Problem 13 - Master Resource Book in JEE Main Physics

Question

According to the theorem of parallel axes, moment of inertia of disc about an axis passing through \(K\) and perpendicular to plane of \(\operatorname{disc}_{,}\) \(=\frac{1}{2} M R^{2}+M R^{2}=\frac{3}{2} M R^{2}\) Total moment of inertia of the system \(=\frac{3}{2} M R^{2}+m(2 R)^{2}+m(\sqrt{2} R)^{2}+m(\sqrt{2} R)^{2}\) \(=3(M+16 m) \frac{R^{2}}{2}\)

Step-by-Step Solution

Answer

The total moment of inertia of the system is \(3(M + 16m)\frac{R^{2}}{2}\).

Step 1: Moment of Inertia of the Disc

According to the parallel axes theorem, the moment of inertia of a disc about an axis through a point \(K\) and perpendicular to the plane is given by the expression \(\frac{1}{2} M R^{2} + M R^{2}\). This simplifies to \(\frac{3}{2} M R^{2}\).

Step 2: Calculate Moment of Inertia of Point Masses

Each point mass contributes to the total moment of inertia based on its distance from the axis of rotation. The mass \(m\) at a distance \(2R\) gives an inertia of \(m(2R)^{2} = 4mR^{2}\). The masses \(m\) each at a distance \(\sqrt{2} R\) contribute \(m(\sqrt{2} R)^{2} = 2mR^{2}\) each.

Step 3: Sum Up All Contributions

Add all the contributions to find the total moment of inertia of the system. This includes \(\frac{3}{2} M R^{2}\) from the disc and \(4mR^{2} + 2mR^{2} + 2mR^{2}\) from the point masses.

Step 4: Simplify the Expression

Combine the terms: \(\frac{3}{2} M R^{2} + 4mR^{2} + 2mR^{2} + 2mR^{2} = \frac{3}{2} M R^{2} + 8mR^{2}\). Further simplify this to get the total moment of inertia as \(3(M + 16m)\frac{R^{2}}{2}\).

Key Concepts

Moment of Inertia

The concept of moment of inertia is fundamental in mechanics, especially when dealing with rotational motion. It is the rotational equivalent of mass and indicates how much torque is needed for a desired angular acceleration. Moment of inertia depends not only on the object's mass but also on how this mass is distributed relative to the axis of rotation. For simpler shapes, the moment of inertia can be found using standard formulas, but for more complex systems, like that described in the exercise, calculations can involve summing contributions from individual components.

• It's an intrinsic property indicating resistance to rotation.
• A higher moment of inertia means more energy is required to spin the object.
• Depends greatly on the distribution of mass around the axis.

Understanding how it works helps in analyzing systems in rotational motion and designing mechanisms that require precise control of spinning components.

Rotation

Rotation refers to the motion where an object spins around an internal or external axis. In our exercise, the disc rotates about an axis perpendicular to its plane, passing through a point, often called point K.
When examining rotational motion, it's essential to differentiate it from linear motion, as forces and energy are differently expressed:

• Rotational movement may occur around various kinds of axes, either fixed or moving.
• Variables like angular velocity and angular acceleration become relevant.
• Torque takes the place of force, being the rotational equivalent.

The kinematics and dynamics of rotation involve unique equations and considerations. In systems like the one in the exercise, rotation influences how mass distribution affects the overall moment of inertia and stability.

Point Masses

A point mass refers to an object whose dimensions are negligible in comparison with the distances involved in a problem. It simplifies calculations in physics, especially when determining the moment of inertia.
In the exercise, each point mass contributes to the total moment of inertia based on its distance from the axis.

• For instance, a mass at distance \(2R\) leads to an inertia contribution of \(4mR^2\).
• Those at distance \(\sqrt{2}R\) contribute \(2mR^2\) each.
• This simplification allows for easy integration of point masses into larger systems.

Point masses are crucial in breaking down complex rotational inertia problems. By evaluating each point mass independently, the combined effect on the system's inertia can be easily calculated.

Distance from Axis

The distance from the axis of rotation is a critical variable when evaluating the moment of inertia of a system. It directly impacts the inertia of individual masses, according to the formula \(I = m r^2\) where \(I\) is the moment of inertia, \(m\) is mass, and \(r\) is the distance from the axis.
In the discussed exercise:

• A mass twice as far from the axis will contribute four times more to the inertia.
• The effective distance can alter the inertia impact of the same mass significantly.
• Accurate assessments of distance ensure precise determination of rotational characteristics.

Understanding the role of distance is vital for predicting how an object will behave when rotated, affecting energy requirements and system stability. Students often find this concept challenging, yet it's indispensable for accurate problem-solving in physics.