Problem 52 - Master Resource Book in JEE Main Physics

Question

Moment of inertia of the square plate about an axis passing through the centre and perpendicular to its plane is $$ I=\frac{m\left(a^{2}+a^{2}\right)}{12}=\frac{m a^{2}}{6} $$

Step-by-Step Solution

Answer

The moment of inertia is \(\frac{ma^2}{6}\).

Step 1: Understanding the Problem

We need to find the moment of inertia of a square plate with side length \(a\) and mass \(m\) about an axis that is perpendicular to its plane and passes through its center.

Step 2: Moment of Inertia Formula

The moment of inertia \(I\) of a square plate about the axis through its center and perpendicular to the plane is given by combining the moments of inertia about two perpendicular axes in the plane. For each axis, the moment is \(\frac{1}{12}ma^2\) because the square is symmetric. Thus, we sum them.

Step 3: Sum of Moments

Calculate \(I\) using both perpendicular axes: \[I = \frac{m(a^2 + a^2)}{12} = \frac{m(2a^2)}{12}\]

Step 4: Simplify the Expression

Simplify the expression by dividing the fraction:\[I = \frac{m(2a^2)}{12} = \frac{ma^2}{6}\]

Key Concepts

Square Plate

A square plate in this context refers to a flat object with four equal sides and right-angled corners. It is a two-dimensional geometric shape, which simplifies many calculations involving physical properties like the moment of inertia. The symmetry of the square is crucial as it equally divides mass along its axes.

• The side length of the square is often denoted as "\(a\)".
• The mass of the square plate is represented by "\(m\)".
• This symmetry aids in the mathematical evaluation of its mechanical properties.

The simplicity of a square makes it a vital model in physics to explore concepts such as balance and rotational dynamics.

Perpendicular Axis

The perpendicular axis theorem is pivotal when finding the moment of inertia for a quadratic shape such as a square plate. It implies that the total moment of inertia about an axis perpendicular to the plane of the object, like our square plate, can be determined by summing the moments of inertia about two perpendicular axes which lie in the same plane, generally \( x \) and \( y \) axes.

• The axis through which the moment of inertia is calculated is "perpendicular" to the square plate.
• This axis intersects the center of mass of the plate.

In essence, it states: \[ I_{z} = I_{x} + I_{y} \] where \( I_{x} \) and \( I_{y} \) are the moments of inertia around the two perpendicular axes.

Center of Mass

The center of mass of a body, such as our square plate, is the point where the entire mass of the object is considered to be concentrated for translational motion. For symmetrical objects, like a square plate, it lies at the geometric center, making calculations easier and the structure balanced.

• Center of mass occurs at the intersection of the plate's diagonals.
• For uniform mass distribution, it simplifies the analysis of rotational motion.

This concept is essential in calculating moments of inertia because the rotation often happens around this point in balanced objects.

Axis Through Center

In the context of the moment of inertia exercise, the axis through the center refers to an imaginary line that goes through the center of mass of the square plate and is perpendicular to the plate. This specific axis helps simplify the problem using symmetrical properties.

• The moment of inertia calculation around this axis takes advantage of the plate's uniform structure.
• Moments about this axis incorporate all symmetrically aligned sections of the plate equally.

This axis is critical in analyzing real-world physical systems such as rotating machinery or balancing objects.

Symmetrical Object

A symmetrical object, like our square plate, has balanced and uniform dimensions or properties around its center. Symmetry eases the calculation of physical properties like the moment of inertia because evenly distributed mass simplifies the math.

• Symmetrical objects have equal properties, like mass distribution, around their central axes.
• This leads to consistent calculations around any set of perpendicular axes.
• In physics, symmetry allows using simplified equations to predict motion and balance.

This inherent quality allows for predictable and uniform behavior when the object is in motion or static, making it a prime subject for study in both theoretical and applied physics.