Model No:SPH-M005
Determination of C, couple per unit twist of suspension wire
The couple per unit twist (C), also known as the torsional rigidity of a cylindrical suspension wire, is given by the formula:
C = (π * η * r?) / (2 * l)
Where the key variables are defined as follows:
Determination of C, couple per unit twist of suspension wire.
Step 1: Consider a Cylindrical Wire: Imagine a solid wire of length 'l' and radius 'r' clamped rigidly at its upper end and twisted through an angle '?' at its lower free end.
Step 2: Shearing Strain in a Hollow Layer: Imagine splitting the cylinder into coaxial hollow cylindrical layers. For a layer of radius 'x' and thickness 'dx', the shearing strain (?) at the distance 'x' from the central axis is given by: ? = (x * ?) / l
Step 3: Shearing Stress and Restoring Force: Using the definition of the rigidity modulus [? = Shearing Stress / Shearing Strain], the shearing force (dF) acting on the cross-sectional area element (2?x * dx) of this layer is: dF = (? * x * ? / l) * 2?x * dx = (2? * ? * ? / l) * x² * dx
Step 4: Torque on the Layer: The deflecting moment (torque) of this force about the central longitudinal axis is equal to force multiplied by its distance from the axis (dF * x): d? = (2? * ? * ? / l) * x³ * dx
Step 5: Total Twisting Couple (?): To find the total twisting couple for the entire solid wire, integrate d? from the center (x = 0) to the outer boundary (x = r):
? = ? [from 0 to r] (2? * ? * ? / l) * x³ * dx = (2? * ? * ? / l) * [r? / 4] = (? * ? * ? * r?) / (2 * l)
Step 6: Couple Per Unit Twist (C): By setting a unit angle of twist (? = 1 radian), the restoring twisting couple simplifies directly to the torsional rigidity value: C = (? * ? * r?) / (2 * l)
Note on Hollow Wires: For a hollow cylindrical tube with an internal radius of r? and an external radius of r?, the formula evaluates to: C = [? * ? * (r?? - r??)] / (2 * l)
This is the benchmark laboratory instrument used for highly precise dynamic determinations. It consists of a long, hollow brass tube (the needle) suspended horizontally at its geometric center by the experimental wire. Inside this tube, two solid and two hollow cylinders of identical lengths fit cleanly end-to-end. By deliberately swapping the relative positions of the solid and hollow cylinders, the moment of inertia (I) of the system is modulated without changing the total suspended mass. The time period of oscillation (T) is accurately clocked for both distinct layouts, allowing for the computational isolation of C and ? without needing to measure the mass or inner dimensions of the main outer tube.
This instrument evaluates the torsional parameters statically without relying on periodic oscillation. The experimental wire is suspended vertically, with its lower end clamped tightly to a heavy cylinder or axle system. A heavy load is appended at the base to ensure the wire remains perfectly straight and taut. Upward and downward twisting couples are applied manually via standard weights hung across a system of tangential cords and low-friction pulleys. The resulting angle of twist (?) is monitored via a mirror-and-scale setup paired with a reading telescope or direct pointers passing over circular scales.
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