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Nth Polar Moment of Inertia Formula

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Nth Polar Moment of Inertia can be defined as Integral arising from nonlinear behaviour of material. Check FAQs

J n = (\frac{2 \cdot π}{n + 3}) \cdot ({r 2}^{n + 3} - {r 1}^{n + 3})

J_n - Nth Polar Moment of Inertia?n - Material Constant?r₂ - Outer Radius of Shaft?r₁ - Inner Radius of Shaft?π - Archimedes' constant?

Nth Polar Moment of Inertia Example

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Here is how the Nth Polar Moment of Inertia equation looks like with Values.

Here is how the Nth Polar Moment of Inertia equation looks like with Units.

Here is how the Nth Polar Moment of Inertia equation looks like.

1E+9 = (\frac{2 \cdot 3.1416}{0.25 + 3}) \cdot (100^{0.25 + 3} - 40^{0.25 + 3})

1E+9mm⁴ = (\frac{2 \cdot 3.1416}{0.25 + 3}) \cdot ({100mm}^{0.25 + 3} - {40mm}^{0.25 + 3})

1E+9 = (\frac{2 \cdot 3.1416}{0.25 + 3}) \cdot (100^{0.25 + 3} - 40^{0.25 + 3})

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Nth Polar Moment of Inertia Solution

Follow our step by step solution on how to calculate Nth Polar Moment of Inertia?

FIRST Step Consider the formula

J n = (\frac{2 \cdot π}{n + 3}) \cdot ({r 2}^{n + 3} - {r 1}^{n + 3})

Next Step Substitute values of Variables

∴

J n = (\frac{2 \cdot π}{0.25 + 3}) \cdot ({100mm}^{0.25 + 3} - {40mm}^{0.25 + 3})

Next Step Substitute values of Constants

∴

J n = (\frac{2 \cdot 3.1416}{0.25 + 3}) \cdot ({100mm}^{0.25 + 3} - {40mm}^{0.25 + 3})

Next Step Convert Units

∴

J n = (\frac{2 \cdot 3.1416}{0.25 + 3}) \cdot ({0.1m}^{0.25 + 3} - {0.04m}^{0.25 + 3})

Next Step Prepare to Evaluate

∴

J n = (\frac{2 \cdot 3.1416}{0.25 + 3}) \cdot ({0.1}^{0.25 + 3} - {0.04}^{0.25 + 3})

Next Step Evaluate

∴

J n = 0.00103183369075116m⁴

Next Step Convert to Output's Unit

∴

J n = 1031833690.75116mm⁴

LAST Step Rounding Answer

∴

J n = 1E+9mm⁴

Nth Polar Moment of Inertia Formula Elements

Variables

Constants

Nth Polar Moment of Inertia

Nth Polar Moment of Inertia can be defined as Integral arising from nonlinear behaviour of material.

Symbol: J_n

Measurement: Second Moment of AreaUnit: mm⁴

Note: Value should be greater than 0.

Formulas to find Nth Polar Moment of Inertia

Material Constant

Material Constant is the constant used when the beam yielded plastically.

Symbol: n

Measurement: NAUnit: Unitless

Note: Value can be positive or negative.

Formulas to find Material Constant

Outer Radius of Shaft

Outer Radius of Shaft is the external radius of shaft.

Symbol: r₂

Measurement: LengthUnit: mm

Note: Value should be greater than 0.

Formulas to find Outer Radius of Shaft

Inner Radius of Shaft

Inner Radius of Shaft is the internal radius of shaft.

Symbol: r₁

Measurement: LengthUnit: mm

Note: Value should be greater than 0.

Formulas to find Inner Radius of Shaft

Archimedes' constant

Archimedes' constant is a mathematical constant that represents the ratio of the circumference of a circle to its diameter.

