Cable Rotational Torque Calculation Around Its Central Axis 

This example demonstrates how to calculate the rotational torque required to rotate a cable around its central axis using millimeters (mm) for practical applications.

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General Formula for Rotational Torque 

The torque required to rotate a cable can be calculated using the formula: 
T = (π × D⁴ × G × θ) ÷ 32 

Where: 
- T: Rotational torque (Nm) 
- D: Cable diameter (mm, converted to meters in calculations) 
- G: Shear modulus of the cable material (Pa) 
- θ: Rotation angle (radians) 

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Example Calculation 

Let's calculate the torque for a cable with the following specifications: 

- Cable Diameter (D): 50 mm (convert to meters = 0.05 m) 
- Shear Modulus (G): 2 × 10⁹ Pa (e.g., copper material) 
- Rotation Angle (θ): 0.1 radians (approximately 5.73°) 

Step-by-Step Calculation 

1. **Calculate the Polar Moment of Inertia (I)** 
  The formula for the polar moment of inertia is: 
  I = (π × D⁴) ÷ 32 

  Convert the cable diameter D = 50 mm to meters: 
  D = 50 mm = 0.05 m 

  Substituting values: 
  I = (3.1416 × 0.05⁴) ÷ 32 
  I = 3.07 × 10⁻⁷ m⁴ 

2. **Calculate the Shear Stress (τ)** 
  The shear stress is given by: 
  τ = G × θ 

  Substituting values: 
  τ = (2 × 10⁹) × 0.1 
  τ = 2 × 10⁸ Pa 

3. **Calculate the Torque (T)** 
  Finally, the torque is calculated using: 
  T = I × τ 

  Substituting values: 
  T = (3.07 × 10⁻⁷) × (2 × 10⁸) 
  T = 61.4 Nm 

Result 
The torque required to rotate this cable by 0.1 radians (5.73°) is 61.4 Nm.

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Practical Applications 

This calculation is critical for: 
- Designing cable winding systems for spools or drums. 
- Ensuring safe rotational control in automated systems. 
- Preventing mechanical stress during cable installation.