How to Calculate Inertia for Standard Shapes

Inertia, specifically rotational inertia or moment of inertia, not only depends on how much mass an object has, but also on how that mass is distributed relative to the axis of rotation.

Different shapes have different geometries, which causes their mass to be positioned at varying distances from the axis. The farther the mass is from the axis, the greater the resistance to rotational motion.

For example:

• A solid cylinder has more mass concentrated near the axis → lower moment of inertia.

• A hollow cylinder has mass farther from the axis → higher moment of inertia.

• A thin rod rotating about its centre has less resistance than the same rod rotating about one end.

This is why each shape has its own unique formula for moment of inertia—reflecting both its mass and geometry.

Solid Cylinder

A solid cylinder is a common shape in engineering, found in components like rollers, shafts, and flywheels. When it rotates about its central (longitudinal) axis, its moment of inertia is given by:

 I =  ½ MR²

Example

A solid cylinder has:

• Mass M = 8 kg
  
• Radius R = 0.3 m

I = ½ MR²

I = ½ × 8 × (0.3)²

I = 0.5 × 8 × 0.09 = 0.36 kg⋅m²

Hollow Cylinder

A hollow cylinder (also called a cylindrical shell) is a tube-like structure where all the mass is concentrated at a fixed distance from the central axis. When it rotates about this central (longitudinal) axis, the moment of inertia is

 I = MR²

Example

A hollow cylindrical shell (thin-walled) has:

• Mass M = 6 kg
  
• Radius R = 0.5 m
  

For a thin-walled hollow cylinder, all the mass is assumed to be at a fixed distance R from the axis. The moment of inertia is given by:

I = MR²

I = 6 × (0.5)²

I = 6 × 0.25 = 1.5 kg⋅m²

Solid Sphere

 A solid sphere is a perfectly round object with mass distributed uniformly throughout its volume. When it rotates about an axis through its centre (like a spinning ball), its moment of inertia is 

 I = ⅖ MR²

Example

A solid sphere has:

• Mass M = 4 kg
  
• Radius R = 0.2 m

For a solid sphere rotating about its center (through its diameter), the moment of inertia is:

I = ⅖ MR²

I = ⅖ × 4 × (0.2)²

I = ⅖ × 4 × 0.04 

I = 0.064kg⋅m²

Thin Rod, with axis at the centre

A thin, uniform rod has mass distributed evenly along its length. When it rotates about an axis perpendicular to its length and passing through its centre (like a baton spinning in the air), the moment of inertia is

 I = ¹/₁₂ ML²

Example

A thin, uniform rod has:

• Mass M = 2 kg
  
• Length L = 1.5 m

When the rotation axis is at the center of the rod and perpendicular to its length, the moment of inertia is:

I = ¹/₁₂ ML²

I = ¹/₁₂ × 2 × (1.5)²

I = ¹/₁₂ × 2 × 2.25 

I = 0.375kg⋅m²

Thin Rod with axis at the end

A thin, uniform rod with rotation about an axis perpendicular to its length and passing through one end (like a door swinging on a hinge) has a moment of inertia given by:

I = ⅓ ML²

Example

A thin rod of length 2 m and mass 3 kg is rotating about one of its ends perpendicular to its length. What is the moment of inertia?

I = ⅓ ML²

Where:

• M 3  kg
  
• L = 2 m
  

I = ⅓ × 3 × (2)²

I = 1×4 = 4 kg⋅m²

So, the moment of inertia is 4 kg·m².

The larger the moment of inertia, the harder it is to rotate the object.
Units of moment of inertia: kg⋅m²

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