Problem 74 - Thomas Calculus

Question

Moments of inertia of solid circular eylinder Find the moment of inertia of a solid circular cylinder of radius 1 and height 2 (a) about the axis of the cylinder and (b) about a line through the centroid perpendicular to the axis of the cylinder. (Take \(\delta=1 . )\)

Step-by-Step Solution

Answer

Moment of inertia about the axis is \( \pi \); perpendicular to the axis is \( \frac{7}{6} \pi \).

Step 1: Identify Cylinder Properties

We have a solid circular cylinder with radius \( R = 1 \) and height \( h = 2 \). The density \( \delta = 1 \). These properties will be used in our calculations.

Step 2: Calculate Mass of Cylinder

The mass \( M \) of the cylinder is calculated using the formula \( M = \delta \times \text{Volume} = \delta \times \pi R^2 h \). With \( \delta = 1 \), \( R = 1 \), and \( h = 2 \), we have \( M = \pi \times 1^2 \times 2 = 2\pi \).

Step 3: Moment of Inertia about the Axis of the Cylinder

For a solid cylinder, the moment of inertia \( I_z \) about its axis is given by \( I_z = \frac{1}{2} M R^2 \). Substituting \( M = 2\pi \) and \( R = 1 \), we get \( I_z = \frac{1}{2} \times 2\pi \times 1^2 = \pi \).

Step 4: Moment of Inertia about Centroid Perpendicular to Axis

The moment of inertia \( I_y \) about a line through the centroid and perpendicular to the axis is given by \( I_y = \frac{1}{4} M R^2 + \frac{1}{12} M h^2 \). Substituting \( M = 2\pi \), \( R = 1 \), and \( h = 2 \), we calculate: \( I_y = \frac{1}{4} \times 2\pi \times 1^2 + \frac{1}{12} \times 2\pi \times 2^2 = \frac{1}{2} \pi + \frac{2}{3} \pi = \frac{7}{6} \pi \).

Key Concepts

Understanding the Cylinder

A cylinder is a three-dimensional geometric shape that consists of two parallel circular bases connected by a curved surface. Imagine a soup can; that is a perfect example of a cylinder.
Key characteristics of a cylinder include:

• Radius (\( R \)): The distance from the center to the edge of the base circle. In this case, it is 1 unit.
• Height (\( h \)): The distance between the two bases. Here, it is 2 units.
• Volume: Calculated as \( \pi R^2 h \), which tells us how much space it occupies.

These properties are essential when calculating mass and moment of inertia, both of which are necessary to understand the dynamics of a cylinder in motion.

Defining Mass in Context

Mass is a measure of the amount of matter in an object. For solid objects like cylinders, we determine the mass using density and volume.
For the given cylinder:

• Density (\( \delta \)): This is given as 1 unit, simplifying calculations.
• Volume: Found by \( \pi \times R^2 \times h \)
• Formula for Mass: \( M = \delta \times \text{Volume} \)

Therefore, the mass can be calculated as \( M = 2\pi \). This mass is a critical factor when determining other physical characteristics like the moment of inertia.

Centroid - The Center of Mass

The centroid of a shape is its geometric center or balance point. For simple symmetrical shapes like a cylinder, the centroid lies along its axis of symmetry, midway between its top and bottom surfaces.
The centroid is essential because:

• It helps locate where the mass is concentrated.
• For a cylinder, it is located at a height of \( \frac{h}{2} \).
• It determines the rotational characteristics related to moments of inertia.

Understanding the centroid allows us to calculate the moment of inertia about axes passing through this center.

Axis and Its Role in Rotational Dynamics

An axis provides a point or line around which the body rotates. In the context of a cylinder, different axes can provide varying resistance to rotation, measured by the moment of inertia.
Considerations for a cylinder:

• Axis along the cylinder: Usually runs through the center, parallel to the height, making rotation straightforward.
• Perpendicular through centroid: Another critical axis is when it passes through the centroid but perpendicular to the height, which provides information on different rotational dynamics.

For our exercise, the moments of inertia are calculated about these axes to understand how resistant the cylinder is to rotation.