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Polar Modulus of Hollow Shaft Formula

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The Polar Modulus of the shaft section is equal to the ratio of the polar moment of inertia to the radius of the shaft. It is denoted by Zp. Check FAQs

Z p = \frac{π \cdot (({d o}^{4}) - ({d i}^{4}))}{16 \cdot d o}

Z_p - Polar Modulus?d_o - Outer Diameter of Shaft?d_i - Inner Dia of Shaft?π - Archimedes' constant?

Polar Modulus of Hollow Shaft Example

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Here is how the Polar Modulus of Hollow Shaft equation looks like with Values.

Here is how the Polar Modulus of Hollow Shaft equation looks like with Units.

Here is how the Polar Modulus of Hollow Shaft equation looks like.

0.0045 = \frac{3.1416 \cdot ((700^{4}) - ({0.688}^{4}))}{16 \cdot 700}

0.0045m³ = \frac{3.1416 \cdot (({700mm}^{4}) - ({0.688m}^{4}))}{16 \cdot 700mm}

0.0045 = \frac{3.1416 \cdot ((700^{4}) - ({0.688}^{4}))}{16 \cdot 700}

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Polar Modulus of Hollow Shaft Solution

Follow our step by step solution on how to calculate Polar Modulus of Hollow Shaft?

FIRST Step Consider the formula

Z p = \frac{π \cdot (({d o}^{4}) - ({d i}^{4}))}{16 \cdot d o}

Next Step Substitute values of Variables

∴

Z p = \frac{π \cdot (({700mm}^{4}) - ({0.688m}^{4}))}{16 \cdot 700mm}

Next Step Substitute values of Constants

∴

Z p = \frac{3.1416 \cdot (({700mm}^{4}) - ({0.688m}^{4}))}{16 \cdot 700mm}

Next Step Convert Units

∴

Z p = \frac{3.1416 \cdot (({0.7m}^{4}) - ({0.688m}^{4}))}{16 \cdot 0.7m}

Next Step Prepare to Evaluate

∴

Z p = \frac{3.1416 \cdot (({0.7}^{4}) - ({0.688}^{4}))}{16 \cdot 0.7}

Next Step Evaluate

∴

Z p = 0.00450074035006326m³

LAST Step Rounding Answer

∴

Z p = 0.0045m³

Polar Modulus of Hollow Shaft Formula Elements

Variables

Constants

Polar Modulus

The Polar Modulus of the shaft section is equal to the ratio of the polar moment of inertia to the radius of the shaft. It is denoted by Zp.

Symbol: Z_p

Measurement: VolumeUnit: m³

Note: Value should be greater than 0.

Formulas to find Polar Modulus

Outer Diameter of Shaft

Outer Diameter of Shaft is the length of the longest chord of the surface of the hollow circular shaft.

Symbol: d_o

Measurement: LengthUnit: mm

Note: Value should be greater than 0.

Formulas to find Outer Diameter of Shaft

Inner Dia of Shaft

The Inner Dia of Shaft is defined as the Diameter that is length of the longest chord inside the hollow shaft.

Symbol: d_i

Measurement: LengthUnit: m

Note: Value can be positive or negative.

Formulas to find Inner Dia 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 to find Polar Modulus

Go Polar Modulus

Z p = \frac{J}{R}

Go Polar Modulus of Solid Shaft

Z p = \frac{π \cdot d^{3}}{16}

Go Polar Modulus using Maximum Twisting Moment

Z p = (\frac{T}{τ max})

Other formulas in Polar Modulus category

Go Diameter of Solid Shaft with known Polar Modulus

d = {(\frac{16 \cdot Z p}{π})}^{\frac{1}{3}}

Go Inner diameter of Hollow Shaft using Polar Modulus

d i = {(({d o}^{4}) - (\frac{Z p \cdot 16 \cdot d o}{π}))}^{\frac{1}{4}}

Go Polar Moment of Inertia using Polar Modulus

J = R \cdot Z p

Go Polar Moment of Inertia of Solid Shaft

J = \frac{π \cdot d^{4}}{32}

How to Evaluate Polar Modulus of Hollow Shaft?

Polar Modulus of Hollow Shaft evaluator uses Polar Modulus = (pi*((Outer Diameter of Shaft^4)-(Inner Dia of Shaft^4)))/(16*Outer Diameter of Shaft) to evaluate the Polar Modulus, The Polar Modulus of Hollow Shaft formula is defined as the ratio of the polar moment of inertia to the radius of the shaft. It is also called as torsional section modulus. It is denoted by Z p. Polar Modulus is denoted by Z_p symbol.

How to evaluate Polar Modulus of Hollow Shaft using this online evaluator? To use this online evaluator for Polar Modulus of Hollow Shaft, enter Outer Diameter of Shaft (d_o) & Inner Dia of Shaft (d_i) and hit the calculate button.

FAQs on Polar Modulus of Hollow Shaft

What is the formula to find Polar Modulus of Hollow Shaft?

The formula of Polar Modulus of Hollow Shaft is expressed as Polar Modulus = (pi*((Outer Diameter of Shaft^4)-(Inner Dia of Shaft^4)))/(16*Outer Diameter of Shaft). Here is an example- 0.004501 = (pi*((0.7^4)-(0.688^4)))/(16*0.7).

How to calculate Polar Modulus of Hollow Shaft?

With Outer Diameter of Shaft (d_o) & Inner Dia of Shaft (d_i) we can find Polar Modulus of Hollow Shaft using the formula - Polar Modulus = (pi*((Outer Diameter of Shaft^4)-(Inner Dia of Shaft^4)))/(16*Outer Diameter of Shaft). This formula also uses Archimedes' constant .

What are the other ways to Calculate Polar Modulus?

Here are the different ways to Calculate Polar Modulus-

Polar Modulus=Polar Moment of Inertia/Radius of Shaft
Polar Modulus=(pi*Dia of Shaft^3)/16
Polar Modulus=(Torque/Maximum Shear Stress)

Can the Polar Modulus of Hollow Shaft be negative?

No, the Polar Modulus of Hollow Shaft, measured in Volume cannot be negative.

Which unit is used to measure Polar Modulus of Hollow Shaft?

Polar Modulus of Hollow Shaft is usually measured using the Cubic Meter[m³] for Volume. Cubic Centimeter[m³], Cubic Millimeter[m³], Liter[m³] are the few other units in which Polar Modulus of Hollow Shaft can be measured.

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