Question

The M.I of a disc of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) about an axis passing through the centre \(\mathrm{O}\) and perpendicular to the plane of disc is \(\left(\mathrm{MR}^{2} / 2\right)\). If one quarter of the disc is removed the new moment of inertia of disc will be..... \(\\{\mathrm{A}\\}\left(\mathrm{MR}^{2} / 3\right)\) \(\\{B\\}\left(M R^{2} / 4\right)\) \(\\{\mathrm{C}\\}(3 / 8) \mathrm{MR}^{2}\) \(\\{\mathrm{D}\\}(3 / 2) \mathrm{MR}^{2}\)

Short answer

The new moment of inertia of the disc after one quarter has been removed is \(I' = \frac{3MR^2}{8}\).

Step-by-step solution

Moment of Inertia of Full Disc

The initial moment of inertia (I) of the full disc with mass M and radius R is given by: \(I = \frac{MR^2}{2}\)

Determine the Mass and Radius of the Quarter Disc

Since one quarter of the disc is removed, the mass (m) and radius (r) of the removed quarter disc are: \(m = \frac{M}{4}\) (1/4th of the mass) \(r = R\) (radius remains the same)

Moment of Inertia of Quarter Disc

The moment of inertia (i) of the removed quarter disc can be calculated using the same formula: \(i = \frac{mr^2}{2}\) Substitute the values of m and r: \(i = \frac{\left(\frac{M}{4}\right)R^2}{2}\)

Simplify the Moment of Inertia of Quarter Disc

Simplify the expression to get the moment of inertia of the quarter disc: \(i = \frac{MR^2}{8}\)

New Moment of Inertia of the Disc

The new moment of inertia (I') of the remaining disc can be obtained by subtracting the quarter disc's moment of inertia from the full disc's moment of inertia: \(I' = I - i\) substitute the values: \(I' = \frac{MR^2}{2} - \frac{MR^2}{8}\)

Simplify the New Moment of Inertia of the Disc

Simplify the expression to get the new moment of inertia of the disc: \(I' = \frac{4MR^2 - MR^2}{8}\) \(I' = \frac{3MR^2}{8}\)

Match with the Correct Option

Now, match the obtained new moment of inertia with the given options: The new moment of inertia of the disc is \(\frac{3MR^2}{8}\), which matches with option C. Therefore, the correct answer is option C. \(I' = \frac{3MR^2}{8}\).

Key concepts

Disc Mechanics

In the world of physics, especially when dealing with disc mechanics, understanding the behavior of discs is crucial. A disc is a flat, circular object, and its mechanical properties are essential in various applications, from vehicles to machinery. When analyzing a disc, one usually considers its mass, radius, and how these factors influence its moment of inertia—an important concept that measures how much resistance the disc shows to rotational motion. This resistance plays a key role in calculating rotational dynamics. The full disc, specifically, has a uniformly distributed mass, making it relatively simple to calculate the moment of inertia when considering the full shape. In cases like our exercise, where a part of the disc is removed, understanding the mechanics of the disc becomes crucial to determining changes in its moment of inertia. By working through the mass and radius calculations, we can see how these play out in practice.

Rotational Dynamics

Rotational dynamics is the study of the motion of rotating bodies and is a central theme in physics. When we talk about rotational dynamics in the context of a disc, we focus on how the disc rotates around its center, and how any changes to its shape or mass affect this rotation. The moment of inertia is a pivotal concept here—it quantifies the rotational inertia of a body. For a solid disc, this is calculated through the formula: \( I = \frac{MR^2}{2} \). This formula tells us that both mass and radius are directly involved in the resistance to rotation. In scenarios where parts of the disc are removed, like in our exercise, this affects the rotational dynamics, as reducing mass decreases the moment of inertia. It's essential to remember that the distribution of mass is always around the same axis, which makes the calculations easier to manage in this context.

Mass Distribution

The concept of mass distribution is particularly important when dealing with moments of inertia. In a uniform disc, mass is evenly spread out, which simplifies calculations. When a section, such as a quarter of a disc, is removed, the distribution changes, and so does the inertia. In our exercise, the removed part still has its mass distributed evenly within its quarter section, but it modifies the overall balance of the disc. This change necessitates a reevaluation of the remaining disc's properties. We calculate the remaining disc's new moment of inertia by considering how the mass has been redistributed—by removing \( \frac{M}{4} \), the disc's original balance shifts, affecting its resistance to rotational movement.

Physics Problem Solving

Solving physics problems often requires a structured approach, starting with a clear understanding of the problem and the concepts involved. In our example, the goal is to find the new moment of inertia after removing a portion of the disc. To tackle this, follow these steps:

• Determine the initial moment of inertia;
• Evaluate the parameters (mass and radius) for the removed section;
• Calculate the moment of inertia for the removed portion;
• Modify the full disc's moment of inertia by subtracting the calculated inertia of the removed part.

These steps help simplify the problem, providing a clear path to the solution. Each task in this process solidifies an understanding of fundamental mechanics principles, providing a solid foundation for more complex physics topics.