Moment of Inertia about the z-axis

The moment of inertia of a uniform disc about its z-axis can be calculated using the formula:

Iz = (1/2) * MR^2

where M is the mass of the disc and R is its radius.

Moment of Inertia about the line y=x

To calculate the moment of inertia about the line y=x, we need to consider the disc's mass distribution along this line. The distance of any point on the line y=x from the z-axis is given by the formula:

r = (x^2 + y^2)^(1/2)

To find the moment of inertia about the line y=x, we need to integrate the mass distribution along this line. However, since the disc is symmetric about the origin, we can simplify the calculations by considering only a quarter of the disc.

Let's consider a small element of the disc in the first quadrant, with a small area dA. The mass of this element can be calculated as:

dm = (M/πR^2) * dA

The distance of this element from the line y=x is given by:

r = (x^2 + x^2)^(1/2) = √2x

The moment of inertia of this small element about the line y=x is given by:

dI = dm * r^2 = (M/πR^2) * dA * (√2x)^2 = (2M/πR^2) * x^2 * dA

To find the total moment of inertia about the line y=x, we need to integrate this expression over the quarter disc. The limits of integration are from 0 to R, for both x and y.

Iy=x = ∫[0 to R] ∫[0 to R] (2M/πR^2) * x^2 * dy * dx

= (2M/πR^2) * ∫[0 to R] (x^2 * ∫[0 to R] dy) * dx

= (2M/πR^2) * ∫[0 to R] (x^2 * R) * dx

= (2M/πR^2) * R * ∫[0 to R] x^2 * dx

= (2M/πR^2) * R * [x^3/3] [0 to R]

= (2M/πR^2) * R * (R^3/3)

= (2/3) * MR^2

Comparing both moments of inertia:

Iz = Iz=x

(1/2) * MR^2 = (2/3) * MR^2

1/2 = 2/3

This is not true, so we made a mistake somewhere in our calculations.

Correct Calculation for c

The correct calculation for c can be done by considering the moment of inertia about the line y=x in polar coordinates.

In polar coordinates, the equation of the line y=x can be written as:

r * cos(theta) = r * sin(theta)

Dividing both sides by r:

cos(theta) = sin(theta)

Squaring both sides:

cos^2(theta) = sin^2(theta)

Using the trigonometric identity:

1 - sin^2(theta) = sin^2

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