physics

Moment of inertia

Moment of inertia is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Moment of inertia rather than just read about it. In short: The moment of inertia (also known as mass moment of inertia, angular/rotational mass, second moment of mass, or rotational inertia) is a measure of how difficult it is to change the rotation rate of a rigid body about a given axis. It is the ratio between the torque applied and the resulting angular acceleration about that axis.

Key takeaways

Moment of inertia belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Moment of inertia to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Moment of inertia from memory before moving on to harder problems.

Reference excerpt

The moment of inertia (also known as mass moment of inertia, angular/rotational mass, second moment of mass, or rotational inertia) is a measure of how difficult it is to change the rotation rate of a rigid body about a given axis. It is the ratio between the torque applied and the resulting angular acceleration about that axis. It plays the same role in rotational motion as mass does in linear motion. A body's moment of inertia about a particular axis depends on both the mass and its distribution relative to the axis, increasing with mass and distance from the axis. For a point mass, the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis. If a body is forced to rotate in a plane, then only the moment of inertia about an axis perpendicular to the plane, a scalar value, matters. If a body is allowed to rotate in all three dimensions, then its moment is described by a symmetric three-by-three matrix. This symmetric matrix can be orthogonally diagonalized by a set of mutually perpendicular principal axes. Torques around these principal axes act independently of each other.

Introduction When a body is free to rotate around an axis, torque must be applied to change its angular momentum. The amount of torque needed to cause any given angular acceleration (the rate of change in angular velocity) is proportional to the moment of inertia of the body. Moments of inertia may be expressed in units of kilogram metre squared (kg·m2) in SI units and pound-foot-squared (lb·ft2) in imperial or US units, and other relevant units. The moment of inertia plays the role in rotational kinetics that mass (inertia) plays in linear kinetics—both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by m r 2 {\displaystyle mr^{2}} , where r {\displaystyle r} is the distance of the point from the axis, and m {\displaystyle m} is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object. In 1673, Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a compound pendulum. The term moment of inertia ("momentum inertiae" in Latin) was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Euler's second law. The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body. The moment of inertia also appears in momentum, kinetic energy, and in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor. The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axes determine how steering forces on the control surfaces of its wings, elevators and rudder(s) affect the plane's motions in roll, pitch and yaw.

Definition The moment of inertia is defined as the product of mass of section and the square of the distance between the reference axis and the centroid of the section. It is denoted with the symbol I or J.

The moment of inertia I is also defined as the ratio of the net angular momentum L of a system to its angular velocity ω around a principal axis, that is

I = L ω . {\displaystyle I={\frac {L}{\omega }}.}

If the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their outstretched arms or divers curl their bodies into a tuck position during a dive, to spin faster. If the shape of the body does not change, then its moment of inertia appears in Newton's law of motion as the ratio of an applied torque τ on a body to the angular acceleration α around a principal axis, that is

τ = I α . {\displaystyle \tau =I\alpha .}

For a simple pendulum, this definition yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as,

I = m r 2 . {\displaystyle I=mr^{2}.}

… excerpt ends here. Continue reading the full article.

Illustrations

Moment of inertia: Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to conservation of angular momentum. — Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to conservation of angular momentum.

Worked examples

Example 1 — a first encounter with Moment of inertia

Start with the simplest possible case. Write down what Moment of inertia claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Moment of inertia before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Moment of inertia ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Moment of inertia

In research: Moment of inertia appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Moment of inertia in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Moment of inertia is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mechanical quantities, Moment (physics), Rigid bodies, so understanding it makes those chapters shorter.
In everyday life: Look for Moment of inertia outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Moment of inertia in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Moment of inertia means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Moment of inertia out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Moment of inertia in simple terms?

Why does Moment of inertia matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Moment of inertia?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Moment of inertia.