Moment of Inertia Calculator

Understanding Moment of Inertia

The moment of inertia (often written as I) describes how difficult it is to start or stop an object spinning about a particular axis. Just as mass measures resistance to linear acceleration, the moment of inertia measures resistance to angular acceleration. It depends not only on how much mass an object has, but also on how that mass is distributed relative to the axis of rotation.

For two objects with the same mass, the one with more mass spread farther from the axis will have a larger moment of inertia and will be harder to spin up or slow down. This calculator focuses on a few common idealized shapes so you can quickly estimate I and build intuition for how geometry affects rotational motion.

Formulas Used by the Calculator

The calculator supports three standard shapes, each rotating about its central axis or midpoint:

• Solid disk or solid cylinder about its central axis
• Solid sphere about its center
• Thin rod about its center (axis perpendicular to the rod)

In all cases, the moment of inertia has the general form

where m is mass, R is a characteristic length (radius or half the length of a rod), and k is a constant that depends on the shape and the axis.

Solid disk or solid cylinder

For a uniform solid disk or solid cylinder of mass m and radius r, rotating about its central axis:

This result can be derived by splitting the disk into concentric rings and integrating the contributions of each ring to the total moment of inertia.

Solid sphere

For a uniform solid sphere of mass m and radius r, rotating about any diameter through its center:

The coefficient is smaller than for the disk because more of the mass lies closer to the axis, making the sphere easier to spin.

Thin rod about its center

For a thin, uniform rod of length L and mass m, rotating about an axis through its center and perpendicular to its length:

This comes from integrating along the rod's length using its linear mass density. Because the mass is spread out along a line, changing the length has a strong impact on the moment of inertia.

Interpreting the Calculator's Results

The calculator returns a value for I in units of kilogram square meters (kg·m²). A larger value means the object is more resistant to changes in its rotational motion about the chosen axis.

In rotational dynamics, Newton's second law takes the form

where τ is the applied torque and α is the angular acceleration. For a given torque, a larger moment of inertia produces a smaller angular acceleration. Conversely, to achieve the same angular acceleration for an object with a larger I, you must apply more torque.

When you compare results for different shapes or different dimensions:

• Higher I → harder to start or stop spinning, but rotational speed changes more slowly under a given torque.
• Lower I → easier to spin up or slow down, but the system may be more sensitive to small torques.

This is why activities such as figure skating and gymnastics exploit body position. Pulling arms closer to the body reduces the moment of inertia and allows faster spins for the same angular momentum.

Worked Example

Consider a solid disk of mass 2 kg and radius 0.30 m, spinning about its central axis. Using the disk formula:

1. Square the radius: 0.30² = 0.09 m².
2. Multiply by the mass: 2 kg × 0.09 m² = 0.18 kg·m².
3. Multiply by 1/2: (1/2) × 0.18 kg·m² = 0.09 kg·m².

So the moment of inertia is 0.09 kg·m². If you keep the mass the same but double the radius to 0.60 m, the radius squared becomes 0.36 m². Repeating the calculation gives:

The moment of inertia has quadrupled, even though the mass stayed the same. This illustrates the strong dependence of I on the square of the radius: doubling the radius increases I by a factor of four.

You can repeat the same process for a solid sphere or a thin rod. For instance, a 2 kg solid sphere with radius 0.30 m has

Notice that this value is smaller than for the disk with the same mass and radius, again because the mass of the sphere is distributed differently.

Shape Comparison

The table below summarizes the formulas used for each supported shape and axis. This can help you see how the constant in front of m r² or m L² affects the resulting moment of inertia.

*For objects with the same mass and characteristic size.

Assumptions and Limitations

The calculator is designed for quick estimates and teaching, not for safety-critical engineering design. It relies on several simplifying assumptions:

• Rigid bodies: Each object is treated as perfectly rigid, with no deformation during rotation.
• Uniform density: Mass is assumed to be evenly distributed throughout the object (no hollow sections, added weights, or internal components).
• Standard axes only: The axis of rotation is fixed as described for each shape (e.g., center of the disk, center of the rod). Off-center or tilted axes are not handled.
• No parallel-axis shifts: The tool does not automatically apply the parallel-axis theorem. If your actual axis is offset from the center, you must adjust the result manually.
• Idealized geometry: Real objects may have holes, cutouts, variable thickness, or tapered sections that are not captured by these simple shapes.
• Educational use: Values are approximate and intended for learning, quick checks, or early-stage concept work. They are not a substitute for detailed engineering calculations, standards, or professional review.

For more complex shapes, composite assemblies, or axes that do not pass through the center, you may need to break the object into simpler parts, find the moment of inertia for each part, and then combine them using superposition and the parallel-axis theorem.

Frequently Asked Questions

What is moment of inertia in simple terms?

Moment of inertia measures how strongly an object resists changes in its rotational motion about a particular axis. It plays the same role in rotational motion that mass plays in linear motion. A large value means the object is "reluctant" to speed up, slow down, or change its spin direction.

Which shape has the largest moment of inertia for the same mass and size?

For the shapes in this calculator and the same mass and radius, a solid disk has a larger moment of inertia than a solid sphere because more of the disk's mass is located farther from the axis. However, other shapes such as hoops or thin rings (not included here) can have even larger values of I for a given mass and radius.

How does changing radius or length affect the moment of inertia?

For the supported shapes, the moment of inertia grows with the square of the characteristic length. Doubling the radius of a disk or sphere, or the length of a rod (about its center), multiplies I by four, assuming the mass stays the same. This is why redistributing mass farther from the axis has such a dramatic effect on rotational behavior.

Can I use these results for off-center or tilted axes?

No. The formulas here apply only to the specific central axes described for each shape. If you need the moment of inertia about a different axis, you must use more advanced tools or apply the parallel-axis theorem and, if necessary, integration tailored to your geometry.