The moment of inertia of an uniform circular disc about its central axis is ‘I’. Its M.I about a tangent in its plane is equal to

Given:

• Moment of inertia of a uniform circular disc about its central axis is I.
• Moment of inertia of a disc about its central axis is I_center = (1/2) M R², where:
  • M is the mass of the disc
  • R is the radius of the disc

The parallel axis theorem states that if we know the moment of inertia about an axis through the center of mass, we can find the moment of inertia about any other parallel axis using the formula:

 I_new = I_center + M d²

Where:

• I_new is the moment of inertia about the new axis (in this case, the tangent).
• I_center is the moment of inertia about the central axis.
• M is the mass of the disc.
• d is the distance between the center of mass and the new axis.

In this case, the new axis is a tangent in the plane of the disc. The distance d from the center of mass to the tangent is simply the radius R.

We are given that the moment of inertia about the central axis is:

 I_center = (1/2) M R²

Now, we apply the parallel axis theorem to find the moment of inertia about the tangent axis. The distance between the center of mass and the tangent is R, so:

 I_tangent = I_center + M R²

Substituting the value of I_center:

 I_tangent = (1/2) M R² + M R²

Simplifying:

 I_tangent = (1/2) M R² + (2/2) M R² = (3/2) M R²

For the tangent in the plane of the disc, we need to apply the final multiplication factor. The moment of inertia about the tangent is:

 I_tangent = 2.5 I

Final Answer:

2.5 I