Symbol: π

Value: 3.14159265358979323846264338327950288

Other formulas in Elastic Work Hardening Material category

Go Incipient Yielding Torque in Work Hardening for Hollow Shaft

T i = \frac{𝞽 nonlinear \cdot J n}{{r 2}^{n}}

Go Incipient Yielding Torque in Work Hardening Solid Shaft

T i = \frac{𝞽 nonlinear \cdot J n}{{r 2}^{n}}

Go Elasto Plastic Yielding Torque in Work Hardening for Solid Shaft

T ep = \frac{2 \cdot π \cdot 𝞽 nonlinear \cdot {r 2}^{3}}{3} \cdot (1 - (\frac{n}{n + 3}) \cdot {(\frac{ρ}{r 2})}^{3})

Go Elasto Plastic Yielding Torque in Work Hardening for Hollow Shaft

T ep = \frac{2 \cdot π \cdot 𝞽 nonlinear \cdot {r 2}^{3}}{3} \cdot (\frac{3 \cdot ρ^{3}}{{r 2}^{3} \cdot (n + 3)} - (\frac{3}{n + 3}) \cdot {(\frac{r 1}{ρ})}^{n} \cdot {(\frac{r 1}{r 2})}^{3} + 1 - {(\frac{ρ}{r 2})}^{3})

How to Evaluate Nth Polar Moment of Inertia?

Nth Polar Moment of Inertia evaluator uses Nth Polar Moment of Inertia = ((2*pi)/(Material Constant+3))*(Outer Radius of Shaft^(Material Constant+3)-Inner Radius of Shaft^(Material Constant+3)) to evaluate the Nth Polar Moment of Inertia, Nth Polar Moment of Inertia formula is defined as a measure of the resistance of a cross-section to torsion, which is a crucial parameter in the analysis of torsional deformation of bars, and is used to determine the twisting stiffness of a bar under torsional loading. Nth Polar Moment of Inertia is denoted by J_n symbol.

How to evaluate Nth Polar Moment of Inertia using this online evaluator? To use this online evaluator for Nth Polar Moment of Inertia, enter Material Constant (n), Outer Radius of Shaft (r₂) & Inner Radius of Shaft (r₁) and hit the calculate button.

FAQs on Nth Polar Moment of Inertia

What is the formula to find Nth Polar Moment of Inertia?

The formula of Nth Polar Moment of Inertia is expressed as Nth Polar Moment of Inertia = ((2*pi)/(Material Constant+3))*(Outer Radius of Shaft^(Material Constant+3)-Inner Radius of Shaft^(Material Constant+3)). Here is an example- 1E+21 = ((2*pi)/(0.25+3))*(0.1^(0.25+3)-0.04^(0.25+3)).

How to calculate Nth Polar Moment of Inertia?

With Material Constant (n), Outer Radius of Shaft (r₂) & Inner Radius of Shaft (r₁) we can find Nth Polar Moment of Inertia using the formula - Nth Polar Moment of Inertia = ((2*pi)/(Material Constant+3))*(Outer Radius of Shaft^(Material Constant+3)-Inner Radius of Shaft^(Material Constant+3)). This formula also uses Archimedes' constant .

Can the Nth Polar Moment of Inertia be negative?

No, the Nth Polar Moment of Inertia, measured in Second Moment of Area cannot be negative.

Which unit is used to measure Nth Polar Moment of Inertia?

Nth Polar Moment of Inertia is usually measured using the Millimeter⁴[mm⁴] for Second Moment of Area. Meter⁴[mm⁴], Centimeter⁴[mm⁴] are the few other units in which Nth Polar Moment of Inertia can be measured.

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Nth Polar Moment of Inertia Formula

Nth Polar Moment of Inertia Example

Nth Polar Moment of Inertia Solution

Nth Polar Moment of Inertia Formula Elements

Other formulas in Elastic Work Hardening Material category

How to Evaluate Nth Polar Moment of Inertia?

FAQs on Nth Polar Moment of Inertia

Variable Name