Moment of inertia of a disc about an axis which is tangent and parallel to its plane is I. then the moment of inertia of disc about a tangent,, but perpendicular to its plane wil be _________

To determine the moment of inertia of a disc about an axis that is tangent and perpendicular to its plane, we can use the parallel axis theorem. Let's break this down step by step.

Understanding the Moment of Inertia

Moment of inertia is a measure of an object's resistance to rotational motion about a given axis. For a solid disc with a uniform mass distribution, the moment of inertia about its central axis (which is perpendicular to its plane) is given by the formula:

I = (1/2) m r²

where m is the mass of the disc and r is its radius.

Using the Parallel Axis Theorem

The parallel axis theorem states that if you know the moment of inertia of a body about an axis through its center of mass, you can find the moment of inertia about any parallel axis by adding the product of the mass and the square of the distance between the two axes.

The theorem can be expressed as:

I = I_cm + m d²

Here, I_cm is the moment of inertia about the center of mass axis, m is the mass, and d is the distance between the center of mass axis and the new axis.

Applying It to Our Disc

In your case, we have two scenarios:

• The moment of inertia of the disc about an axis that is tangent and parallel to its plane is I.
• We need to find the moment of inertia about an axis that is tangent but perpendicular to its plane.

Let's denote the moment of inertia about the tangent and parallel axis as I_t, and about the tangent and perpendicular axis as I_{\perp}.

Calculating I_perpendicular

Using the parallel axis theorem:

I_{\perp} = I_t + m d²

In this case, I_t is given as I. The distance d is equal to the radius of the disc, r, because the perpendicular axis is at the edge of the disc, while the parallel axis is at the center.

Thus, we can write:

I_{\perp} = I + m r²

Substituting Values

Now, we know that the moment of inertia about the center of mass is:

I = (1/2) m r²

Substituting this into our equation gives:

I_{\perp} = I + m r² = \left(\frac{1}{2} m r²\right) + m r² = \frac{3}{2} m r²

Final Result

Therefore, the moment of inertia of the disc about a tangent axis that is perpendicular to its plane is:

I_{\perp} = 3I

In summary, the moment of inertia about the tangent and perpendicular axis is three times that of the tangent and parallel axis, providing a clear understanding of how the orientation of the axis affects rotational inertia.