8. Static Torsion — Rigidity Modulus of a Wire by the Static Twist Method¶

Module

Elasticity of Solids · Module 2

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Aim¶

To determine the rigidity (shear) modulus of the material of a wire by measuring its static angle of twist under a known applied couple.

Theoretical Background¶

Unlike the torsional pendulum, which infers the rigidity modulus from a dynamic (oscillating) measurement, the static torsion method measures it directly and statically: one end of the wire is clamped rigidly, while the other end is attached to a horizontal disc or wheel fitted with a pulley of known radius. A cord wound around the pulley and passed over a frictionless guide carries a weight pan, so that a known torque (couple) can be applied to the wire, twisting it through a measurable angle.

For an isotropic elastic wire obeying Hooke's law in shear, the angle of twist per unit applied torque is a constant, called the torsional compliance, that depends on the wire's rigidity modulus, its length, and — very sensitively — its radius (to the fourth power), exactly as in the dynamic torsional-pendulum case, since both ultimately derive from the same relation between shear stress, shear strain and the twist gradient along the wire.

By applying a series of known torques and reading the corresponding twist angle on a graduated circular scale (using a pointer or lamp-and-scale arrangement for high precision), a straight-line graph of twist angle against applied torque can be obtained; its slope gives the torsional compliance and hence the rigidity modulus directly, without any timing measurement at all.

This static method complements the torsional-pendulum method: it avoids any assumptions about a symmetric mass distribution or oscillatory damping, but places correspondingly greater demands on precisely reading small angles.

Governing Formulas¶

Static twist under an applied couple C:

\[ \theta = \dfrac{2 C l}{\pi \eta r^{4}} \]

Rigidity modulus:

\[ \eta = \dfrac{2 C l}{\pi r^{4} \theta} \]

where the applied couple \(C = F R_{\text{arm}}\) (\(F\) = weight applied, \(R_{\text{arm}}\) = pulley radius), \(l\) the length of the wire between the clamp and the disc, and \(r\) its radius. \(\theta\) is measured in radians.

Labelled apparatus layout — original schematic prepared for this guide (not a photograph). A wire clamped rigidly at the top and attached at the bottom to a horizontal disc fitted with a pulley of known radius; a cord over the pulley carries a weight pan, and a pointer with a graduated circular scale reads the resulting twist angle.

Simulator Controls¶

Control	Symbol	Range	Unit
Applied weight	F	50 – 500	g
Pulley (arm) radius	R_arm	2 – 8	cm
Wire length	l	30 – 100	cm
Wire radius	r	0.20 – 0.80	mm

Readouts:

Quantity	Symbol	Unit
Angle of twist	θ	degrees

Procedure¶

Launch the Static Torsion panel; set the pulley radius, wire length and wire radius to their default values.
Increase the applied weight F in equal steps and record the twist angle θ shown on the readout at each step.
Plot θ against the applied couple C = F·R_arm; verify a straight line through the origin.
Unload in the same steps and check for any residual twist (permanent set) once the load is fully removed.
Calculate the rigidity modulus η from the slope of the θ vs C graph, using the known l and r.

Observation Table¶

Table: Twist angle with increasing applied couple

S. No.	F (g)	R_arm (cm)	C = F·R_arm (g·cm)	θ (°)	θ (rad)
1
2
3
4
5
6

Graph¶

Plot Twist angle, θ (y-axis) against Applied couple, C (x-axis).

Expected trend: a straight line through the origin (θ ∝ C).

θ vs C, expected to be a straight line through the origin; slope = 2l/(πηr⁴)

15×15 grid — choose your own scale per square; plot from your observation table

Calculation¶

Compute the slope of the θ (in radians) vs C graph: slope = 2l/(πηr⁴).
Rearrange to find η = 2l/(πr⁴ × slope), using the known l and r.
Check for and note any residual (non-zero) twist angle after full unloading, which would indicate the elastic limit was exceeded.

Result¶

Table: Rigidity modulus of the wire material

l (cm)	r (mm)	Slope (rad per g·cm)	η (dyne/cm²)
1

Maximum Permissible Error¶

As in the torsional pendulum, the wire radius r enters raised to the fourth power, so its measurement error (from a screw gauge) dominates the total uncertainty in η by far.

Precautions¶

Wind the cord over the pulley without slipping and ensure it leaves the pulley horizontally so the effective arm radius is exactly R_arm.
Take readings only after the twist has settled and any oscillation has died out.
Do not exceed the elastic limit of the wire — if a residual twist remains after full unloading, discard the run and repeat with smaller loads.

Self-Check Quiz¶

1. If the pulley (arm) radius R_arm is doubled while the applied weight F stays the same, the applied couple C:

A) Doubles
B) Halves
C) Stays the same
D) Quadruples

Answer: A — C = F·R_arm, so doubling the arm radius doubles the couple for the same weight.

2. The static torsion method differs from the torsional pendulum method chiefly in that it:

A) Uses a different material
B) Measures the twist directly under a steady applied torque rather than inferring the torsional constant from an oscillation period
C) Cannot determine the rigidity modulus at all
D) Requires no wire

Answer: B — Both methods find the same torsional constant C = πηr⁴/2l, but the static method measures θ directly under a known steady torque instead of timing oscillations.

3. A residual twist remaining after the load is fully removed indicates that:

A) The wire behaved perfectly elastically
B) The elastic limit of the wire was exceeded during the run
C) The pulley radius was measured incorrectly
D) Nothing is wrong; this is expected

Answer: B — A perfectly elastic wire returns exactly to zero twist on full unloading; any residual twist is evidence of permanent (plastic) deformation.

Viva-Voce Questions¶

Derive, in outline, why the twist angle is directly proportional to the applied couple for an elastic wire.
Why does the twist angle depend on the fourth power of the wire radius?
What experimental precautions ensure that the applied couple is accurately F·R_arm and nothing else?
How would you check, using only your own loading/unloading data, whether the wire remained within its elastic limit?
Compare the static torsion method with the torsional pendulum method for determining η — what are the relative advantages of each?
Why is it important for the cord to leave the pulley exactly horizontally?