AP Physics 1118 views·Updated Jun 9, 2026·2 pages

Understanding Moment of Inertia for Common Shapes

priscilla@cilla

Moment of inertia is a crucial concept in rotational motion,...

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$Moment of Inertia Newton's 2nd Law for rotation: ang. acceler.d= $\frac{T_{net}}{I}$ $\underset{inertia}{\leftarrow}$ common shapes obje$

Moment of Inertia Basics

When objects rotate, Newton's Second Law takes a special form: angular acceleration = net torque ÷ moment of inertia $α = τ/I$ . The moment of inertia (I) represents an object's rotational inertia - how much it resists changes in rotational motion.

Different shapes have specific formulas for calculating moment of inertia. For example, a thin rod rotating about its center has I = (1/12)ML², while the same rod rotating about its end has I = (1/3)ML². Notice how the value increases when rotation occurs farther from the center of mass!

Other common shapes include a plane $I = (1/12)Ma² about center$ , a cylinder/disk $I = (1/2)MR² about center$ , and a solid sphere $I = (2/5)MR² about diameter$ . The formula always includes both the object's mass and the square of a distance measurement.

Think About It: Why does moment of inertia depend on both mass AND the distribution of that mass? The further mass is from the rotation axis, the harder it is to start or stop rotation!

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$Moment of Inertia Newton's 2nd Law for rotation: ang. acceler.d= $\frac{T_{net}}{I}$ $\underset{inertia}{\leftarrow}$ common shapes obje$

Calculating Moment of Inertia

For systems with multiple objects, the total moment of inertia equals the sum of each individual part's moment of inertia. This can be written as I = m₁R₁² + m₂R₂² + ... for objects at different distances from the axis.

When all mass is at the same distance from the rotation axis (like in a cylindrical hoop), the formula simplifies to I = MR², where M is the total mass and R is the radius to all mass points.

Let's see how to apply these formulas with an example: For a solid cylinder with mass 4 kg, radius 0.5 m, and length 0.8 m rotating about its central axis, we use I = (1/2)mR². Plugging in the values: I = (1/2)(4 kg)(0.5 m)² = 0.5 kg·m². This tells us exactly how much the cylinder resists rotational acceleration.

Pro Tip: When solving moment of inertia problems, first identify the shape and rotation axis before selecting the correct formula. The axis location makes a huge difference in the final value!

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AP Physics 1118 views·Updated Jun 9, 2026·2 pages

Understanding Moment of Inertia for Common Shapes

priscilla@cilla

Add to Google sources

Moment of inertia is a crucial concept in rotational motion, representing an object's resistance to changes in its rotation. Think of it as the rotational equivalent of mass, determining how easily an object can be accelerated in a spinning motion.

of 2

$Moment of Inertia Newton's 2nd Law for rotation: ang. acceler.d= $\frac{T_{net}}{I}$ $\underset{inertia}{\leftarrow}$ common shapes obje$

Moment of Inertia Basics

Think About It: Why does moment of inertia depend on both mass AND the distribution of that mass? The further mass is from the rotation axis, the harder it is to start or stop rotation!

of 2

$Moment of Inertia Newton's 2nd Law for rotation: ang. acceler.d= $\frac{T_{net}}{I}$ $\underset{inertia}{\leftarrow}$ common shapes obje$

Calculating Moment of Inertia

When all mass is at the same distance from the rotation axis (like in a cylindrical hoop), the formula simplifies to I = MR², where M is the total mass and R is the radius to all mass points.

Pro Tip: When solving moment of inertia problems, first identify the shape and rotation axis before selecting the correct formula. The axis location makes a huge difference in the final value!