Polar Moment of Inertia Calculator
Calculate polar moment of inertia for circular beams under torsion
Calculate Polar Moment of Inertia
Radius of the circular cross-section
Results
Physics Formula
Solid Circle: J = πR⁴/2
Calculation: J = π × (25.00)⁴ / 2 = 613592 mm⁴
Torsion Applications
Shear Stress: τ_max = T×R/J
Angle of Twist: φ = TL/(JG)
Where T = torque, L = length, G = shear modulus, ρ = radial distance
Engineering Analysis
Example: Steel Shaft Design
Problem Setup
Scenario: Design a solid steel shaft with 5 cm diameter
Given: D = 50 mm, R = 25 mm
Question: What is the polar moment of inertia?
Solution Steps
1. Apply solid circle formula: J = πR⁴/2
2. Substitute values: J = π × (25 mm)⁴ / 2
3. Calculate: J = π × 390,625 / 2 = 613,592 mm⁴
4. Convert to cm⁴: J = 61.36 cm⁴
Result: This shaft can handle significant torsional loads!
Engineering Examples
Steel Shaft (25mm)
Small steel transmission shaft
Hollow Pipe (60/40mm)
Steel pipe with wall thickness
Drive Shaft (2 inch)
Automotive driveshaft
Turbine Shaft (10cm)
Power generation shaft
Key Engineering Concepts
Polar Moment of Inertia
Resistance to torsional deformation
Shear Stress
Stress due to torsional loading
Angle of Twist
Angular deformation under torque
Torque
Applied twisting moment
Essential Formulas
Solid Circle
J = πR⁴/2 = πD⁴/32
For solid circular shafts
Hollow Circle
J = π(R⁴ - Rᵢ⁴)/2
For hollow circular shafts
Shear Stress
τ = Tρ/J
Maximum at outer radius
Angle of Twist
φ = TL/(JG)
Angular deformation
Understanding Polar Moment of Inertia
What is Polar Moment of Inertia?
The polar moment of inertia (J) is a geometric property that quantifies how a cross-section's area is distributed relative to an axis. It determines a shaft's resistance to torsional deformation.
Why It Matters in Engineering
Higher polar moments of inertia mean lower shear stresses and smaller angles of twist for the same applied torque, making the shaft more suitable for high-torque applications.
Shape Comparison
Solid Circle
Maximum strength for given diameter
Hollow Circle
Weight savings with reduced strength
Design Trade-off
Hollow sections optimize strength-to-weight ratio
Engineering Applications
Automotive
Driveshafts, axles, steering columns
Power Generation
Turbine shafts, generator rotors
Industrial
Machine spindles, transmission shafts
Aerospace
Control rods, helicopter rotors
Design Considerations
Material Properties
Shear modulus affects twist angle
Safety Factors
Account for dynamic loading and fatigue
Critical Speed
Avoid resonance frequencies
Manufacturing
Consider machining and assembly constraints