Problem 14 - Classical Mechanics

Question

What is the height- - to-diameter ratio of a right cylinder such that the inertia ellipsoid at the center of the cylinder is a sphere?

Step-by-Step Solution

Answer

The height-to-diameter ratio is \( \frac{\sqrt{3}}{2} \).

Step 1: Understanding the Inertia Tensor for a Cylinder

The moment of inertia for a solid right cylinder around its central axis is given by the formula: \( I_z = \frac{1}{2} m r^2 \), where \( m \) is the mass and \( r \) is the radius. For the axes perpendicular to the central axis: \( I_x = I_y = \frac{1}{12} m (3r^2 + h^2) \), where \( h \) is the height of the cylinder.

Step 2: Defining a Sphere in Terms of Inertia

For the inertia ellipsoid to be a sphere, all moments of inertia must be equal: \( I_x = I_y = I_z \). This implies \( \frac{1}{12} m (3r^2 + h^2) = \frac{1}{2} m r^2 \).

Step 3: Simplifying the Equality

Cancel out the mass \( m \) from both sides of the equation: \( \frac{1}{12} (3r^2 + h^2) = \frac{1}{2} r^2 \). Multiply through by 12 to clear the fraction: \( 3r^2 + h^2 = 6r^2 \).

Step 4: Solving for the Height

Isolate \( h^2 \) in the equation: \( h^2 = 6r^2 - 3r^2 \), which simplifies to \( h^2 = 3r^2 \). Taking the square root on both sides gives us \( h = r \sqrt{3} \).

Step 5: Finding the Height-to-Diameter Ratio

The diameter \( d \) of the cylinder is \( 2r \). Therefore, the height-to-diameter ratio is \( \frac{h}{d} = \frac{r \sqrt{3}}{2r} = \frac{\sqrt{3}}{2} \).

Key Concepts

Inertia Tensor

The inertia tensor is a mathematical representation that captures how mass is distributed in a three-dimensional object. This distribution affects how the object will rotate about various axes. In essence, the inertia tensor is a 3x3 matrix that contains moments of inertia and products of inertia for the object. For a simple shape like a right cylinder, these values reveal how it resists rotational acceleration along different axes.

For a cylinder, the moment of inertia along its central axis (often labeled as the z-axis) is given by the equation:

• \( I_z = \frac{1}{2} m r^2 \)

Here, \( m \) is the total mass of the cylinder and \( r \) is the radius. On the other hand, for axes perpendicular to its central axis, which are usually labeled as the x and y axes, the equation becomes:

• \( I_x = I_y = \frac{1}{12} m (3r^2 + h^2) \)

This set of equations is vital because it helps us analyze rotational dynamics and is key to understanding the inertia ellipsoid.

Inertia Ellipsoid

The inertia ellipsoid is a conceptual tool used to visualize the distribution of inertia in an object. It helps in understanding how an object will behave when exposed to rotational forces. Very simply, it's a 3D shape that represents the moments of inertia about various axes within the object.

For an object like a cylinder, the inertia ellipsoid appears when you scale this shape according to the principal moments of inertia. If the ellipsoid is a perfect sphere, it means all three principal moments of inertia (\( I_x, I_y, I_z \)) are equal. This special condition is what the exercise aimed to achieve through correct proportions of the cylinder. By setting all moments of inertia equal, the geometry of the cylinder is such that its mass is perfectly balanced in all directions.

This insight allows engineers and physicists to predict rotational stability and balance, which is especially useful in designing objects that rotate, such as wheels or satellite components.

Height-to-Diameter Ratio

The height-to-diameter ratio is a significant geometrical feature, particularly in the case of a right cylinder. This ratio expresses how tall a cylinder is relative to its width. In many engineering applications, adjusting this ratio can significantly change the physical properties and the rotational dynamics of the cylinder.

In the problem at hand, the goal is to find this ratio so that the inertia ellipsoid becomes a sphere. From the solutions, we found that setting all the moments such that \( I_x = I_y = I_z \) is possible. Solving this gives us:

• Height \( h = r \sqrt{3} \)
• Diameter \( d = 2r \)
• Height-to-Diameter Ratio \( \frac{\sqrt{3}}{2} \)

This ratio provides a balanced rotation, making the cylinder's resistance to rotational motion equal in all directions about its center. This configuration makes it appear isotropic in terms of rotational inertia.

Right Cylinder

A right cylinder is a common geometric shape, characterized by its circle bases being aligned directly above each other. That means its sides are perpendicular to these bases. Its geometry can be defined simply through two parameters: its radius (\( r \)) and its height (\( h \)).

This shape is frequently encountered in both theoretical and practical applications, from mechanical systems such as pistons to concepts in geometry and physics.

• The central axis, where the circular bases are centered, plays a pivotal role in calculating its moment of inertia.
• In physics, a right cylinder is often used to simplify problems due to its symmetrical properties.

The problem of balancing the inertia ellipsoid to become a sphere, by adjusting the height-to-diameter ratio of a right cylinder, illustrates its importance in achieving 'dynamic symmetry'. This concept is critical for applications involving rotating machinery and vehicle wheels.