Question

The moment of inertia of a hollow sphere of mass \(\mathrm{M}\) and inner and outer radii \(\mathrm{R}\) and \(2 \mathrm{R}\) about the axis passing through its centre and perpendicular to its plane is \(\\{\mathrm{A}\\}(3 / 2) \mathrm{MR}^{2}\) \\{B \(\\}(13 / 32) \mathrm{MR}^{2}\) \(\\{\mathrm{C}\\}(31 / 35) \mathrm{MR}^{2}\) \(\\{\mathrm{D}\\}(62 / 35) \mathrm{MR}^{2}\)

Short answer

The moment of inertia of a hollow sphere with mass M and inner radius R and outer radius 2R about an axis passing through its center and perpendicular to its plane is \(\frac{14}{15} MR^2\).

Step-by-step solution

Find the moment of inertia of both spheres if they were solid

To find the moment of inertia for the solid sphere of radius R and 2R we will use the solid sphere moment of inertia formula \(I = \frac{2}{5}MR^2\). 1. For the smaller sphere of radius R and mass (M/3): \[I_{smaller} = \frac{2}{5} \frac{M}{3} R^2\] 2. For the larger sphere of radius 2R and mass (2M/3): \[I_{larger} = \frac{2}{5}\frac{2M}{3}(2R)^2\]

Find the moment of inertia for the hollow sphere

Now that we have the moments of inertia for both solid spheres, we can subtract the moment of inertia of the smaller solid sphere from that of the larger solid sphere: \[I_{hollow} = I_{larger} - I_{smaller}\] \[I_{hollow} = \frac{2}{5}\frac{2M}{3}(2R)^2 - \frac{2}{5} \frac{M}{3} R^2\]

Simplify the equation

To simplify the equation, first combine the constants: \[I_{hollow} = \frac{2M}{15}(8R^2 - R^2)\] \[I_{hollow} = \frac{2M}{15}(7R^2)\] Now, further simplification gives: \[I_{hollow} = \frac{14}{15}MR^2\] Comparing the simplified equation to the given choices, we can see that none of the choices match the derived equation. This implies that the provided choices are incorrect.

Key concepts

Hollow Sphere

A hollow sphere is an interesting object to study when it comes to rotational inertia because it combines two distinct radii for its inner and outer surfaces. Unlike a solid sphere, a hollow sphere does not have a uniform mass distribution. Instead, the mass is spread out over a shell that encases an empty space. This creates a complex scenario for calculating its moment of inertia.

The moment of inertia depends greatly on how mass is distributed with respect to the axis of rotation. In the case of a hollow sphere, this means considering a larger outer radius and a smaller inner radius.

Subtracting the inertia of the material that would fill the inner radius from the total inertia is necessary to find the result for a hollow configuration. This gives us insight into how the mass arrangement impacts rotational characteristics.

Solid Sphere

A solid sphere is one of the simplest forms to calculate moment of inertia, due to its uniform mass distribution. When a solid sphere rotates around an axis passing through its center, all its particles are equidistant from the axis, which simplifies calculations.

For a solid sphere with mass \(M\) and radius \(R\), the moment of inertia is given by \(I = \frac{2}{5}MR^2\). This formula reflects the symmetric mass distribution throughout the sphere, meaning no mass is concentrated at or away from any point.

Hence, when dealing with a solid sphere, this uniformity is crucial in calculating the moment of inertia accurately. This formula allows us to predict how the sphere will behave when it spins.

Mass Distribution

Mass distribution plays a vital role in determining the moment of inertia because it defines how the mass is spread in relation to the axis of rotation. In physical objects, especially spheres, the mass isn't always uniformly distributed.

In a hollow sphere, the mass is distributed along the shell; it isn't concentrated or dispersed uniformly throughout the volume, as it would be in a solid sphere. This mainly affects how the hollow sphere resists changes in rotational motion.

The formula for moment of inertia takes into account this distribution: the farther the mass is from the axis of rotation, generally, the greater the contribution to the moment of inertia. Hence, different mass distributions lead to different rotational behaviors for objects with the same mass but different forms.

Axis of Rotation

The axis of rotation is a crucial factor in calculating moments of inertia, as it is the imaginary line around which the object spins. Depending on where this axis is located, the moment of inertia can vary significantly, even for objects with identical shape and mass.

• For a solid sphere, any axis passing through its center provides symmetric distribution, simplifying inertia calculations.
• For a hollow sphere, the axis going through its center and perpendicular to its plane considers the hollow nature of the sphere, which is pivotal in calculations.

This concept explains why certain rotations might feel easier or harder – as the axis changes, so does the distribution of mass and the corresponding inertia, influencing the rotation greatly.