Problem 567 - JEE Physics

Question

Let I be the moment of inertia of a uniform square plate about an axis \(\mathrm{AB}\) that passes through its centre and is parallel to two of its sides \(\mathrm{CD}\) is a line in the plane of the plate that passes through the centre of the plate and makes an angle of \(\theta\) with \(\mathrm{AB}\). The moment of inertia of the plate about the axis \(\mathrm{CD}\) is then equal to.... \(\\{\mathrm{A}\\} \mathrm{I}\) \(\\{B\\} I \sin ^{2} \theta\) \(\\{C\\} I \cos ^{2} \theta\) \(\\{\mathrm{D}\\} I \cos ^{2}(\theta / 2)\)

Step-by-Step Solution

Answer

The moment of inertia of the plate about the axis CD is equal to I (option A).

Step 1: Identify the given information

In this problem, we have a uniform square plate with moment of inertia I about an axis AB passing through its center. We are also given another axis CD that passes through the center of the plate and makes an angle θ with AB. Our task is to find the moment of inertia of the plate about the axis CD.

Step 2: Set up axes and find initial moments of inertia

Let's set our coordinate axes by placing the origin at the center of the square plate. Assume the plate has sides of length "a" and mass "m", and the axis AB is parallel to y-axis while CD makes an angle θ with the y-axis. The moment of inertia for the plate about the y-axis is given as \(I_y = \frac{1}{12}ma^2\). Since AB is parallel to the y-axis, we can say that I = \(I_y = \frac{1}{12}ma^2\). Now, we need to find the moment of inertia about the x-axis passing through the center of the plate. The moment of inertia for the plate about the x-axis is given as \(I_x = \frac{1}{12}ma^2\).

Step 3: Apply the Perpendicular Axis Theorem

The Perpendicular Axis Theorem states that the moment of inertia about an axis perpendicular to the plane of an object is equal to the sum of the moments of inertia about two mutually perpendicular axes lying in the plane of the object and passing through the same point. In our case, this point is the center of the square plate. Applying this theorem, we have: \(I_{CD} = I_x\cos^2\theta + I_y\sin^2\theta\)

Step 4: Substitute the values and simplify

We already know the values of \(I_x\) and \(I_y\) from the previous step. Now, we will substitute their values in the equation and simplify it. \(I_{CD} = \left(\frac{1}{12}ma^2\right)\cos^2\theta + \left(\frac{1}{12}ma^2\right)\sin^2\theta\) Factor out \(\frac{1}{12}ma^2\): \(I_{CD} = \frac{1}{12}ma^2(\cos^2\theta + \sin^2\theta)\) We know that \(\cos^2\theta + \sin^2\theta = 1\). So, \(I_{CD} = \frac{1}{12}ma^2\) Since we are given that the moment of inertia I about the axis AB is equal to \(\frac{1}{12}ma^2\), our final answer becomes: \(I_{CD} = I\) So, the correct option is: \(\\{\mathrm{A}\\} \mathrm{I}\)

Key Concepts

Perpendicular Axis Theorem

The Perpendicular Axis Theorem is a valuable tool in rotational dynamics. It helps simplify the calculation of the moment of inertia for objects with certain symmetries. This theorem is applicable to flat objects lying in a plane. According to the Perpendicular Axis Theorem, the moment of inertia \[ I_z \]about an axis perpendicular to the plane (let's call it the \(z\)-axis) is the sum of the moments of inertia about two perpendicular axes (\(x\) and \(y\)) that lie in the plane. In mathematical terms:\[ I_z = I_x + I_y \]This theorem allows us to utilize known moments of inertia about simpler axes to find the moment of inertia about a more complex axis. While straightforward, it is crucial to ensure that the perpendicular axis indeed passes through the common point of the other two axes and the object plane. This approach simplifies many complex calculations in rotational dynamics, especially for symmetrical objects like plates.

uniform square plate

A uniform square plate is a two-dimensional object with equal length sides and uniform mass distribution. For our purposes, the square plate has sides of length \(a\) and total mass \(m\). In problems involving rotational dynamics, understanding the distribution of mass is vital. A key aspect of a uniform plate is that the mass is evenly distributed across the entire surface. This uniformity simplifies calculations for moments of inertia, as we can employ standard formulas.For example, the moment of inertia about an axis passing through the center and parallel to one side can be determined using:\[ I = \frac{1}{12}ma^2 \]This formula comes in handy when applying the Perpendicular Axis Theorem or when analyzing rotational motion.

rotational dynamics

Rotational dynamics is the study of the motion of rotating bodies and the forces and torques that cause or change this motion. In rotational dynamics, concepts like torque, angular momentum, and moment of inertia play crucial roles.

• Moment of Inertia: This measures how resistant an object is to changes in its rotational motion. It depends on both the mass of the object and its distribution relative to the axis of rotation.
• Angular Momentum: Analogous to linear momentum, it depends on the rotational inertia and angular velocity of a rotating object.
• Torque: The rotational equivalent of force, which causes changes in the rotational motion.

In our problem, understanding these concepts helps us determine how a square plate behaves when a force acts on it or when it rotates about different axes. Through the lens of these principles, we can quantify and predict how objects will move or respond to their environments, which is essential in physics and engineering applications.