Polar Moment Of Inertia Calculator

Polar Moment of Inertia Formula:

\[ J = \frac{T \times L_{shaft}}{\theta \times G_{Torsion}} \]

Torque (T):

N·m

Length of Shaft (L_shaft):

Angle of Twist (θ):

rad

Modulus of Rigidity (G_Torsion):

Polar Moment of Inertia (J):

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1. What is Polar Moment of Inertia?

Polar Moment of Inertia is the moment of inertia of a cross-section with respect to its polar axis, which is an axis at right angles to the plane of the cross-section. It is a crucial property in torsion calculations for shafts and other structural elements.

2. How Does the Calculator Work?

The calculator uses the polar moment of inertia formula:

\[ J = \frac{T \times L_{shaft}}{\theta \times G_{Torsion}} \]

Where:

\( J \) — Polar Moment of Inertia (m⁴)
\( T \) — Torque (N·m)
\( L_{shaft} \) — Length of Shaft (m)
\( \theta \) — Angle of Twist (rad)
\( G_{Torsion} \) — Modulus of Rigidity (Pa)

Explanation: This formula calculates the polar moment of inertia based on the applied torque, shaft length, angle of twist, and material's modulus of rigidity.

3. Importance of Polar Moment of Inertia

Details: Polar moment of inertia is essential for designing shafts and other components subjected to torsional loads. It helps determine the stress distribution and angular deflection under applied torque.

4. Using the Calculator

Tips: Enter torque in Newton-meters, length in meters, angle of twist in radians, and modulus of rigidity in Pascals. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between polar moment of inertia and moment of inertia?
A: Moment of inertia relates to bending resistance, while polar moment of inertia relates to torsional resistance about the polar axis.

Q2: What are typical units for polar moment of inertia?
A: The SI unit is meters to the fourth power (m⁴), though mm⁴ or in⁴ are also commonly used.

Q3: How does shaft geometry affect polar moment of inertia?
A: For circular cross-sections, \( J = \frac{\pi d^4}{32} \) where d is the diameter. Different shapes have different formulas.

Q4: What is modulus of rigidity?
A: Modulus of rigidity (G) is the measure of a material's resistance to shear deformation, given by the ratio of shear stress to shear strain.

Q5: When is this formula applicable?
A: This formula applies to homogeneous, isotropic materials with linear elastic behavior under pure torsion.