Length Of Shaft Given Polar Moment Of Inertia And Modulus Of Rigidity Calculator

Formula Used:

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

Modulus of Rigidity (C):

Angle of Twist (θ):

degrees

Polar Moment of Inertia (J_shaft):

m⁴

Torque Exerted on Wheel (τ):

N·m

Length of Shaft (L):

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1. What is the Length of Shaft Formula?

The formula calculates the length of a shaft based on its material properties and applied torque. It's derived from the torsion formula and is essential in mechanical engineering for designing shafts that can withstand specific torque loads without excessive twisting.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: The formula relates the shaft length to its material properties (modulus of rigidity), geometric properties (polar moment of inertia), and the applied torque and resulting twist angle.

3. Importance of Shaft Length Calculation

Details: Accurate shaft length calculation is crucial for designing mechanical systems where torque transmission is involved. It ensures that shafts can handle the required torque without excessive deformation or failure.

4. Using the Calculator

Tips: Enter all values in the appropriate units. Modulus of rigidity in Pascals, angle of twist in degrees, polar moment of inertia in meters to the fourth power, and torque in Newton-meters. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is Modulus of Rigidity?
A: Modulus of Rigidity (also called shear modulus) is a material property that measures its resistance to shearing deformation. It's defined as the ratio of shear stress to shear strain.

Q2: What is Polar Moment of Inertia?
A: Polar Moment of Inertia is a measure of an object's ability to resist torsion. For circular shafts, it's calculated as \( J = \frac{\pi d^4}{32} \) where d is the diameter.

Q3: How does angle of twist affect shaft design?
A: The angle of twist determines how much a shaft will rotate under applied torque. Excessive twist can lead to misalignment and vibration in mechanical systems.

Q4: What are typical values for Modulus of Rigidity?
A: For steel: ~80 GPa, for aluminum: ~26 GPa, for brass: ~40 GPa. The exact value depends on the specific alloy and treatment.

Q5: When is this formula most applicable?
A: This formula is most accurate for circular shafts made of homogeneous, isotropic materials undergoing elastic deformation within their proportional limit.