Question

The moment of inertia of a thin disc about axes \( a, b, c, d \) are \( I_1, I_2, I_3 \) and \( I_4 \) respectively, as shown in figure. If the moment of inertia about an axis passing through the centre and perpendicular to the plane of the disc is I then,

• A. \( I = I_1 + I_2 \)
• B. \( I = I_3 + I_4 \)
• C. \( I = I_1 + I_3 \)
• D. \( I = I_1 + I_2 + I_3 + I_4 \)

Hint:

For highly symmetric 2D objects like a circular disc or a square plate, the moment of inertia about any diametrical axis in the plane of the object is constant. This means any pair of perpendicular (or even non-perpendicular, but symmetrically equal) coplanar diametrical axes will sum up to give the perpendicular polar moment of inertia.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The Perpendicular Axis Theorem is applicable to laminar objects like thin discs.
It states that the moment of inertia of a planar body about an axis perpendicular to its plane (\(I_z\)) is equal to the sum of its moments of inertia about two mutually perpendicular axes lying in its plane (\(I_x\) and \(I_y\)) that intersect at the same point.
For a symmetrical disc, the moment of inertia about any diameter is the same.
Step 2: Key Formula or Approach:
1. Perpendicular Axis Theorem: \(I_z = I_x + I_y\).
2. Symmetry of Disc: If \(x\) and \(y\) are diameters, \(I_x = I_y = \frac{1}{4} MR^2\).
3. Given axes \(a, b, c, d\) are all coplanar and pass through the center.
Step 3: Detailed Explanation:
Looking at the diagram, axes \(a, b\) are perpendicular diameters. Axes \(c, d\) are another set of perpendicular diameters.
Step A: According to the theorem, for any two perpendicular axes in the plane (\(a \perp b\)), their sum equals the perpendicular MOI.
Thus, \(I = I_1 + I_2\). (Option A is correct).
Step B: Similarly, axes \(c\) and \(d\) are shown to be perpendicular in the diagram (intersecting at \(90^\circ\)).
Thus, \(I = I_3 + I_4\). (Option B is correct).
Step C: In a uniform disc, all diametrical axes are equivalent.
\(I_1 = I_2 = I_3 = I_4 = I_{dia} = \frac{1}{2} I\).
If \(I_1 = \frac{1}{2} I\) and \(I_3 = \frac{1}{2} I\), then \(I_1 + I_3 = I\).
This is true even if axes \(a\) and \(c\) are not perpendicular, provided they are diameters. However, the Perpendicular Axis Theorem specifically requires the sum of perpendicular pairs.
Since the disc is highly symmetric, the sum of ANY two perpendicular diameters will be \(I\).
Options A, B, and C are valid representations.
Step 4: Final Answer:
By applying the perpendicular axis theorem and exploiting circular symmetry, it is shown that the sum of moments of inertia about any two perpendicular diameters equals the polar moment of inertia.