Understanding Angle of Twist
When a torque is applied to a shaft, the shaft rotates by an angle called the angle of twist. For a straight, circular shaft made of a linear-elastic, homogeneous material, this angle can be found directly from the shaft's geometry and material properties, without any physical testing.
The formula
The standard torsion formula relates the angle of twist θ (in radians) to the applied torque, the shaft's geometry, and its material:
θ = T · L / (J · G)
- T — applied torque, in newton-meters (N·m)
- L — length of shaft over which the torque acts, in meters (m)
- J — polar moment of inertia of the cross-section, in m⁴; for a solid circular shaft of diameter d, J = π · d⁴ / 32
- G — shear modulus (modulus of rigidity) of the shaft material, in pascals (Pa)
The result comes out in radians; multiply by 180/π to convert to degrees, which is usually easier to picture. This calculator also reports the maximum shear stress at the shaft's outer surface, τ = T · r / J, where r = d/2 is the outer radius — a useful check that the shaft is staying within its elastic range.
Working with units
- This calculator takes torque in N·m, length in m, diameter in mm (converted internally to meters), and shear modulus in GPa (converted to pascals) via the material selector.
- Typical shear modulus values: steel ≈ 79 GPa, aluminum alloys ≈ 26 GPa, brass ≈ 40 GPa, titanium alloys ≈ 44 GPa, copper ≈ 48 GPa.
- For a hollow shaft, replace J with π · (do⁴ − di⁴) / 32 using the outer and inner diameters; this calculator assumes a solid circular cross-section.
Knowing the limits
This formula assumes the shaft is straight, has a uniform circular cross-section along its length, and stays within the linear-elastic range of the material — meaning the shear stress never exceeds the proportional limit. It does not apply to non-circular cross-sections (which warp out of plane under torsion and need a shape-specific formula), tapered shafts, or shafts loaded beyond yield.
Frequently Asked Questions
What is the formula for angle of twist?
How do I find the polar moment of inertia J?
What shear modulus should I use?
Does this formula work for non-circular shafts?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |