Polar Moment of Inertia Lab Simulator

What Is the Polar Moment of Inertia Lab Simulator?

How to Use the Simulator

1. Read the four cards before you touch anything. They are already answering for a 25 mm solid steel shaft at 300 N.m: Polar moment of inertia 3.835 cm^4, Shear stress at the surface 97.785 MPa, Twist per metre 5.652 deg/m and Stiffness per unit of material 1.0000x.
2. Read the sentence under each card, because that is where the panel shows its hand. They read pi (D^4 - d^4) / 32 -- the material never enters it, ok, 39.1% of yield, T L / (J G); double the diameter and this falls by exactly 16 and exactly 1 + k^2; a solid shaft is the baseline.
3. Move Outer diameter D one rung right, from 25 mm to 30 mm. The first card goes to 7.952 cm^4 and the stress card falls to 56.588 MPa. Two rungs further, at 50 mm, they are 61.359 cm^4 and 12.223 MPa — sixteen times and one eighth of the 25 mm readings, exactly.
4. Put the diameter back and drag Bore, as d / D instead. At bore 0.80 D the first card drops to 2.264 cm^4, the fourth climbs to 1.6400x, and the grid row for the bore wall comes alive at 132.500 MPa where it had been printing 0 Pa.
5. Watch the stress card while you do that. It goes up, from 97.785 MPa to 165.625 MPa, because the same torque now rides on a smaller section. Boring a shaft out buys stiffness per unit of metal and charges for it at the skin.
6. Now move Material through all six names. The first two cards do not move at all. The twist card runs 5.652 deg/m, 8.537 deg/m, 10.027 deg/m, 10.826 deg/m, 17.577 deg/m and 3830.876 deg/m, and the torsional rigidity row moves with it.
7. Push Applied torque T to its top rung. At 3.18310e+6 N.m on a 25 mm shaft the stress card reads 1037.529 GPa over OVER by 4150.12x -- would have failed, and the card changes appearance. The first card has not moved off 3.835 cm^4, because the section does not know what you hung on it.
8. Look at the top band whenever the bore is not zero. The shading is a straight ramp from nothing on the axis out to the surface, and the hatched disc is the metal that has gone. The grid row for mid-radius is exactly half the surface figure on every setting, which is that ramp in one number.
9. Read the caption under the middle band before you believe the twist. It states the factor the angle is drawn at, so a clause such as twist shown at 10x means the drawn angle is ten times the real one, and twist too small to draw means the line beside the dashed one is straight.
10. Press Reset when you have finished. It puts the four sliders back to 25 mm, solid, 300 N.m and steel, and replays the opening settle so the scene is drawn in rather than cut to.

Dragging is the right way to find out which readings move together and a poor way to land on a particular state, because all four controls step by whole rungs. The five preset buttons above exist for that reason, and the written polar moment of inertia guide is where the argument is made from scratch and the problems are worked with every step shown.

One thing this panel cannot do is take a diameter, a bore or a torque that is not on a rung, and the two inverse questions are beyond it altogether. The polar moment of inertia calculator takes typed figures, and it will run the relation backwards as well: the outer diameter behind a stiffness you need, or the largest bore that still leaves it.

Worked Example: The Same Shaft, Solid and Bored to 80 Percent

Nothing in the tables below was worked out here. Each column is a setting the lab was put into through its own sliders and then read back from, cell by cell, so every string in them is the panel’s own. The first table is the pair this section is named after.

One: one bar, one torque, and the middle taken out of it

Press A 25 mm solid steel shaft at 300 N.m and then The same shaft bored out to 80 percent. Only the bore slider moves between them, so the four control rows are the same except one. This table is transposed, because the comparison worth making runs down a column.

Three rows move in the direction a reader expects and one does not. The section falls, the twist rises and the fourth card rises; the stress card rises too, from 97.785 MPa to 165.625 MPa. That last one is half the lesson of this page, and leaving it out would turn a trade into a free gift.

Read the two kept-ratio rows together. The section keeps 0.5904 of its solid value while the metal in the face keeps only 0.3600, and the fourth card is those two divided: 1.6400x. Nothing in that sentence mentions steel, and nothing in it would change if the bar were aluminium.

The bore-wall row is the one to look at twice. It prints the literal 0 Pa in the solid column, because on a solid shaft the bore wall is the axis, where the radius is nothing. Bored to 80 per cent it reads 132.500 MPa, which is eight tenths of the surface figure, exactly as a straight ramp demands.

Two: the five shafts the buttons load

These are the presets with the readings they produce. Read the stress and verdict columns across: four of the five sit under the reference and the last one does not, on a shaft that was sized against it.

