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,
Show 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.
Step 1: Understanding the Question:
We are given a thin circular disc and four coplanar axes \( a, b, c, d \) passing through its center \( O \). We need to determine the relationships between their respective moments of inertia \( I_1, I_2, I_3, I_4 \) and the polar moment of inertia \( I \) (about the axis perpendicular to the disc through its center). Step 2: Key Formula or Approach:
1. Perpendicular Axis Theorem: For a planar laminar body, the moment of inertia about an axis perpendicular to its plane (\( I_z \)) is equal to the sum of the moments of inertia about any two mutually perpendicular axes in its plane (\( I_x \) and \( I_y \)) intersecting at the same point:
\[ I_z = I_x + I_y \]
2. Symmetry of a Circular Disc: Due to the circular symmetry of the disc, the moment of inertia about any diameter is identical:
\[ I_{\text{diameter}} = \frac{1}{4} M R^2 \] Step 3: Detailed Explanation:
Let the mass of the disc be \( M \) and its radius be \( R \).
- The moment of inertia about the perpendicular axis is:
\[ I = \frac{1}{2} M R^2 \]
- Since all four axes \( a, b, c, d \) are diametrical axes lying in the plane of the disc, by symmetry, their individual moments of inertia are equal:
\[ I_1 = I_2 = I_3 = I_4 = \frac{1}{4} M R^2 \]
Let's evaluate each option:
- Option (A): The axes \( a \) and \( b \) are mutually perpendicular in the plane of the disc. By the Perpendicular Axis Theorem:
\[ I = I_1 + I_2 \]
This is correct.
- Option (B): The axes \( c \) (horizontal) and \( d \) (vertical) are mutually perpendicular in the plane of the disc. By the Perpendicular Axis Theorem:
\[ I = I_3 + I_4 \]
This is correct.
- Option (C): Although axes \( a \) and \( c \) are not perpendicular, since \( I_1 = I_3 = \frac{1}{4} M R^2 \), their sum is:
\[ I_1 + I_3 = \frac{1}{4} M R^2 + \frac{1}{4} M R^2 = \frac{1}{2} M R^2 = I \]
Thus, numerically, \( I = I_1 + I_3 \) is also correct due to the circular symmetry of the disc. Step 4: Final Answer:
Options (A), (B), and (C) are all correct.
