Question

One solid sphere \(\mathrm{A}\) and another hollow sphere \(\mathrm{B}\) are of the same mass and same outer radii. The moment of inertia about their diameters are respectively \(\mathrm{I}_{\mathrm{A}}\) and \(\mathrm{I}_{\mathrm{B}}\) such that... \(\\{\mathrm{A}\\} \mathrm{I}_{\mathrm{A}}=\mathrm{I}_{\mathrm{B}}\) \(\\{\mathrm{B}\\} \mathrm{I}_{\mathrm{A}}>\mathrm{I}_{\mathrm{B}}\) \(\\{\mathrm{C}\\} \mathrm{I}_{\mathrm{A}}<\mathrm{I}_{\mathrm{B}}\) \(\\{\mathrm{D}\\}\left(\mathrm{I}_{\mathrm{A}} / \mathrm{I}_{\mathrm{B}}\right)=(\mathrm{d} \mathrm{A} / \mathrm{dB})\) (radio of their densities)

Short answer

The correct answer is Option C: \(\mathrm{I}_{\mathrm{A}}<\mathrm{I}_{\mathrm{B}}\).

Step-by-step solution

Determine the moment of inertia formula for a solid sphere

For a solid sphere (sphere A), the moment of inertia _(I)_ about any diameter can be found using the following formula: \[I_A = \frac{2}{5} m_A R_A^2\] where, - \(I_A\) = moment of inertia of sphere A - \(m_A\) = mass of sphere A - \(R_A\) = radius of sphere A

Determine the moment of inertia formula for a hollow sphere

For a hollow sphere (sphere B), the moment of inertia _(I)_ about any diameter can be found using the following formula: \[I_B = \frac{2}{3} m_B R_B^2 \] where, - \(I_B\) = moment of inertia of sphere B - \(m_B\) = mass of sphere B - \(R_B\) = radius of sphere B

Use the given information

According to the exercise, both spheres have the same mass and outer radii. So, \(m_A = m_B\) and \(R_A = R_B\). We can replace the mass and radii in the formulas we found in Steps 1 and 2 with the given information.

Compare the moment of inertia of both spheres

Now, we need to compare the moment of inertia of both spheres after replacing mass and radii in the formulas from Steps 1 and 2. For sphere A: \[I_A = \frac{2}{5} m_A R_A^2 = \frac{2}{5} m_B R_B^2\] For sphere B: \[I_B =\frac{2}{3} m_B R_B^2\] From the above two equations, it becomes clear that: \[\frac{2}{5} m_B R_B^2 < \frac{2}{3} m_B R_B^2\] Therefore, we can conclude that: \[ \mathrm{I}_{\mathrm{A}}<\mathrm{I}_{\mathrm{B}}\] Thus, the correct answer is

Option C

\(\mathrm{I}_{\mathrm{A}}<\mathrm{I}_{\mathrm{B}}\).

Key concepts

Solid Sphere

A solid sphere is a perfectly round three-dimensional shape where all the mass is distributed uniformly throughout its entire volume. In practical terms, think of it like a basketball that is completely filled with material without any cavity or hollow at its center.
The moment of inertia, which measures how resistant an object is to rotational motion, is lower for a solid sphere compared to a hollow sphere with the same mass and radius. This is because the mass in a solid sphere is concentrated closer to the axis of rotation, making it easier to spin.
For a solid sphere with mass \(m\) and radius \(R\), the moment of inertia about any of its diameters is given by the formula: \[I_A = \frac{2}{5} m R^2\] This means that if you increase the mass or radius, the moment of inertia will increase accordingly, since the product of these values gets larger.

Hollow Sphere

A hollow sphere is like a shell, where the mass is distributed along the hollow structure without any significant mass in the center. Picture a soccer ball or a thin metal ball, where the solid material forms a thin exterior layer.
For a hollow sphere, the moment of inertia is greater than that of a solid sphere with the same mass and radius, because the mass is distributed further from the axis of rotation. This makes it more resistant to changes in rotational motion.
The mathematical expression for the moment of inertia of a hollow sphere with mass \(m\) and radius \(R\) is: \[I_B = \frac{2}{3} m R^2\] This formula shows how the mass and radius again directly influence the inertia. The larger the radius or mass, the higher the moment of inertia.

Radius

The radius of a sphere is the distance from its center to any point on its surface. It plays a crucial role in determining the moment of inertia. In the context of the problem, both spheres—solid and hollow—have the same radius.
The radius influences the moment of inertia formula because it is squared in both cases. So, the impact of any change in radius is more pronounced than changes in mass because radius is directly related to the rotational characteristics of the sphere.
Understanding how radius affects rotational dynamics can be visually helpful; imagine how the ease of rotating an object changes drastically as you make it larger while keeping its mass the same.

Mass

Mass is a measure of the amount of matter in an object and is directly proportional to the moment of inertia. In this exercise, both spheres have equal mass, which simplifies the process of comparing their moments of inertia.
In terms of physics, mass represents the inertia of an object—its resistance to changes in its state of motion. This characteristic is crucial when calculating the moment of inertia, as seen in the formulas for both solid and hollow spheres:

• For a solid sphere: \(I_A = \frac{2}{5} m R^2\)
• For a hollow sphere: \(I_B = \frac{2}{3} m R^2\)

Thus, even with equal mass, a solid sphere will tend to be easier to rotate about its diameter compared to a hollow sphere of the same radius.

Density

Density plays an implicit role when considering spheres of different types but the same mass and radius. It is defined as the mass per unit volume of an object. For equal mass, a solid sphere will naturally have a higher density compared to its hollow counterpart because all the mass is packed into a smaller volume.
While the task doesn't require explicit calculations of density, understanding this concept can explain why solid and hollow spheres behave differently under rotation. A solid sphere's higher density means the mass is closer to the center, reducing its moment of inertia relative to a hollow sphere.
In problems involving comparative inertia, density becomes a key underlying factor, subtly influencing the dynamics through the distribution of mass.