LaCoste Gravimeter

LaCoste gravimeter is a portable instrument used to measure gravitational acceleration. It is used in oil and gas exploration. A schematic diagram of the LaCoste gravimeter is shown above. Its components are relatively simple: a rod of length l, a spherical test mass of mass m, and a spring with a spring constant k.

The gravitational acceleration is measured by the slight rotation of the rod. The entire assembly must be calibrated when in static equilibrium. For this reason, it is a relative gravitational measurement system.

One can perform a dynamics analysis to find the equations that relate the rod deflection to the gravitational acceleration felt by the test mass and to the amount of stretch the spring experiences. This can be done simply by assuming a massless rod, a massless spring, and no relative motion at the contact point C between the rod and a support mechanism attached to the vertical wall.

A more detailed - thus realistic - analysis, however requires that one pay attention to

1. Relative motion at the contact point C. This will contribute to the energy dissipation by the torque and the sliding at the contact point.

2. The mass of the spring. As the spring deforms, its centre of mass changes and thus has its own kinetic energy.

3. The mass of the rod. As the rod rotates, it has its own kinetic energy and potential energy.

Such detailed analysis will prove complicated using Newton's laws of motion since the considerations for the energy conservation and mechanical forces acting on the system become separate. One needs also to pay attention to the angular momentum of the sytem. A better way is to use Lagrangian mechanics.