Mass Moment of Inertia Calculator
Mass Moment of Inertia Calculator
The mass moment of inertia calculator is a powerful tool used in physics and engineering to determine an object’s resistance to rotational acceleration. It’s crucial for understanding how objects behave when they rotate or spin.
Imagine a figure skater spinning on ice. When they pull their arms close to their body, they spin faster. This is because their mass moment of inertia decreases, allowing them to rotate more quickly. When they extend their arms, their spin slows down due to an increased mass moment of inertia.Mass Moment of Inertia Chart
| Shape | Formula | Variables |
|---|---|---|
| Solid Sphere | I = 2/5 M R² | M = Mass, R = Radius |
| Hollow Sphere | I = 2/3 M R² | M = Mass, R = Radius |
| Solid Cylinder (rotation axis) | I = 1/2 M R² | M = Mass, R = Radius |
| Thin Rod (end axis) | I = 1/3 M L² | M = Mass, L = Length |
| Rectangular Plate (perpendicular axis) | I = 1/12 M (a² + b²) | M = Mass, a & b = Side lengths |
Mass Moment of Inertia Formula
The general formula for mass moment of inertia is:
I = ∫ r² dmWhere I is the moment of inertia, r is the perpendicular distance from the axis of rotation to the mass element, and dm is the differential mass element.
Let’s take a simple example of a solid rod rotating about its center.
The formula for this specific case is:
I = 1/12 M L²If we have a rod with a mass of 2 kg and a length of 3 meters, the calculation would be:
I = 1/12 2 3² = 1.5 kg·m²How to calculate mass moment of inertia?
To calculate the mass moment of inertia, follow these steps:
- Identify the object’s shape and axis of rotation.
- Choose the appropriate formula from the chart.
- Measure or determine the necessary variables (mass, length, radius, etc.).
- Plug the values into the formula and solve.
For instance, let’s calculate the moment of inertia for a solid sphere with a mass of 5 kg and a radius of 0.1 m:
- Shape: Solid Sphere
- Formula: I = 2/5 M R²
- Variables: M = 5 kg, R = 0.1 m
- Calculation: I = 2/5 5 0.1² = 0.004 kg·m²
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