Saurabh Koranglekar
Last Activity: 5 Years ago
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.