Rows one and three are the fourth power with nothing hidden in it. Same steel, same torque, and the diameter doubled from 25 mm to 50 mm: the section goes from 3.835 cm^4 to 61.359 cm^4 and the twist from 5.652 deg/m to 353.257 mdeg/m. Those are exactly sixteen times and exactly one sixteenth, and the preset title is that claim.

Rows one and four differ by one slider and three readings. The polar moment is the same string, the surface stress is the same string, and the verdict is the same string; only the twist changes, from 5.652 deg/m to 17.577 deg/m. That single comparison is the headline claim of this subject made testable.

Row five is the honest one. A 400 mm shaft at the torque a 1000 megawatt set makes at 50 hertz reads 253.303 MPa, flagged OVER by 1.01x -- would have failed, on a shaft whose published size was chosen against that very figure. The reason is rounding, and it is picked apart further down this page.

Formula and Symbol Reference

The band at the head of this page carries the section property with the stress relation beside it, because the first on its own is what makes this quantity confusing. Every reading on the panel is one of those two relations, the twist that follows from them, or a ratio between two sections.

The symbol column below uses the panel’s own names, spelled the way the lab’s own strip spells them in plain ASCII. Where the figure in the torque slider comes from — a force and a lever arm — is the torque guide; and for which of the two letters belongs to which quantity, the physics symbols guide keeps the list.

If you read one row twice, make it the fifth. Every stress figure the panel prints is quoted at a radius counted outward from the centreline, which puts the surface reading at D/2 and the axis reading at nothing at all. Nearly every wrong answer in this subject is that one length taken from the wrong place.

Reading the Diagram: a Stress Ramp, a Twisted Line and a Log Axis

The drawing is laid out as three bands and each band caption strip is reserved before anything is placed in it, so no part of the picture is ever painted across another. The top band carries the section, the middle band carries the twist, and the bottom band carries the one axis that is shared.

The ramp starts at exactly zero and that is the point of the whole band. Shear stress is the torque times the distance from the axis divided by the section, so it is strictly proportional to that distance: nothing on the axis, half the surface value at mid-radius, and the bore ratio times the surface value at the bore wall. The panel prints all three as numbers, so the picture can be checked rather than trusted.

The middle band never draws an angle without saying what scale it is at. The twist the sliders can reach runs from 287.481 ndeg/m to 35624586017.214 deg/m, which is eighteen distinct decades, so a true drawing is impossible across almost all of it. The lab magnifies or reduces by a power of ten to bring the angle into a band it can draw, and the caption carries the factor.

There are fifteen of those clauses, not two. They cover eighteen distinct decades of twist between them: one for an angle drawn true, thirteen magnifying and reducing powers of ten (four up and nine down), and twist too small to draw on 294 of the 3,072 settings, where the line beside the dashed one is simply straight. No setting anywhere draws a twisted line without a clause.

The bottom band is nine decades because it has to hold everything the sliders can reach. The smallest polar moment they produce is 43.756 mm^4 and the largest 251327.412 cm^4, and every one of the nine decades carries a tick. The three markers on it are this section, the solid shaft of the same outer diameter, and the solid shaft with the same polar moment — and on a solid setting all three land together.

The Physics: Why the Fourth Power, and Why Hollow Wins

Leave the panel on the opening shaft and ask what the outer metal is doing that the core is not. Two things grow with the radius at the same time: metal further out has to shear through a greater distance for the same wind-up, and whatever it pushes back with acts on a longer arm about the centre. Multiply those together and each ring contributes as the square of its radius, which is why the defining integral weights area by that square — and why, taken over a disc, it collapses onto the fourth power of the diameter.

Every consequence is in that exponent. Double the diameter and the section is multiplied by exactly sixteen, so the twist falls by sixteen and the surface stress by eight, whatever the torque, the length or the metal. Press the third preset and read it off: 3.835 cm^4 becomes 61.359 cm^4, 5.652 deg/m becomes 353.257 mdeg/m, and 97.785 MPa becomes 12.223 MPa.

The same exponent is why hollow wins. Bore a concentric hole of ratio k and the section keeps exactly 1 - k^4 of its solid value while the metal in the face keeps only 1 - k^2, and the ratio of those two is exactly 1 + k^2. The bore ladder below is that identity walked rung by rung.

Divide the last two ratio columns into each other on any row and the sixth column falls out. At bore 0.80 D that is 0.5904 over 0.3600, which the panel prints as 1.6400x. The reason is the ramp: the metal near the axis was barely carrying anything, so removing it costs the fourth power of the ratio and saves only the square.

And the honest other half is in the third column. The surface stress climbs every rung, from 97.785 MPa solid to 284.341 MPa at bore 0.90 D, where the verdict finally flips to OVER by 1.14x -- would have failed. The gain is per unit of material, never per shaft, and the bottom rung of that ladder is a section that has stopped being a shaft.

The sharpest result on the panel needs no material at all, and the material slider is how you prove it. Hold the opening shaft and walk all six names: the first two cards print the same two strings every time.

