choosing $QR$ as the axis means that the mass extending toward vertex $P$ is positioned at the largest average perpendicular distance possible.
Because more mass elements are situated at greater perpendicular distances ($r_i$) relative to the base axis $QR$ compared to any other axis option, the mathematical summation $\sum m_i r_i^2$ reaches its absolute maximum value here.
Step 4: Final Answer:
The moment of inertia is maximum about the axis QR, which corresponds to option (C).
