Calculate the total rotational inertia of a ring sandwiched betwenn two discs. Use a mass of 1.50 kg for all three objects. The two discs have a radius of 0.333 m, and the ring has inner and outer radii of 0.125m and 0.225 m respectively
The Correct Answer and Explanation is:
The total rotational inertia of the system comprising a ring sandwiched between two discs is approximately 0.216 kg·m².
Explanation
Rotational inertia, also known as the moment of inertia, is a measure of an object’s resistance to angular acceleration about a rotational axis. It depends on the mass distribution relative to that axis. In this scenario, we have two discs and one ring, each with a mass of 1.50 kg, and we need to calculate their combined moment of inertia.
- Moment of Inertia for Discs:
Each disc is treated as a solid cylinder. The formula for the moment of inertia ( I ) of a disc about its central axis is given by:
[
I_{\text{disc}} = \frac{1}{2} m r^2
]
where ( m ) is the mass of the disc and ( r ) is its radius. Substituting the values:
[
I_{\text{disc}} = \frac{1}{2} \times 1.50 \, \text{kg} \times (0.333 \, \text{m})^2 \approx 0.083 \, \text{kg·m}^2
]
Since there are two identical discs, the total moment of inertia from the discs is:
[
I_{\text{total discs}} = 2 \times I_{\text{disc}} \approx 0.166 \, \text{kg·m}^2
] - Moment of Inertia for the Ring:
The ring has an inner radius ( R_{\text{inner}} = 0.125 \, \text{m} ) and an outer radius ( R_{\text{outer}} = 0.225 \, \text{m} ). The moment of inertia of a ring is given by:
[
I_{\text{ring}} = \frac{1}{2} m (R_{\text{outer}}^2 + R_{\text{inner}}^2)
]
Substituting the values:
[
I_{\text{ring}} = \frac{1}{2} \times 1.50 \, \text{kg} \times ((0.225 \, \text{m})^2 + (0.125 \, \text{m})^2) \approx 0.050 \, \text{kg·m}^2
] - Total Rotational Inertia:
Adding the contributions from both the discs and the ring, we find:
[
I_{\text{total}} = I_{\text{total discs}} + I_{\text{ring}} \approx 0.166 + 0.050 \approx 0.216 \, \text{kg·m}^2
]
In summary, the combined rotational inertia of the system is approximately 0.216 kg·m², reflecting the overall resistance to rotational acceleration based on the masses and geometries of the components.