Two columns are frozen and two are not. The polar moment and the surface stress are byte-identical down all six rows, because neither contains a material; the twist and the torsional rigidity move with the modulus, which is the only place a material enters. Those six modulus figures are the source’s own tabulated typical values at room temperature, and none was checked independently here.

One bridge is worth stating, and then it stops. The mass moment of inertia of a uniform shaft is exactly this quantity multiplied by a density and a length, which is why one is in metres to the fourth power and the other in kilogram metres squared. That is an identity rather than a figure: neither source here gives a density, so no kilogram metre squared number appears on this page at all. The quantity itself, for nine common shapes, belongs to the moment of inertia guide and to the moment of inertia simulator.

Where It Breaks Down

Every state the four sliders can reach is answered, and the arithmetic is not where this lab can let you down. What it cannot do is model a real shaft, and what it refuses to do is treat its own idealisations as facts. The first item below is the one that matters more than all the others put together.

The model never notices that the shaft has broken

Read this one before any of the others. Torque and stress stay in strict proportion here however large either grows, so the panel answers settings that no piece of metal would survive. Measured against the 250 MPa reference, 1,230 of the 3,072 states the sliders reach are past it — 40.04 per cent of the grid — and the worst corner prints 218239.979 GPa of shear stress. The stress card says so state by state and changes appearance when it does; the polar moment card never will, because a property of the shape is simply indifferent to whether the metal held.

The 250 MPa is an assumption about one shaft, not a property of steel

It is the figure a published sample calculation states for the steel of its own example shaft, inside its own sizing arithmetic, and this lab uses it only as a fixed reference to measure a stress against. Any actual component arrives with a certificate for the metal it was cut from, a heat treatment history and a safety factor somebody chose on purpose. Nothing on this page is a design allowable, and the percentages beside the stress card are comparisons rather than verdicts on anybody’s part.

The six shear moduli are published typical values, not properties of your metal

They come from a table headed as typical values at room temperature, each carrying its own citation marker in the source, and none of them was independently checked in this slot. No temperature dependence, no alloy and no heat treatment is claimed for any of them. They matter only for the twist card and the torsional rigidity row, and the material slider prints the attribution underneath so that it travels with the number. The stretching modulus is a different constant again, and the Young’s modulus calculator is where that one lives.

A published rounding is not an exact figure, and the turbine preset proves it

The sizing calculation behind that preset computes a radius of 0.200877 m and quotes 0.200 m, or a diameter of 400 mm; its factor-of-five case quotes 0.343 m and 690 mm, where the literal diameter is 0.6870 m and its own radius doubles to 686 mm. Because stress goes as one over the radius cubed, rounding the radius down pushes the stress up: the 400 mm shaft reads 253.303 MPa, which is 1.3212 per cent above the 250 MPa it was sized against. It is not a safe shaft, it is a rounded one, which is why the source then applies a factor of safety of five.

Everything on this panel is a circle

Twist a section that is not round and it warps out of its own plane, so this quantity stops being the torsional constant and a torsion constant takes over the job. The cleared source states that outright, in a sentence quoted whole in the references below and paraphrased in one line under the drawing. Nothing here offers a figure for a square bar, a splined hub or a keyed shaft, because nothing here sources one.

Twisting, and nothing else at all

Nothing in this model is bent, stretched or pushed sideways, and the shaft itself has no keyway cut into it, no shoulder, no press fit and no change of diameter along its length. Features like those are where shafts actually break, almost never in the plain barrel between them. No figure is offered for that here, for one reason: a stress concentration factor is precisely what neither cleared source carries, and a number made up to fill the hole would conceal it instead of marking it.

Everything is per metre, and the bore is on the axis

There is no length slider, because the twist is exactly proportional to the length and a per-metre figure is therefore the general one. There is no eccentric bore either, no cross-drilling and no seam: the hole is assumed concentric and smaller than the shaft, which is what makes the kept-ratio rows exact. A section with any of those features is a different calculation this lab does not do.

This shear stress is not the shear stress in a fluid

The symbol is shared and the quantity is not. Here it is an elastic shear stress in a twisted solid, fixed by a geometry and a torque; in a fluid it is set by a velocity gradient and a viscosity, and nothing on this page applies to it. The viscosity guide is where that other one lives, and the viscosity simulator is the panel for it.

The sliders are ladders, and the lab runs one way only

All four controls step by whole rungs, so the lab reaches 3,072 states and nothing in between. It also runs forwards only: you set a section and a load and read what follows. The polar moment of inertia calculator takes typed figures and runs the relation in all three directions, including the largest bore that still leaves a stiffness you have to keep.

