Animated Solution for Physics - Rotational Motion: The solid cylinder of length $80\text{ cm}$ and mass $M$ has a radius of $20\text{ cm}$. Calculate the density of the material used, if the moment of inertia of the cylinder about an axis $CD$ parallel to $AB$ as shown in figure is $2.7{\text{ kg m}}^2$.

Imagine you are looking at a solid cylinder. The problem gives us its length $L=80\text{ cm}$ (which is $0.8\text{ m}$) and its radius $r=20\text{ cm}$ (which is $0.2\text{ m}$). We are also given the moment of inertia about a specific axis $CD$, which is $I_{CD}=2.7{\text{ kg m}}^2$.

If we look closely at the provided diagram, the axis $AB$ is the central longitudinal axis passing right through the middle of the cylinder. The axis $CD$ is parallel to $AB$ and is located at a distance of $L/2$ from it.

To find the moment of inertia about the axis $CD$, we need to bridge the gap between the central axis $AB$ and the new axis $CD$. This is exactly where the Parallel Axis Theorem comes into play. It states that the moment of inertia about any axis is equal to the moment of inertia about a parallel axis passing through the center of mass, plus the product of the mass and the square of the perpendicular distance between the axes.

Mathematically, this is written as:

For a solid cylinder, the moment of inertia about its central longitudinal axis $AB$ is identical to that of a solid disk:

The distance $d$ between the two axes is given in the diagram as $L/2$. Substituting these into our master equation, we get:

Now, let's plug in the numerical values provided in the problem. We know $I_{CD}=2.7$, $r=0.2\text{ m}$, and $L=0.8\text{ m}$.

Let's break down the math step-by-step to avoid any silly mistakes:

Dividing both sides by $0.18$, we find the mass of the cylinder:

Density

The ultimate goal is to find the density of the material. Density ($\rho$) is defined as mass divided by volume. For a cylinder, the volume is the area of the base times the height, which is $\pi r^{2}L$.

Substituting our known values:

Calculating this gives us:

To match the format of the options, we can write this in scientific notation:

$$\rho =1.49\times 10^2{\text{ kg/m}}^3$$

This perfectly matches option (d). The beauty of this problem lies in carefully reading the diagram to identify the distance between the axes and then systematically applying the Parallel Axis Theorem.