Modulus of Rigidity of Shaft Given Torque Transmitted and Polar Moment of Inertia Calculator

Formula Used:

\[ C = \frac{\tau \times L}{\theta \times J_{shaft}} \]

Torque Exerted on Wheel (τ):

N·m

Length of Shaft (L):

Angle of Twist (θ):

rad

Polar Moment of Inertia (J_shaft):

m⁴

Modulus of Rigidity (C):

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1. What is Modulus of Rigidity?

Modulus of Rigidity, also known as shear modulus, is the elastic coefficient when a shear force is applied resulting in lateral deformation. It gives us a measure of how rigid a body is and its ability to resist deformation under shear stress.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ C = \frac{\tau \times L}{\theta \times J_{shaft}} \]

Where:

\( C \) — Modulus of Rigidity (Pa)
\( \tau \) — Torque Exerted on Wheel (N·m)
\( L \) — Length of Shaft (m)
\( \theta \) — Angle of Twist (rad)
\( J_{shaft} \) — Polar Moment of Inertia of shaft (m⁴)

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

3. Importance of Modulus of Rigidity Calculation

Details: Calculating the modulus of rigidity is crucial for designing mechanical components that undergo torsional stress, such as shafts, springs, and structural elements. It helps engineers determine material suitability and predict deformation under torsional loads.

4. Using the Calculator

Tips: Enter torque in Newton-meters, length in meters, angle of twist in radians, and polar moment of inertia in meters to the fourth power. All values must be positive and non-zero.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of Modulus of Rigidity?
A: Modulus of Rigidity measures a material's resistance to shearing deformation. Higher values indicate greater stiffness and resistance to shear stress.

Q2: How does Modulus of Rigidity differ from Young's Modulus?
A: Young's Modulus measures resistance to linear deformation under tensile or compressive stress, while Modulus of Rigidity measures resistance to angular deformation under shear stress.

Q3: What are typical values of Modulus of Rigidity for common materials?
A: Steel: ~80 GPa, Aluminum: ~26 GPa, Copper: ~45 GPa, Rubber: ~0.0001-0.001 GPa (highly variable).

Q4: Why is angle of twist measured in radians?
A: Radians are used because they represent the ratio of arc length to radius, making them a natural unit for angular measurements in engineering calculations.

Q5: What factors affect the accuracy of this calculation?
A: Material homogeneity, temperature effects, measurement accuracy of input parameters, and assumptions about uniform circular cross-section can affect accuracy.