Torsion

Unit 7 Macro Cohesion

Torsion

Juan Juan
13 min read
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Torsion is the rotational deformation that occurs when a material is twisted about its longitudinal axis by an applied torque, causing shear stress to distribute across its cross-section. It is the Macro Cohesion force in Superphysics’ Solid Relationalities — the large-scale, structure-forming action at the boundary between the aetherspace and space.

Unlike friction (linear/displacement), torsion is rotational and twist-inducing — it curls the material around its axis, creating coherent angular displacement that binds the structure into a helical configuration.

The Equation: Torsion Fundamentals

Torsional Shear Stress (Internal Distribution)

τ=TrJ \tau = \frac{T r}{J}

Where:

Symbol Meaning Unit
( \tau ) Shear stress at radius ( r ) ( \text{Pa} )
( T ) Applied torque ( \text{N} \cdot \text{m} )
( r ) Radial distance from axis ( \text{m} )
( J ) Polar moment of inertia ( \text{m}^4 )

Maximum Shear Stress (At Outer Surface)

τmax=TcJ \tau_{\text{max}} = \frac{T c}{J}

Where:

Symbol Meaning Unit
( c ) Outer radius of shaft ( \text{m} )
( \tau_{\text{max}} ) Maximum shear stress ( \text{Pa} )

Polar Moment of Inertia (Geometric Resistance)

J=π2r4 J = \frac{\pi}{2} r^4

For a solid circular shaft. For a hollow shaft:

J=π2(ro4−ri4) J = \frac{\pi}{2} (r_o^4 - r_i^4)

Where ( r_o ) is outer radius and ( r_i ) is inner radius.

Angle of Twist (Torsional Deformation)

ϕ=TLGJ \phi = \frac{T L}{G J}

Where:

Symbol Meaning Unit
( \phi ) Angle of twist ( \text{rad} )
( L ) Length of shaft ( \text{m} )
( G ) Shear modulus ( \text{Pa} )
( J ) Polar moment of inertia ( \text{m}^4 )

Torsional Strain Energy (Stored Cohesion)

U=12Tϕ=T2L2GJ U = \frac{1}{2} T \phi = \frac{T^2 L}{2 G J}

This energy is stored in the twisted material and released when torque is removed.

Torsional Rigidity (Resistance to Twist)

kt=GJL k_t = \frac{G J}{L}

Where ( k_t ) is the torsional stiffness. Higher ( k_t ) means greater resistance to angular displacement.

Key Properties of Torsion

Property Explanation
Rotational Material twists about its longitudinal axis
Helical Cross-sections rotate progressively along length
Cohesive Binds structure into a coherent twisted form
Shear-dominated Internal stresses are primarily shear
Radius-dependent Stress increases linearly with distance from axis
Length-dependent Longer shafts twist more for same torque
Material-dependent ( G ) and ( J ) determine torsional resistance

Torsional Failure Modes

Material Type Failure Mode
Ductile Shear yield; twists permanently
Brittle Fractures along 45° helical plane
Composite Delamination and fiber breakage

Torsional Pendulum (Oscillatory Cohesion)

Tperiod=2πIkt T_{\text{period}} = 2\pi \sqrt{\frac{I}{k_t}}

Where ( I ) is moment of inertia and ( k_t ) is torsional stiffness. The period depends on rotational inertia and restoring torque.

Wave Propagation (Torsional Waves)

vt=Gρ v_t = \sqrt{\frac{G}{\rho}}

Where:

Symbol Meaning Unit
( v_t ) Torsional wave velocity ( \text{m}/\text{s} )
( G ) Shear modulus ( \text{Pa} )
( \rho ) Density ( \text{kg}/\text{m}^3 )

Real-Life Examples

Example How Torsion Works
1. Driveshaft Transmits torque from engine to wheels, twisting along length
2. Torsion bar suspension Twisted metal bar stores energy, returns to shape
3. Screwdriver Twisting motion drives screw into material
4. Wringing a towel Torsion squeezes water out through twisting
5. Drilling a well Drill string twists to transmit torque to bit
6. Guitar string tuning Tuning pegs apply torsion to adjust tension
7. Torsional pendulum clock Twisted suspension regulates timekeeping
8. Bridge cables Torsional forces from wind cause twisting oscillations
9. Turbine shaft Transmits rotational power under torsional load
10. DNA supercoiling Biological torsion stores energy in twisted strands

Summary

Question Answer
What is torsion? Rotational deformation from twisting about a longitudinal axis
What law describes it? Torsion Formula: ( \tau = \frac{T r}{J} ); Angle of Twist: ( \phi = \frac{T L}{G J} )
What type of force? Macro Cohesion (rotational, Rule 2 + 4)
What drives it? Applied torque; angular displacement about axis
Real examples? Driveshafts, torsion bars, screwdrivers, wringing towels, DNA supercoiling
What’s its displacement counterpart? Friction (Macro Displacement)

Bottom line: Torsion is the rotational, helical twisting of a material about its axis — the Macro Cohesion force that pairs with friction’s linear sliding resistance in Superphysics’ Solid Relationalities.

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