Question

A thin uniform circular disc has mass \(M\) and radius \(a\). (a) Prove that its moment of inertia about an axis perpendicular to its plane and passing through its centre is \(\frac{1}{2} M a^{2}\). (b) Prove that the moment of inertia of the same disc about a diameter is \(\frac{1}{4} \mathrm{Ma}^{2}\). This is an example of the general result for planar bodies that the moment of inertia of the body about an axis perpendicular to the plane is equal to the sum 211of the moments of inertia about two perpendicular axes lying in the plane: in an obvious notation $$ I_{z}=\int r^{2} d m=\int\left(x^{2}+y^{2}\right) d m=\int x^{2} d m+\int y^{2} d m=I_{y}+I_{x} $$

Short answer

(a) \[ I_z = \frac{1}{2} M a^2 \] (b) \[ I_{\text{diameter}} = \frac{1}{4} M a^2 \]

Step-by-step solution

Understand the Problem Statement

We need to prove two properties of the moment of inertia for a thin uniform circular disc: (a) about an axis perpendicular to its plane and passing through its center and (b) about a diameter.

- Define Moment of Inertia (Part A)

The moment of inertia of a thin uniform circular disc about an axis perpendicular to its plane and passing through its center is given by: \[ I_z = \frac{1}{2} M a^2 \]

- Use Polar Coordinates for Part A

Consider a small mass element, dm, at a distance r from the center of the disc in polar coordinates (r, θ). The area element is given by: \[ dm = \frac{M}{\text{Area}} \times dA = \frac{M}{\theta a^2} \times r \text{ d}r \text{ d}\theta \]

- Integrate for Moment of Inertia (Part A)

Using the definition of moment of inertia, integrate for the entire disc: \[ I_z = \frac{M}{\theta a^2} \times \theta \times \frac{a^2 \times r^2}{2} \] Simplifying: \[ I_z = \frac{1}{2} M a^2 \]

- Define Moment of Inertia (Part B)

The moment of inertia of the disc about its diameter is to be proven as: \[ I_{\text{diameter}} = \frac{1}{4} M a^2 \]

- Use Perpendicular Axes Theorem for Part B

For any planar body, the perpendicular moment of inertia is the sum of the moments about two perpendicular in-plane axes. \[ I_z = I_x + I_y \]

- Apply Symmetry to Circular Disc

By symmetry, the moment of inertia about any diameter should be the same. Thus, \[ I_x = I_y \] and \[ I_z = 2 I_{\text{diameter}} \]

- Conclude for Part B

Since we know \[ I_z = \frac{1}{2} M a^2 \] and \[ I_z = 2 I_{\text{diameter}} \], solving gives: \[ I_{\text{diameter}} = \frac{1}{4} M a^2 \]

Key concepts

circular disc

A circular disc is a flat, round object with uniform thickness and mass distribution. The key points about a circular disc include:

• It has a constant mass density across its surface.
• Its geometry makes it useful for studying rotational dynamics and inertia.
• To find its moment of inertia, one must consider small mass elements within the disc.

These characteristics simplify the calculation of physical properties like moment of inertia.

perpendicular axis theorem

The perpendicular axis theorem applies to planar bodies and relates the moment of inertia about an axis perpendicular to the plane to moments of inertia about two perpendicular axes in the plane.
For a thin circular disc, the perpendicular axis theorem states:
- \[ I_{z} = I_{x} + I_{y} \]
where:
- \[ I_{z} \] is the moment of inertia about the perpendicular axis through the center of the disc.
The theorem simplifies calculations, making it easier to relate the moments of inertia along different axes.

symmetry in physics

Symmetry plays a crucial role in physics, allowing for simplified calculations and deeper understanding of physical phenomena.
For a circular disc:
- Due to its symmetrical shape, moments of inertia about any diameter are equal (i.e., \[I_{x} = I_{y}\]).
- This symmetry, combined with the perpendicular axis theorem, simplifies determining moments of inertia for different axes.
Symmetry ensures that complex calculations can often be broken down into simpler, more manageable parts.

planar bodies

In physics, a planar body is an object that lies entirely within a single plane.
Examples include:
- Circular discs.
- Rectangular sheets.
Key features of planar bodies include:
- Moments of inertia are easier to calculate using formulas that apply to two-dimensional shapes.
- They simplify learning and visualization of concepts like rotational dynamics.
Planar bodies serve as excellent models for understanding more complex three-dimensional objects.

polar coordinates

Polar coordinates are a two-dimensional coordinate system where each point is determined by a distance and angle.
- Using polar coordinates (\[(r, \theta)\]), spatial calculations become more intuitive.
- For a circular disc, mass elements are considered at a distance \[r\] from the center.
- Small area elements in polar coordinates are given by \[dA = r \, dr \, d\theta\].
This system simplifies integrals needed to find moments of inertia, especially for circles and discs. Polar coordinates provide a natural framework for dealing with rotational symmetry and related concepts.