Line \(\mathrm{PQ}\) is parallel to \(\mathrm{y}\)-axis and moment of inertia of a rigid body about PQ line is given by \(\mathrm{I}=2 \mathrm{x}^{2}-12 \mathrm{x}+27\), where \(\mathrm{x}\) is in meter and \(\mathrm{I}\) is in \(\mathrm{kg}-\mathrm{m}^{2}\). The minimum value of \(\mathrm{I}\) is : (1) \(27 \mathrm{~kg} \mathrm{~m}^{2}\) (2) \(11 \mathrm{~kg} \mathrm{~m}^{2}\) (3) \(17 \mathrm{~kg} \mathrm{~m}^{2}\) (4) \(9 \mathrm{~kg} \mathrm{~m}^{2}\)

Understanding quadratic equations is key to solving many mathematical problems, including finding the minimum moment of inertia in this exercise. A quadratic equation is usually written as \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. The shape of the graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \(a\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards. This forms the basis for finding the vertex, where the minimum or maximum value of the quadratic function occurs. The given quadratic function representing the moment of inertia is: \[ I = 2x^2 - 12x + 27 \] which is already in the standard quadratic form \(ax^2 + bx + c\). Here, \(a = 2\), \(b = -12\), and \(c = 27\). The sign of \(a\) (which is positive) tells us that the parabola opens upwards and thus has a minimum point.

The vertex of a parabola is crucial in finding the minimum or maximum value of a quadratic function. When we talk about the vertex of a parabola described by \(ax^2 + bx + c\), we refer to the point where the function reaches its extreme value (either minimum or maximum). For a parabola that opens upwards (\(a > 0\)), this vertex represents the minimum value. The x-coordinate of the vertex can be calculated using the formula: \[ x_{\text{vertex}} = -\frac{b}{2a} \] In our problem: \[ a = 2 \] \[ b = -12 \] Substituting these values in the formula: \[ x_{\text{vertex}} = -\frac{-12}{2 * 2} = 3 \] This x-coordinate signifies where the minimum value of the moment of inertia occurs. Then, to find the minimum moment of inertia \(I\), we substitute \(x = 3\) back into the quadratic equation: \[ I = 2(3)^2 - 12(3) + 27 = 18 - 36 + 27 = 9 \] Thus, the minimum moment of inertia is \(9 \text{ kg m}^2\).

Moment of inertia is a physical concept that measures the rotational inertia of a rigid body—how resistant it is to rotational motion about a particular axis. It depends on both the mass distribution and the geometry of the object. In this exercise, the moment of inertia about a line parallel to the y-axis (line PQ) is given by a quadratic function: \[ I = 2x^2 - 12x + 27 \] Moment of inertia plays a crucial role in physics and engineering, particularly in the design and analysis of rotating systems such as wheels, gears, and flywheels. The importance of finding the minimum moment of inertia lies in optimization problems where energy efficiency and rotational dynamics are critical considerations. As calculated, this minimum value is \(9 \text{ kg m}^2\). This value helps in understanding the behavior of the rigid body when subjected to rotational forces and also aids in designing more efficient systems. Understanding these foundational concepts ensures a clearer grasp of such physics problems and their real-world applications.