Question

Two discs of the same material and thickness have radii \(0.2 \mathrm{~m}\) and \(0.6 \mathrm{~m}\) their moment of inertia about their axes will be in the ratio \(\\{\mathrm{A}\\} 1: 81\) \(\\{\mathrm{B}\\} 1: 27\) \(\\{C\\} 1: 9\) \(\\{\mathrm{D}\\} 1: 3\)

Short answer

The ratio of the moments of inertia for the two discs is 1:81. Since both discs have the same material and thickness, their masses are proportional to the square of their radii. Using the formula for the moment of inertia of a disc (\(I = \frac{1}{2}mr^2\)), we calculate the moments of inertia for both discs and determine the ratio \(\frac{I_1}{I_2} = \frac{1}{81}\). The correct answer is \(\mathrm{A} \: 1: 81\).

Step-by-step solution

Formula for the moment of inertia of a disc

Recall the formula for the moment of inertia (I) of a solid disc about its central axis: \[I = \frac{1}{2}mr^2\] where m is the mass and r is the radius of the disc.

Calculate the moments of inertia for both discs

Since both discs have the same material and thickness, their masses will be proportional to the square of their radii, i.e., \(m_2 = k(0.2)^2\) and \(m_1 = k(0.6)^2\) , where k is a constant of proportionality. Now we can calculate the moments of inertia for both discs using the given radii and the proportional masses: \(I_1 = \frac{1}{2}(k(0.2)^2)(0.2)^2 = \frac{k}{50}\) \(I_2 = \frac{1}{2}(k(0.6)^2)(0.6)^2 = \frac{9k}{10}\)

Calculate the ratio of the moments of inertia

We are interested in finding the ratio \(\frac{I_1}{I_2}\). Using the values we found in the previous step, we can write: \[\frac{I_1}{I_2} = \frac{\frac{k}{50}}{\frac{9k}{10}} = \frac{1}{81}\]

Find the matching option

The ratio we obtained is 1:81, which matches option A. Therefore, the correct answer is: \(\boxed{\mathrm{A}\: 1: 81}\).

Key concepts

Solid Disc

A solid disc is a three-dimensional circular object, typically flat, with uniform thickness and density. This means that the material composition is consistent throughout the disc, making it an idealized object for various physics calculations. Discs are foundational in rotational mechanics, particularly in understanding concepts like angular velocity and moment of inertia.
When considering a solid disc in physics, the key assumption is that the entire mass of the disc is distributed evenly across its surface. This uniform mass distribution is crucial for simplifying calculations. In practical terms, when you spin a solid disc, each point on the disc experiences the same centripetal force, which makes predictive calculations feasible using mathematical formulas like those relating to moment of inertia.

Central Axis

The central axis of a disc refers to an imaginary line that runs perpendicular to the disc's plane and passes through its center. Think of this axis as the point around which the disc spins or rotates.
The central axis is an essential concept in calculating the moment of inertia because it determines the balance and symmetry of the disc during rotation. For a solid disc, the moment of inertia calculation assumes rotation about this central axis, as it's the simplest case and provides symmetry, making calculations accurate and manageable.
The central axis affects the rotational dynamics such as torque and angular acceleration. Understanding this axis is key to grasping phenomena like gyroscopic motion, which relies on the disc maintaining its orientation due to its rotational inertia.

Proportional Mass

In the problem of calculating the moment of inertia for two discs of differing radii, the concept of proportional mass plays a fundamental role. Since both discs have the same thickness and are made from the same material, the mass of each disc is proportional to the square of its radius.
This is derived from the fact that, for a disc, mass is related to the area it covers. The formula for area is \(\pi r^2\), meaning that mass grows with the square of the radius, assuming constant density and thickness. In the exercise, the assumption is used that the mass of each disc can be expressed as \(m = k imes r^2\), where k is a constant.

• This means if one disc has a radius that is three times as long as another disc, its mass will be nine times greater.
• This principle of proportionality is key in solving for the moment of inertia by allowing us to express mass in relation to the radius and another proportionality factor.

Radius of Disc

The radius of a disc is the distance from its center to any point on the edge. It is a fundamental property that not only helps define the shape and size of the disc but also its rotational characteristics.
In calculations involving a disc's moment of inertia, the radius is crucial because it helps determine how much mass is distributed about the central axis. The formula for the moment of inertia of a solid disc is \[I = \frac{1}{2}mr^2\] ,
reflecting that the moment of inertia depends heavily on the square of the radius. This specific relationship highlights how quickly the moment of inertia increases as the disc’s size increases, due to the distribution of mass further from the central axis.
When analyzing two discs, as in the exercise, noting the disparity in radii helps understand why they have different moments of inertia, with a smaller radius corresponding to a smaller moment of inertia. This directly influences the rotational behavior of each disc, which is foundational in mechanical design and analysis.