Problem 591 - JEE Physics

Question

Two identical hollow spheres of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) are joined together and the combination is rotated about an axis tangential to one sphere and perpendicular to the line connecting their centres. The moment of inertia of the combination is \(\\{\mathrm{A}\\} 10 \mathrm{MR}^{2}\) \(\\{\mathrm{B}\\}(4 / 3) \mathrm{MR}^{2}\) \(\\{\mathrm{C}\\}(32 / 3) \mathrm{MR}^{2}\) \(\\{\mathrm{D}\\}(34 / 3) \mathrm{MR}^{2}\)

Step-by-Step Solution

Answer

The total moment of inertia for the system of two identical hollow spheres is \(10 MR^2\), which corresponds to option (A).

Step 1: Moment of Inertia for a Single Sphere

The moment of inertia \(I\) for a single hollow sphere about an axis through its center can be given as \(2/3 MR^2\).

Step 2: Moment of Inertia for the Combined System

Since the axis passes through the edge of one of the spheres and is perpendicular to the line joining the centers, for the sphere with the axis passing through it, the moment of inertia stays as \(2/3 MR^2\). For the other sphere, we have to apply the parallel axis theorem. It states that the moment of inertia about any axis parallel to and a distance d away from the axis through the centre of mass of an object is given by \(I+Md^2\), where \(I\) is the moment of inertia about the axis through the center of mass, \(M\) is the object's mass, and \(d\) is the distance from the center of mass to axis. In our case, \(d\) is equal to the diameter of the sphere which is \(2R\), thus for the other sphere, it is \(2/3 MR^2 + M(2R)^2\).

Step 3: Calculate the Total Moment of Inertia

To obtain the total moment of inertia for the entire system, we must sum the moment of inertia of both spheres. So, the total moment of inertia is \(2/3 MR^2 + 2/3 MR^2 + 4MR^2 = 10 MR^2\).

Step 4: Checking the Options

Comparing the result with the given options, we find that the total moment of inertia is equal to option (A) \(10 MR^2\), which is the correct answer.

Key Concepts

Parallel Axis Theorem

The parallel axis theorem is a valuable tool in rotational dynamics. It allows us to calculate the moment of inertia of a body about any axis, given we know its moment of inertia about an axis through its center of mass. The formula for this theorem is:

• If the distance between the two axes is described as "d", then the moment of inertia about the new axis is given by \(I + Md^2\), where \(I\) is the moment of inertia about the center of mass axis, and \(M\) is the mass of the object.

This theorem is particularly useful in situations where complex systems rotate about axes that do not pass through their center of mass. In the given exercise, the theorem helps in finding the moment of inertia of one of the hollow spheres as it is not rotating about its own center. By applying the theorem, we shift the axis from the central axis to a tangential axis by using the distance between the centers of the two spheres.

Hollow Sphere

Understanding the moment of inertia of a hollow sphere is essential for solving problems involving rotational dynamics. A hollow sphere, as opposed to a solid sphere, has all its mass distributed only around its spherical shell rather than throughout its volume. This unique mass distribution leads to a different formula for the moment of inertia.

• The moment of inertia of a hollow sphere about an axis through its center is given by \( \frac{2}{3}MR^2 \), where \(M\) is the mass and \(R\) is the radius.

This difference arises because the mass is concentrated further away from the axis, contributing more to the rotational inertia compared to a solid sphere with the same total mass and radius. In the given exercise, this formula is crucial as it serves as the starting point for applying the parallel axis theorem to one of the spheres.

Tangential Axis

Rotating about a tangential axis means that the axis of rotation is on the surface of the object and is parallel to a line through the center of mass. In the context of the hollow spheres, a tangential axis is essential in determining how the moment of inertia changes from its standard center-of-mass-aligned form.

• When you consider the moment of inertia around a tangential axis, you use the parallel axis theorem to include the additional distance from the center to the new axis.
• In the problem at hand, this axis does not pass through the center of one of the hollow spheres, but rather through the surface, which adds an extra term of \( M(2R)^2 \) to the moment of inertia formula for that sphere.

The concept of a tangential axis helps you analyze situations where objects rotate about unconventional axes, which shift their rotational properties.

Rotational Dynamics

Rotational dynamics is the study of objects in a rotational motion. Similar to how linear dynamics deals with forces and motion in a straight line, rotational dynamics deals with torques and rotational motion. Understanding the moment of inertia is a cornerstone of this topic as it quantifies how a mass distribution resists rotational acceleration.

• The moment of inertia depends on both the mass of the object and its shape or mass distribution relative to the axis of rotation.
• In many problems, including the one described, calculating the total moment of inertia about the defined axis is crucial to predicting how the system will behave under rotational influence.
• Applying the concepts of rotational dynamics, we can determine how fast or slow an object will begin to spin given a certain torque, or how the distribution of mass affects such motions.

Thus, rotational dynamics provides the framework for understanding and predicting rotational motion in various contexts, as demonstrated by calculating the moment of inertia for combined systems like the hollow spheres in the exercise.