Angle of Twist for Shaft Given Polar Moment of Inertia and Modulus of Rigidity Calculator

Angle of Twist Formula:

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

Torque Exerted on Wheel (τ):

N·m

Length of Shaft (L):

Modulus of Rigidity (C):

Polar Moment of Inertia (J):

m⁴

Angle of Twist (θ):

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1. What is the Angle of Twist Formula?

The angle of twist formula calculates the angular displacement of a shaft when subjected to torque. It provides a measure of how much a shaft twists under applied torsional load, which is crucial for designing mechanical systems and ensuring structural integrity.

2. How Does the Calculator Work?

The calculator uses the angle of twist formula:

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

Where:

\( \theta \) — Angle of twist (radians)
\( \tau \) — Torque exerted on wheel (N·m)
\( L \) — Length of shaft (m)
\( C \) — Modulus of rigidity (Pa)
\( J_{shaft} \) — Polar moment of inertia of shaft (m⁴)

Explanation: The formula calculates the angle through which the fixed end of a shaft rotates with respect to the free end when torque is applied.

3. Importance of Angle of Twist Calculation

Details: Accurate angle of twist calculation is essential for designing shafts in mechanical systems, ensuring they can withstand torsional loads without excessive deformation, and preventing failure in rotating machinery.

4. Using the Calculator

Tips: Enter torque in N·m, length in meters, modulus of rigidity in Pa, and polar moment of inertia in m⁴. All values must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of modulus of rigidity?
A: Modulus of rigidity measures a material's resistance to deformation under shear stress. Higher values indicate stiffer materials that twist less under the same torque.

Q2: How does shaft length affect the angle of twist?
A: Longer shafts experience greater angle of twist for the same applied torque, as twist is directly proportional to length.

Q3: What is polar moment of inertia?
A: Polar moment of inertia measures a shaft's resistance to torsional deformation. It depends on the cross-sectional shape and size of the shaft.

Q4: When is this formula applicable?
A: This formula applies to circular shafts with uniform cross-section under pure torsion within the elastic limit of the material.

Q5: What are typical units for angle of twist?
A: Angle of twist is typically measured in radians, but can be converted to degrees by multiplying by 180/π.