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The Eöt-Wash Group:
Laboratory Tests of Gravitational and sub-Gravitational Physics

How does a Torsion Balance work so well?

Torsion balances have the unique feature that they are sensitive to the angle between the forces on their test bodies. In particular, if the the forces on the test bodies are parallel there will be no torque on the balance, even if the forces have different magnitudes. (This explains why torsion balances can be used to make measurements at the part in 10¹³ level even though no part of the balance is itself good to that precision.)

To see this, consider the simple torsion pendulum, shown below, that contains 2 test bodies held by a massless frame hanging from a fine fiber.The vector connecting the test bodies is r=r₁-r₂.

The torque on the torsion pendulum produced by forces acting on two test bodies is:

T = r₁ x F₁ + r₂ x F₂

The fiber must hang along the vector F₁ + F₂.Otherwise, the net force on the pendulum could not be zero, and it could never be at rest. The component of the torque parallel to the fiber will twist the fiber. This component is proportional to

(F₁ + F₂)·T = r₁ · F₁ x (F₁ + F₂) + r₂ · F₂ x (F₁ + F₂).

(F₁ + F₂)·T = r₁ · F₁ x F₂ - r₂ · F₁ x F₂.

where we used rules about scalar triple products to rearrange the expression.

Then,

T = (r · F₁ x F₂) /(F₁ + F₂).

So, the torque on the fiber depends on the angle between the two forces. There is no torque unless there is a non-zero angle between the forces, so torsion balance experiments are null experiments that detect any non-parallelism of the forces on the test bodies.

In our experiments, we test the equivalence principle by comparing the ratio of gravitational forces on the test bodies to the ratio of inertial forces on the test bodies caused by the rotation of the earth. If the ratios aren't the same, then the inertial force and gravity are not coupling to the same thing in the test bodies. And, because the ratios of the components of one vector to those ofthe other aren't the same, there will be an angle between the forces which wewill see as a torque on the torsion fiber.

Now, in the real world there are complications. We tacitly assumed that the gravitational field was uniform. In a non-uniform field, even if the test bodies are identical, there will be an angle between the forces on the test bodies. We must make the local gravitational field free from gradients, and we must make the pendulum as insensitive to gradients as possible.

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© 1987-2009 Eöt-Wash Group. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation, DOE or NASA. Trouble? Comments? Contact cah49#at#u.washington.edu