Three things the drawing gives up on a narrow screen

Below a drawn radius of 26 pixels the word marking the central axis is dropped, although the cross is still there and the caption still says the stress rises from zero on the axis. The label on the equivalent-solid marker shortens in four steps and is dropped if none fits, although its stem stays and the panel keeps the figure. And below a 160 pixel drawing the bands would begin switching off rather than drawing something unexplained. On a narrow screen take the figures from the cards, which are unaffected.

Where the Polar Moment of Inertia Is Actually Used

Deciding whether to bore a driveshaft out

Load The same shaft bored out to 80 percent and the whole decision is on screen in two cards. The section keeps 0.5904 of its value on 0.3600 of the metal, so the fourth card reads 1.6400x; the stress card goes from 97.785 MPa to 165.625 MPa. Tubing wins that trade, which is why so little of a driveshaft, a bicycle spindle or an aircraft control rod is solid metal — and why the wall that is left then gets argued over on its own account.

Sizing a shaft from the stiffness it has to have

Move the diameter slider one rung and watch how little it takes. From 25 mm to 30 mm the section goes from 3.835 cm^4 to 7.952 cm^4, which is more than double for a fifth more diameter, and the twist falls from 5.652 deg/m to 2.726 deg/m. The fourth power is the cheapest lever in the whole problem, which is why it is pulled first.

Reproducing a published turboset sizing

Load A 1000 MW turbine shaft at 50 Hz: 400 mm solid at 3.18310e+6 N.m, which is the torque a 1000 megawatt set makes at 50 hertz. Getting from a power rating to that torque is one step, since power is torque times angular speed, and the power guide is where it lives. The panel prints 253.303 MPa for the rounded shaft, which is why the published calculation then applies a factor of safety.

Choosing a metal when the stiffness is already fixed

The material slider is a specification in one card. Switch the opening shaft from steel to aluminium and the section stays 3.835 cm^4, the surface stress stays 97.785 MPa, and only the twist changes, from 5.652 deg/m to 17.577 deg/m. If a design is limited by stress, the metal is nearly irrelevant; if it is limited by wind-up, the metal is most of the answer.

Checking a stress against a published yield figure

The sentence under the stress card turns a section into something arguable. 310.989 MPa on a 17 mm spindle arrives with OVER by 1.24x -- would have failed attached, which is far more useful than the stress alone. The reference it is measured against is one sample calculation assumption about one shaft, so read the percentage as a comparison and never as a sign-off on a real part.

Catching a figure somebody else quoted

The commonest practical use of this panel is as a check. Set the diameter, the bore and the torque the figure was supposedly taken at and see whether the cards agree, and if the quoted stress has no radius attached to it, the grid rows show how much room there is: 0 Pa on the axis, 48.892 MPa at mid-radius and 97.785 MPa at the surface of one ordinary shaft.

Telling this quantity apart from the one it is named after

Spinning a wheel or a rotor up to speed is a question about the mass moment of inertia, which cannot be answered without knowing how much metal there is and where; the moment of inertia calculator is the page for that one. There is no mass control anywhere on this panel, because the quantity it prints contains no mass at all. The two are not interchangeable, and nothing this panel displays can be turned into the other without information it was never given.

Where to Go Next

A panel can put you in a state and let you move one thing at a time, and that is the end of what a panel is for. The polar moment of inertia guide is where the subject itself is written out: why the fourth power dominates, what boring a shaft out really buys and costs, and why this quantity is not the one it is named after, with the problems worked step by step.

One post sits very close to this one and is not the same thing. The moment of inertia guide owns the mass moment of inertia in kilogram metres squared, for nine common shapes, which is about how hard a body is to spin up rather than how hard a section is to twist.

Five more pick up what this lab uses but does not explain. The torque guide is the quantity on the torque slider, the moment of a force guide is the same idea under its other name, the power guide turns a turbine rating in megawatts into the figure the torque slider wants, the physics symbols guide settles which letter stands for which quantity, and the mechanics formula sheet is the one-page reference.

Six more again sit one step further out. The types of forces guide is where a shear force sits among the others, Hooke's law and the spring constant guide are the stretching half of elasticity, the equilibrium guide is what has to balance before any of this applies, the angular momentum guide is the rotating body in motion, and the viscosity guide is the other shear stress entirely.

On the tool side, the polar moment of inertia calculator takes typed diameters, bores and torques and runs the relation in all three directions, the moment of inertia calculator is the mass quantity, the torque calculator works a torque from a force and a lever arm, the Nm to ft-lb converter changes its unit, the Young’s modulus calculator is the stretching modulus, and the angular momentum calculator is the body in motion.

Six neighbouring pictures sit beside this one: the moment of inertia simulator, the torque simulator, the Hooke’s law simulator, the equilibrium simulator, the angular momentum simulator and the viscosity simulator. Beyond them there is the simulation library, an index of calculators, and the blog for everything written out at length.