A circular disc X of radius R is made from an iron pole of thickness t, and another disc Y of radius 4R is made from an iron plate of thickness t/4. then the relation between the moment of inertia I and IY| is ?

To understand the relationship between the moments of inertia of the two discs, X and Y, we need to apply the formula for the moment of inertia for a solid disc. The moment of inertia (I) of a disc about its central axis is given by the formula:

Moment of Inertia Formula

The formula is:

I = (1/2) * M * R²

Where:

• I = moment of inertia
• M = mass of the disc
• R = radius of the disc

Calculating Moment of Inertia for Disc X

For disc X with radius R and thickness t:

• The volume of disc X can be calculated as:

V_X = π * R² * t

• Since the density (ρ) of iron remains constant, the mass (M_X) is:

M_X = ρ * V_X = ρ * π * R² * t

• Now substituting M_X into the moment of inertia formula:

I_X = (1/2) * (ρ * π * R² * t) * R² = (1/2) * ρ * π * t * R^4

Calculating Moment of Inertia for Disc Y

Now, let's consider disc Y, which has a radius of 4R and thickness t/4:

• The volume of disc Y is:

V_Y = π * (4R)² * (t/4) = π * 16R² * (t/4) = 4π * R² * t

• The mass (M_Y) for disc Y is therefore:

M_Y = ρ * V_Y = ρ * 4π * R² * t

• Substituting M_Y into the moment of inertia formula gives us:

I_Y = (1/2) * (ρ * 4π * R² * t) * (4R)² = (1/2) * (ρ * 4π * R² * t) * 16R² = 32 * (1/2) * ρ * π * t * R^4

Finding the Relation Between I_X and I_Y

Now, we can compare I_X and I_Y:

I_X = (1/2) * ρ * π * t * R^4

I_Y = 32 * (1/2) * ρ * π * t * R^4

From this, we observe:

I_Y = 32 * I_X

Final Thoughts

The moment of inertia of disc Y is 32 times greater than that of disc X. This significant difference arises due to the larger radius and mass of disc Y, despite its thinner thickness. Understanding these relationships can help in various applications, especially in mechanical systems where rotational dynamics are involved.