Question

A ring of mass \(\mathrm{M}\) and radius \(\mathrm{r}\) is melted and then molded in to a sphere then the moment of inertia of the sphere will be..... \(\\{\mathrm{A}\\}\) more than that of the ring \\{B \\} Less than that of the ring \(\\{\mathrm{C}\\}\) Equal to that of the ring \\{D\\} None of these

Short answer

The moment of inertia of the sphere is less than that of the ring. The correct answer is (B) Less than that of the ring.

Step-by-step solution

Calculate the moment of inertia of the ring

The moment of inertia of a ring with mass \(M\) and radius \(r\) is given by the formula: \(I_\text{ring} = Mr^2\) In our case, we have the mass (\(M\)) and radius (\(r\)) of the ring, so we can directly apply this formula: \(I_\text{ring} = M r^2\)

Calculate the moment of inertia of the sphere

When the ring is melted and molded into a sphere, its moment of inertia will be given by the formula: \(I_\text{sphere} = \frac{2}{5}M R^2\) Since the mass and radius of the sphere are both equal to the mass and radius of the ring, we can substitute those values: \(I_\text{sphere} = \frac{2}{5}M r^2\)

Compare the moments of inertia

Now we have both moments of inertia and can compare them: \(I_\text{ring} = M r^2\) and \(I_\text{sphere} = \frac{2}{5}M r^2\) Start by dividing both sides by \(M r^2\): \(\frac{I_\text{ring}}{M r^2} = 1\) and \(\frac{I_\text{sphere}}{M r^2} = \frac{2}{5}\) Now we compare these two ratios, noticing that \(\frac{2}{5} < 1\) This means that \(I_\text{sphere} = \frac{2}{5}M r^2 < M r^2 = I_\text{ring}\) Hence, the moment of inertia of the sphere is less than that of the ring.

Answer

The correct answer is (B) Less than that of the ring.

Key concepts

Ring and Sphere Transformation

When discussing the transformation from a ring to a sphere, it is important to understand the concept of redistributing mass. Initially, we have a ring, which is essentially a thin, circular band of a specified mass \( M \) and radius \( r \).

• The mass of the ring is uniformly distributed along its circumference.
• The moment of inertia of the ring is determined by how far its mass is from its rotational axis, which is given by the formula \( I_\text{ring} = Mr^2 \).

When this ring is melted and reshaped into a sphere, the mass is redistributed more evenly across all three dimensions, rather than along a plane. The sphere, having the same mass as the original ring, will have a different distribution of its inertia. The moment of inertia for a sphere is derived from the formula \( I_\text{sphere} = \frac{2}{5}MR^2 \), reflecting its symmetrical distribution of mass around its center.Although both the ring and the sphere have the same mass and radius, their moments of inertia differ due to the differing mass distributions. The sphere's moment of inertia is considerably reduced, highlighting the effects of geometry on rotational motion.

Physics JEE Problem

The JEE (Joint Entrance Examination) is a highly standardized test in India, which includes challenging problems in physics to test a student's understanding of various concepts. One type of problem you might encounter involves moments of inertia, such as the one given in the exercise: a ring that is molded into a sphere.

• Such problems test your ability to understand and manipulate formulas for moment of inertia.
• They also require you to apply theoretical physics concepts to practical changes in shape and mass distribution.
• Knowing how to derive and compare different inertia values is crucial for solving these problems correctly.

In this case, the problem revolves around understanding the differences in inertia as a shape changes from a 1-dimensional distribution (the ring) to a 3-dimensional one (the sphere). This transformation requires recognizing and applying the appropriate formulas for each shape to determine which has the larger moment of inertia. For students, mastering this process is essential to score well in the examination.

Rotational Motion Concepts

Rotational motion is a fundamental topic in mechanics within physics. It extends the idea of linear motion to objects that rotate around an axis, meaning concepts like inertia adapt to consider distributed mass about a rotational center.Understanding rotational motion begins with the concept of the moment of inertia, a measure of an object's resistance to changes in its rotational state.

• The moment of inertia depends not just on mass but also on the shape and axis of rotation.
• For simple geometric shapes, specific formulas calculate the moment of inertia, such as \( Mr^2 \) for a ring and \( \frac{2}{5}MR^2 \) for a sphere.

The shift from linear to rotational motion principles necessitates understanding how an object's geometry affects its dynamics. With rotational motion problems, especially in contexts like examinations, it’s important to grasp these concepts and equations thoroughly. A clear understanding helps predict how an object will behave when forces are applied to it, which is central to many practical and theoretical physics applications.