Problem 566 - JEE Physics

Question

Two disc of same thickness but of different radii are made of two different materials such that their masses are same. The densities of the materials are in the ratio \(1: 3\). The moment of inertia of these disc about the respective axes passing through their centres and perpendicular to their planes will be in the ratio. \(\\{\mathrm{A}\\} 1: 3\) \\{B\\} \(3: 1\) \\{C\\} \(1: 9\) \(\\{\mathrm{D}\\} 9: 1\)

Step-by-Step Solution

Answer

The ratio of the moments of inertia of the two discs is \({B}\\ \, 3:1 \).

Step 1: Determine the formula for the moment of inertia of a disc

The moment of inertia of a disc about an axis passing through its center and perpendicular to its plane is given by the formula: \[I = \frac{1}{2}MR^2\] Where: \(I\) is the moment of inertia \(M\) is the mass of the disc \(R\) is the radius of the disc

Step 2: Set up the relationship between the given densities and radii

The density \(\rho\) of a material is defined as the mass per unit volume: \(\rho = \frac{M}{V}\). For a circular disc, the volume can be expressed in terms of the radius and thickness: \(V = \pi R^2 h\), where \(h\) is the thickness. Since both discs have the same mass, we can write: \[\frac{\rho_1}{\rho_2} = \frac{3}{1} = \frac{M_1 R_1^2 h}{M_2 R_2^2 h}\] Since \(M_1 = M_2\) and the thicknesses are equal, we can cancel them out: \[\frac{3}{1} = \frac{R_1^2}{R_2^2}\]

Step 3: Find the ratio of the moments of inertia of the two discs

We want to find the ratio \(\frac{I_1}{I_2}\), where \(I_1 = \frac{1}{2}M_1R_1^2\) and \(I_2 = \frac{1}{2}M_2R_2^2\). Using the known ratio of densities, we can calculate the ratio of moments of inertia: \[\frac{I_1}{I_2} = \frac{\frac{1}{2}M_1R_1^2}{\frac{1}{2}M_2R_2^2} = \frac{R_1^2}{R_2^2}\] Using the relation obtained in Step 2 for \(\frac{R_1^2}{R_2^2}\), we can write: \[\frac{I_1}{I_2} = \frac{3}{1} = 3:1\] So the answer to the problem is: \[{B}\\ \, 3:1 \]

Key Concepts

Disc Density

Density is a key property of materials that tells us how much mass is packed into a given volume. For a disc, the density \( \rho \) is defined as the mass \( M \) divided by the volume \( V \). This relationship is expressed by:

• \( \rho = \frac{M}{V} \)

For circular discs, volume can be calculated as \( V = \pi R^2 h \), where \( R \) is the radius and \( h \) is the thickness. This means that even if two discs have the same mass, differences in density will affect their size, specifically radius if thickness is constant.
Understanding density helps us see how different materials can occupy space differently, leading to different physical properties and behaviors.

Mass and Volume Relationship

The relationship between mass and volume is crucial in physics. When we talk about discs made of different materials but having the same mass, we're really exploring how changes in volume reflect the properties of those materials. Mass is constant here, leading to the formula:

• \( M = \rho V \)

For discs with equal thicknesses, the volume focuses on the radius. Hence, if one disc is made from a material three times as dense as the other, the denser material's disc must have a smaller radius to maintain the same mass. This gives rise to a direct way to compare properties based on radius and volume.

Ratio of Densities

In this problem, the two materials have densities in a ratio of \( 1:3 \). This tells us that one material is three times as dense as the other. By setting mass equal for both discs, we can deduce:

• \( \frac{\rho_1}{\rho_2} = \frac{3}{1} \)

With density ratio known and mass same, this allows us to derive a relationship between their radii:

• \( \frac{R_1^2}{R_2^2} = \frac{3}{1} \)

This step simplifies solving the problem, as finding the ratios of any property linked to these will inherently depend on the density ratio.

Circular Disc Properties

Circular discs are defined by properties like radius, thickness, and mass. These features determine calculations like the moment of inertia, which tells us how difficult it is to change the disc's rotational motion. The formula for the moment of inertia \( I \) of a disc is:

• \( I = \frac{1}{2}MR^2 \)

Where \( I \) depends directly on both mass \( M \) and the square of the radius \( R \).
When comparing two discs with equal mass but different radii, the difference in their radius squared directly influences their moments of inertia. As demonstrated, if one disc has a radius that scales in a particular ratio due to density differences, this ratio directly influences the moment of inertia, providing a clear understanding of rotational dynamics.