Torsion balance is shown in Figure 152. W is a calibrated torsion wire of very small diameter, terminating in a support carrying a small mirror and an aluminum bar of negligible mass. A small weight is fastened to one end of the bar, and a similar weight is suspended from a fine wire on the other end. The angular deflection of the suspended system is measured with the aid of a telescope by observing the shift of a scale image reflected by the mirror.
The torque T required to rotate the swinging system (balance beam, weights, and mirror) through an angular deflection 0, with a torsion constant of r, is 7=-8.
The working equation of the torsion balance will be determined therefore by an expression for T in terms of the parameters measured at a particular station and the instrumental constants of the balance.
The gravity force which turns the beam against the torsional resistance of the wire may be divided into two parts, viz.:
- The force arising from the curvature of the equipotential surface passing through the center of gravity of the balance beam, called the curvature quantity, and
- The force arising from the convergence of the equipotential surfaces passing through the hanging weight mass and through the mass on the end of the beam.
- This is called the gradient of gravity.
Curvature of Lines of Force of Gravity.
Consider the direction of the lines of force of gravity for a restricted portion of the earth’s surface where there is no discontinuity in density in the subsurface and where the beds making up the geological section are fiat.
In such an ideal case, the direction and intensity of the lines of gravitational force would be the same at each of a series of closely spaced points across such an area.
The lines of force would be parallel to each other and their direction would be shown by the direction of a plumb line. (See Figure 153a.) No forces would act to produce a twist on a torsion balance placed in such a force.
GRAVITATIONAL METHODS
As indicated in Figure 153b. An equipotential surface which would lie at right angles to the gravity force lines would be flat and horizontal.
If a mass of high density, such as a granite ridge, is introduced into the gravity force field pictured in Figure 153, an anomaly is created and the direction of the lines of force of gravity will be deflected very slightly toward this heavier mass. This will cause a curvature or distortion in the lines of force of gravity, as shown in Figure 154.
A plumb line would show this deflection if we were able to measure the very minute change in direction thus caused.
An equipotential surface which is everywhere at right angles to the direction of gravity force would be arched up above the heavy subsurface mass.
These conditions are illustrated again in the three parts of Figure 155 which show
(a) the lines of force for an area where uniform conditions prevail;
(b) the lines of force which would exist if the mass only were effective
(c) the resultant or deflected force lines, as caused by the combined attraction of the heavier mass and the center of gravity of the earth.
EXPLORATION GEOPHYSICS
A torsion balance placed in a distorted force field will be deflected and twisted by the gravity forces acting on it, as illustrated in Figure 156. It will be understood that the curvatures of forces and equipotential surfaces are greatly exaggerated in the illustrations. The actual curvatures are extremely small, on the order of 10-12 radians per centimeter.
A mass of low density below the surface, such as a salt dome, would create a dispersion of the force field or a warping in the direction of the lines of force of gravity as illustrated in Figure 157. Such a low density arca produces a sag in the equipotential surfaces above it.
A plumb line at a station over a salt dome would be deflected toward the more dense sediments surrounding the dome.
Gravitational Potential and Equipotential Surfaces
To understand the forces acting on a torsion balance, it is necessary to examine the nature of the potential function of the gravity field and, in more detail, the concept of equipotential surfaces of gravity.
If we consider the space around a sphere of mass a mathematical function can be set up, called the potential function, which expresses the gravity potential for every point in this space.
Referring to Figure 158, the force on a unit particle at Pa, due to the spherical mass, is P = -G na, where the minus sign denotes that the force is in the negative direction, or is toward m.
The difference of potential between two points in a field is defined as the negative of the work done by the field in carrying the particle between the points.
Physically, this means that a particle in a force field (force of gravity, in this case) will move from a point of higher to a point of lower potential.