A solid cylinder of uniform density of radius 3.2 cm has mass of 83g. If its length is 14 cm, calculate its moment of inertia about (i) its own axis of rotation passing through the centre,

(ii) an axis passing through its centre & perpendicular to its length.

To calculate the moment of inertia of a solid cylinder, we can use specific formulas that relate to its geometry and mass distribution. The moment of inertia is a measure of how difficult it is to change the rotational motion of an object. For a solid cylinder, the calculations differ based on the axis of rotation. Let’s break this down step by step.

Parameters of the Cylinder

We have a solid cylinder with the following characteristics:

• Radius (r): 3.2 cm = 0.032 m
• Mass (m): 83 g = 0.083 kg
• Length (h): 14 cm = 0.14 m

Moment of Inertia About Its Own Axis

The formula for the moment of inertia (I) of a solid cylinder about its own axis (which runs along the length of the cylinder) is given by:

I = (1/2) * m * r²

Substituting the values:

I = (1/2) * 0.083 kg * (0.032 m)²

Calculating this:

• First, calculate r²: (0.032 m)² = 0.001024 m²
• Then, multiply by mass: 0.083 kg * 0.001024 m² = 0.000085056 kg·m²
• Finally, multiply by 1/2: I = (1/2) * 0.000085056 kg·m² = 0.000042528 kg·m²

Thus, the moment of inertia about its own axis is approximately 0.0000425 kg·m².

Moment of Inertia About an Axis Perpendicular to Its Length

For the moment of inertia about an axis that is perpendicular to the length of the cylinder and passes through its center, the formula is:

I = (1/12) * m * (3r² + h²)

Now, substituting the known values:

I = (1/12) * 0.083 kg * (3 * (0.032 m)² + (0.14 m)²)

Calculating this step-by-step:

• Calculate 3r²: 3 * (0.032 m)² = 3 * 0.001024 m² = 0.003072 m²
• Calculate h²: (0.14 m)² = 0.0196 m²
• Add them together: 0.003072 m² + 0.0196 m² = 0.022672 m²
• Now, multiply by mass: 0.083 kg * 0.022672 m² = 0.001882176 kg·m²
• Finally, multiply by 1/12: I = (1/12) * 0.001882176 kg·m² = 0.000156848 kg·m²

Therefore, the moment of inertia about an axis perpendicular to its length is approximately 0.0001568 kg·m².

Summary of Results

To summarize, we have:

• Moment of inertia about its own axis: 0.0000425 kg·m²
• Moment of inertia about an axis perpendicular to its length: 0.0001568 kg·m²

These calculations illustrate how the distribution of mass in a solid cylinder affects its resistance to rotational motion about different axes. Understanding these principles is crucial in fields such as mechanical engineering and physics.