Angle of Twist

θ = \frac{T L}{G J}

Elastic twist of a circular shaft. Analogous to δ = PL/AE: T ↔ P, G ↔ E, J ↔ A. θ is in radians. Series shafts (same T) add twists; a larger J reduces θ.

$θ$ (rad): Angle of twist
$T$ (N·m): Torque
$L$ (m): Length
$G$ (Pa): Shear modulus
$J$ (m⁴): Polar moment of inertia

Step-by-step solved example

T = 200 N·m, L = 1.00 m, G = 80 GPa, J = 2.50×10⁻⁷ m⁴. Find θ.

θ (radians) = TL/GJ for a prismatic circular shaft.

1. GJ
GJ = (80×10⁹)(2.50×10⁻⁷) = 2.00×10⁴ N·m².
$G J = 2.00 \times 1 0^{4}$
2. Twist
θ = TL/GJ = 200×1 / 2.00×10⁴ = 0.0100 rad.
$θ = 0.0100 rad$

Answer: 0.0100 rad

10 practice questions

0/10 correct

1.θ doubles if L doubles (same T, G, J)

2.Solid-circle polar J is

3.G for steel is about

4.θ in radians times 180/π gives

5.T = 400 N·m, L = 1 m, G = 80 GPa, J = 2.5×10⁻⁷ m⁴. θ =

6.Increasing J

7.The torsion analog of PL/AE is

8.Two circular shafts in series (single torque path)

9.θ = 0.040 rad over L = 2.0 m. Unit twist θ/L =

10.Polar J of a solid 20 mm shaft is