# Relationship of spring constant and torsion constant?

1. May 24, 2014

1. The problem statement, all variables and given/known data

I'm trying to find a relationship between spring and torsion constants for an experiment but am not sure what it is and am struggling to find it.

2. Relevant equations

Torque = Torsion Constant * Angle of Twist

Force = Spring Constant * Extension

Shear Modulus = Force*Length/Area*Extension

3. The attempt at a solution

I have absolutely no idea.

2. May 24, 2014

### SteamKing

Staff Emeritus
The spring constant and the torsion constant measure different actions on a body. The spring constant is fundamentally related to the axial compression of the spring, while the torsion constant measures the effect of twisting on the spring.

AFAIK, the two constants are not related.

3. May 24, 2014

### unscientific

The variables involved are orthogonal to one another ($r$ vs $\phi$), so they are independent of one another.

4. May 24, 2014

### AlephZero

If you are asking about a straight wire, the two material properties (Young's modulus and shear modulus) are independent of each other. There is a formula that connects E, G, and Poisson's ratio for isotropic materials, but any two of those three constants are independent.

If "spring constant" means you are talking about a coil spring, there is a complicated relation between the extension and torsion stiffness which involves the geometry of the spring (turns per unit length, diameter of the wire, and diameter of the coils) as well as the shear modulus of the material. But most real-world springs are meant to be used either in extension or in torsion, but not both together.

5. May 26, 2014

Strange, I was seemed to be made to believe there definitely was one. Talking about a rubber band by the way.

6. Jun 5, 2014

1. The problem statement, all variables and given/known data
Does extension affect the torsion constant in any way?

Known data: Extension, Force, Time Period

2. Relevant equations

Time Period of a Torsion Pendulum = 2PI sqrt(Moment of Inertia/Torsion Constant)

3. The attempt at a solution

Having a very hard time, unsure what to do to figure this out, even with the data.

7. Feb 23, 2017

### her

what is the relationship between the spring constant and the outer diameter of the spring?

8. Feb 23, 2017

### haruspex

The terminology for springs is confusing.
An everyday helical spring under tension or compression works by torsion of the material. As you pull the ends and stretch out the spring, the wire undergoes torsion. If a spring of N coils, fully closed when relaxed, is pulled completely straight (and the ends of the wire not allowed to rotate the applied load) it will have a net strain of 2πN along its length.
To figure out the algebra, it might be easiest to start with a simple loop of wire, cut at some point. If the two ends of the cut are displaced from each other by one wire thickness they now form one loop of a fully compressed spring. Through what angle will the two cut ends have turned on their core axes?

There are also "torsion springs". In this case, a torque is applied along the main axis of the helix, tightening or expanding the spring diameter. As far as the wire is concerned, this is a bending moment, not torsion. See https://en.wikipedia.org/wiki/Torsion_spring.

A third way to use a helical spring is to bend it sideways. This again produces torsion in the wire, but unevenly.