Problem 32 - Master Resource Book in JEE Main Physics

Question

If \(M\) mass of the square plate before cutting the holes, then mass of portion of each hole. $$ m=\frac{M}{16 R^{2}} \times \pi R^{2}=\frac{\pi}{16} M $$ \(\therefore\) Moment of inertia of remaining portion $$ \begin{aligned} I &=I_{\text {square }}-4 l_{\text {hole }} \\ &=\frac{M}{12}\left(16 R^{2}+16 R^{2}\right)-4\left[\frac{m R^{2}}{2}+m(\sqrt{2} R)^{2}\right] \\ &=\frac{M}{12} \times 32 R^{2}-10 \mathrm{mR}^{2} \\ &=\frac{8}{3} M R^{2}-\frac{10 \pi}{16} M R^{2}=\left(\frac{8}{3}-\frac{10 \pi}{16}\right) M R^{2} \end{aligned} $$

Step-by-Step Solution

Answer

The moment of inertia of the remaining plate is \( \left( \frac{8}{3} - \frac{10\pi}{16} \right) M R^2 \).

Step 1: Understanding the Problem

We are given a square metal plate with holes cut out of it. We need to find the moment of inertia of the remaining part of the plate after removing the mass of the holes.

Step 2: Determine Mass of Each Hole

The mass of the entire plate without holes is given as \( M \). The problem states that the mass of each hole, \( m \), is \( \frac{\pi}{16} M \). This was derived using the formula for the mass of the holes \( m = \frac{M}{16 R^2} \times \pi R^2 \).

Step 3: Calculate Initial Moment of Inertia

The moment of inertia \( I_{square} \) for a square plate of side \( 4R \) is \( \frac{M}{12}(16R^2 + 16R^2) = \frac{M}{12} \times 32R^2 \). This simplifies to \( \frac{8}{3}MR^2 \).

Step 4: Calculate Moment of Inertia for Each Hole

Each hole contributes two components to adjust the moment of inertia: first from its own spread at \( \frac{mR^2}{2} \), and second from displacement relative to the center \( m(\sqrt{2} R)^2 \). The total contribution from each hole is therefore \( 4 \times \left[ \frac{mR^2}{2} + m(2R^2)\right] = 10mR^2 \).

Step 5: Subtract Hole Contributions

The moment of inertia for the remaining plate is the initial moment \( \frac{8}{3}MR^2 \) minus the combined contributions of the holes \( 10mR^2 \). Thus, the adjusted moment of inertia is \( \frac{8}{3} M R^2 - 10mR^2 \).

Step 6: Final Adjustments and Simplification

Substituting \( m = \frac{\pi}{16}M \) into \( 10mR^2 \), we have \( 10 \times \frac{\pi}{16} M R^2 = \frac{10\pi}{16}M R^2 \). Then, subtract this from \( \frac{8}{3}MR^2 \) to get the final expression: \( \left( \frac{8}{3} - \frac{10\pi}{16} \right) M R^2 \).

Key Concepts

Rotational Dynamics

Rotational dynamics is an area of physics concerned with the motion of rotating objects and the forces that affect them. Just as linear dynamics deals with the motion of objects and the forces that affect linear motion, rotational dynamics considers objects that spin or rotate around an axis.
Key concepts in rotational dynamics include:

• Torque: This is the rotational equivalent of force. While force causes an object to move linearly, torque causes it to rotate. Torque depends on the magnitude of the force, the distance from the pivot point, and the angle at which it's applied.
• Moment of Inertia: This measures an object's resistance to change in its rotational state. For a rotating object, it is similar to mass in linear motion. Each part of the object contributes to the total moment of inertia based on its mass and distance from the rotation axis.
• Angular Momentum: This is a property of anything that rotates and is conserved in the absence of an external torque, much like linear momentum in translational dynamics.

Understanding these principles is crucial in predicting how objects will react under various rotational influences, such as spinning wheels, rotating planets, or even small gadget components.

Mass Distribution

Mass distribution refers to how mass is spread across an object, which significantly impacts how that object rotates. The way mass is distributed affects the moment of inertia, which defines the object's resistance to changes in its rotation.
In the case of rotational dynamics, the mass further from the rotation axis contributes more to the moment of inertia than mass closer to the axis. For example, a disc with its mass concentrated at the rim will have a larger moment of inertia than one with its mass concentrated at the center.
Some essential aspects to consider include:

• Uniform vs Non-uniform Distribution: An object with a uniform distribution of mass has the same density throughout. A non-uniform distribution means that some areas are denser than others, affecting how the object spins.
• Calculating Mass Distribution: Subtracting mass from certain areas, like drilling holes, changes the distribution. This can decrease the moment of inertia, making the object easier to spin or stop.

Understanding mass distribution allows engineers to optimize objects for stability, energy efficiency, and performance in areas like vehicle design, structural engineering, and aerospace applications.

Mechanical Engineering

Mechanical engineering involves applying principles of physics and materials science for the design and analysis of mechanical systems. This field requires a solid understanding of both linear and rotational dynamics, as well as mass distribution.
1. **The Role of Moment of Inertia:** In engineering, calculating the moment of inertia is crucial for designing anything that rotates, including engines, turbines, and gears. The rotational dynamics are key in determining how much energy is needed to start or stop these systems. 2. **Designing Efficient Systems:** By understanding mass distribution, mechanical engineers can design systems that operate efficiently. Reducing the moment of inertia by adjusting the mass distribution enables lighter and faster-responding mechanical components, improving performance in vehicles and machinery. 3. **Applications Across Fields:** Mechanical engineering principles apply to a diverse range of fields, including transportation, robotics, aerospace, and energy. For example, optimizing the mass distribution of a drone's propellers can enhance its stability and flight duration. Through a clear understanding of these concepts, engineers create innovative designs and solutions driving technological advances across numerous industries.