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Torsion of a cylinder
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Twisting a wire needs torque proportional to angle.
A twisted cylinder develops a restoring torque τ = (πGr⁴/2L)θ. Hollow shafts are almost as strong as solid ones.
Drive shafts in cars are tubes.
Torque scales with r⁴.
In more detail
When a cylinder is twisted, shear stress varies with radius and is greatest at the surface. The torque is proportional to the angle of twist and the shear modulus times the fourth power of the radius.
Real-world uses
- Drive shafts in cars transmit torque by torsion.
- Torsion springs store energy in devices such as clocks and garage doors.
- A torsion pendulum or a drive shaft twists under torque, and the angle of twist depends on the material and the shaft's dimensions.
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Frequently asked questions
Why are hollow shafts used?
They give high strength against torsion with less weight, since material near the axis contributes little.
Does radius matter?
Yes. Torsional stiffness increases with the fourth power of radius.
Why do thin wires twist more easily?
The twist angle grows with length and falls with the fourth power of radius, so thin, long wires twist much more.
Related topics in Fluids & Elasticity
Poisson's ratio · Elastic potential energy density · Bending of beams