physics

Second polar moment of area

Second polar moment of area 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 Second polar moment of area rather than just read about it. In short: The second polar moment of area, also known as "polar moment of inertia" or even "moment of inertia", is a quantity used to describe resistance to torsional deformation (deflection), in objects (or segments of an object) with an invariant cross-section and no significant warping or out-of-plane deformation. It is a constituent of the second moment of area, linked through the perpendicular axis theorem.

Key takeaways

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

Reference excerpt

The second polar moment of area, also known as "polar moment of inertia" or even "moment of inertia", is a quantity used to describe resistance to torsional deformation (deflection), in objects (or segments of an object) with an invariant cross-section and no significant warping or out-of-plane deformation. It is a constituent of the second moment of area, linked through the perpendicular axis theorem. Where the planar second moment of area describes an object's resistance to deflection (bending) when subjected to a force applied to a plane parallel to the central axis, the polar second moment of area describes an object's resistance to deflection when subjected to a moment applied in a plane perpendicular to the object's central axis (i.e. parallel to the cross-section). Similar to planar second moment of area calculations ( I x {\displaystyle I_{x}} , I y {\displaystyle I_{y}} , and I x y {\displaystyle I_{xy}} ), the polar second moment of area is often denoted as I z {\displaystyle I_{z}} . While several engineering textbooks and academic publications also denote it as J {\displaystyle J} or J z {\displaystyle J_{z}} , this designation should be given careful attention so that it does not become confused with the torsion constant, J t {\displaystyle J_{t}} , used for non-cylindrical objects. Simply put, the polar moment of area is a shaft or beam's resistance to being distorted by torsion, as a function of its shape. The rigidity comes from the object's cross-sectional area only, and does not depend on its material composition or shear modulus. The greater the magnitude of the second polar moment of area, the greater the torsional stiffness of the object.

Definition

The equation describing the polar moment of area is a multiple integral over the cross-sectional area, A {\displaystyle A} , of the object.

J = ∬ A r 2 d A {\displaystyle J=\iint _{A}r^{2}\,dA}

where r {\displaystyle r} is the distance to the element d A {\displaystyle dA} . Substituting the x {\displaystyle x} and y {\displaystyle y} components, using the Pythagorean theorem:

J = ∬ A ( x 2 + y 2 ) d x d y {\displaystyle J=\iint _{A}\left(x^{2}+y^{2}\right)dx\,dy}

J = ∬ A x 2 d x d y + ∬ A y 2 d x d y {\displaystyle J=\iint _{A}x^{2}\,dx\,dy+\iint _{A}y^{2}\,dx\,dy}

Given the planar second moments of area equations, where:

I x = ∬ A y 2 d x d y {\displaystyle I_{x}=\iint _{A}y^{2}dx\,dy}

I y = ∬ A x 2 d x d y {\displaystyle I_{y}=\iint _{A}x^{2}dx\,dy}

It is shown that the polar moment of area can be described as the summation of the x {\displaystyle x} and y {\displaystyle y} planar moments of area, I x {\displaystyle I_{x}} and I y {\displaystyle I_{y}}

∴ J = I z = I x + I y {\displaystyle \therefore J=I_{z}=I_{x}+I_{y}}

This is also shown in the perpendicular axis theorem. For objects that have rotational symmetry, such as a cylinder or hollow tube, the equation can be simplified to:

J = 2 I x {\displaystyle J=2I_{x}} or J = 2 I y {\displaystyle J=2I_{y}}

For a circular section with radius R {\displaystyle R} :

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Second polar moment of area

Start with the simplest possible case. Write down what Second polar moment of area 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 Second polar moment of area 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 Second polar moment of area 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 Second polar moment of area

In research: Second polar moment of area 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 Second polar moment of area 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: Second polar moment of area is common in secondary-school and first-year university syllabi. It links to neighbouring topics Continuum mechanics, Mechanical quantities, Moment (physics), so understanding it makes those chapters shorter.
In everyday life: Look for Second polar moment of area 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Second polar moment of area” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Second polar moment of area in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Second polar moment of area 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 Second polar moment of area out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Second polar moment of area in simple terms?

Why does Second polar moment of area 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 Second polar moment of area?

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 Second polar moment of area.