Given:
Radius of the disc (original disc) = R
Mass of the disc (original disc) = M1
Diameter of the circular hole = R1
Rim of the hole passes through the center

To find:
Moment of inertia of the remaining part of the disc about a perpendicular axis passing through the center

Explanation:
The moment of inertia of an object is a measure of its resistance to rotational motion. It depends on the mass distribution of the object and the axis of rotation.

To find the moment of inertia of the remaining part of the disc, we need to consider the moment of inertia of the original disc and subtract the moment of inertia of the hole.

1. Moment of Inertia of the Original Disc:
The moment of inertia of a disc about an axis passing through its center and perpendicular to its plane is given by the formula:

I = (1/2) * M * R^2

Where:
I = Moment of inertia
M = Mass of the disc
R = Radius of the disc

Substituting the given values:

I1 = (1/2) * M1 * R^2

2. Moment of Inertia of the Hole:
The moment of inertia of a disc about an axis passing through its center and perpendicular to its plane is given by the formula:

I = (1/4) * M * R^2

Where:
I = Moment of inertia
M = Mass of the hole (portion removed)
R = Radius of the hole (diameter/2)

Substituting the given values:

I2 = (1/4) * M2 * (R1/2)^2
= (1/4) * M2 * (R1^2/4)
= (1/16) * M2 * R1^2

3. Moment of Inertia of the Remaining Part:
To find the moment of inertia of the remaining part, we subtract the moment of inertia of the hole from the moment of inertia of the original disc:

I3 = I1 - I2
= (1/2) * M1 * R^2 - (1/16) * M2 * R1^2

4. Calculating the Mass of the Hole:
The mass of the hole can be calculated by subtracting the mass of the remaining part from the mass of the original disc:

M2 = M1 - M

Substituting this value back into the equation for I3:

I3 = (1/2) * M1 * R^2 - (1/16) * (M1 - M) * R1^2

5. Simplifying the Equation:
Taking the common denominator of 16:

I3 = (8/16) * M1 * R^2 - (1/16) * (M1 - M) * R1^2
= (8M1R^2 - M1R1^2 + MR1^2) / 16

6. Simplifying Further:
I3 = (8M1R^2 - M1R1^2 + MR1^2) / 16
= (8M1R^2 - M1R1^2 + MR1^2) / 16
= (8M1R^2 + MR1^2 - M1R1^2) /