Polar Moment of Inertia of Shaft given Total Strain Energy Stored in Shaft Calculator

Formula Used:

\[ J_{shaft} = \frac{U \times (2 \times G \times r_{shaft}^2)}{(\tau^2) \times L} \]

Strain Energy in Body (U):

Joule

Modulus of Rigidity of Shaft (G):

Pascal

Radius of Shaft (r_shaft):

Meter

Shear Stress on Surface of Shaft (τ):

Pascal

Length of Shaft (L):

Meter

Polar Moment of Inertia of Shaft (J_shaft):

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1. What is Polar Moment of Inertia?

The Polar Moment of Inertia of a shaft is a measure of the shaft's resistance to torsion. It represents the distribution of the shaft's cross-sectional area relative to its axis and is crucial in calculating the shaft's ability to withstand twisting forces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ J_{shaft} = \frac{U \times (2 \times G \times r_{shaft}^2)}{(\tau^2) \times L} \]

Where:

\( J_{shaft} \) — Polar Moment of Inertia of shaft (m⁴)
\( U \) — Strain Energy in body (Joule)
\( G \) — Modulus of rigidity of Shaft (Pascal)
\( r_{shaft} \) — Radius of Shaft (Meter)
\( \tau \) — Shear stress on surface of shaft (Pascal)
\( L \) — Length of Shaft (Meter)

Explanation: This formula calculates the polar moment of inertia based on the strain energy stored in the shaft and the material properties under torsional loading.

3. Importance of Polar Moment of Inertia Calculation

Details: Accurate calculation of polar moment of inertia is essential for designing shafts and rotating components to ensure they can withstand torsional stresses without excessive deformation or failure.

4. Using the Calculator

Tips: Enter all values in appropriate units. Strain energy in Joules, modulus of rigidity in Pascal, radius and length in meters, and shear stress in Pascal. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of polar moment of inertia?
A: Polar moment of inertia quantifies a shaft's resistance to torsional deformation. Higher values indicate greater resistance to twisting.

Q2: How does shaft radius affect polar moment of inertia?
A: Polar moment of inertia increases with the fourth power of the radius, making radius the most significant factor in torsional stiffness.

Q3: What is the relationship between polar moment of inertia and torsional stiffness?
A: Torsional stiffness is directly proportional to the polar moment of inertia and the modulus of rigidity of the material.

Q4: Can this formula be used for hollow shafts?
A: This specific formula is derived for solid circular shafts. For hollow shafts, different formulas accounting for inner and outer diameters should be used.

Q5: What are typical units for polar moment of inertia?
A: In the SI system, polar moment of inertia is measured in meters to the fourth power (m⁴).