The zeroAength springgravity meter
By LUCIEN LaCOSTE Austin, Texas
took my only formal geophysics coursein 1932at the University of Texas. Arnold Romberg was the teacher. Early in the course,he explained the .theory of the horizontal pendulum and showed that theoretically it has an infinite period when its axis of rotation is vertical. It behaves like a sphere on a perfectly horizontal table - i.e., the sphere stays wherever it is put. Romberg also told us that no verticalseismographhad ever been designed with equally good characteristics. A few days later he sent his entire class to the blackboard, each with a different problem. Mine was to design a new type of vertical seismograph. I gave it my best efforts but found nothing new. However, the problem was interesting and I kept thinking about it. I am an incurable optimist, so I began looking in earnest for a theoretically infinite period in a vertical seismograph just because the horizontal pendulum had one. I started with the simple suspension shown in Figure 2 of the original article which follows this foreword. There are two torques, gravitational and spring, in this system. The condition that will produce an infinite period is that the two torques balance each other exactly for any angle 0. Since the gravitational torque varies as sin 0, the spring torque must also. However, the spring torque is the product of two variables, the pull of the spring and the lever arm. Can we expresssin 8 as a product of two trigonometric functions? Yes,sin 0 = 2 sin 8/2 cos 8/2. Next, can we show that a! = /3 = 0/2? Since we have postulated that the fixed and movable arms of the suspension be equal, Q = /3. A geometry theorem states that 0 is measured by the arc FE which means that CY, which is measured by half the same arc, equals 8/2.
An inspection of Figure 2 shows that the first condition is met, but the second condition is fulfilled only if the force exerted by the spring is proportional to its length. In other words, the initial (unstretched length) of the spring must be zero. The partly dotted straight line in Figure 1 of this foreword shows the required spring characteristic. Since the straight line goes through the origin, the force exerted by the spring is proportional to its length, and its length is zero when the spring is unstressed.Accordingly, I coined the expressionzero-length spring to describe the spring. The term zero-unstressed-lengthspring describes the spring better but is unwieldy. Figure 2 of this forewordis an exampleof a zero-lengthspring. The figure shows two views of a spiral spring. Its two ends are bent to nearly meet at the center of the spiral. All the turns of the spiral and its two endslie in a common plane when the spring is unstressed. Also, in using the spring we will stretch it only in a direction normal to the original plane of the spiral. Then the unstressedlength of the spring is zero, and the spring is a zero-length spring. mce helical springs make better tension springs than spiral springs, it is desirable to use them. Obviously, it is impossible to make a helical spring of zero physicallength becausethe turns cannot all lie in the same plane. However, it is possible (as explained later) to make helical springswhose turns pressagainst
A 11we have to do now is show that sin e/2 varies as lever arm DG and that cos 012 varies as the force exerted by the spring. Editor’ note: In 1934,Dr. Lucien J. B LaCoste s publisheda landmarkpaperon the now-jbmouszero-lengthspring. Becausethis fundamental paper is neithergenerallyavailable nor widelyread. TLE is pleased to reprint it as an appendix to Dr. LaCoste’ s article which givesthe background leading up to his invention. His comments also addressan important point that has been misunderstood(mainly due to faulty descriptionsin the literature) by many geophysicists many years; i.e., gravity meters for with a zero-length springare not inherentlyunstable.Dr. LaCoste was the...
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