Pendulum Underwater ViscosityJ. Costa Leme, Escola EB/S de Lanheses, Viana do Castelo, Portugal

C. Moura, Departamento de Fsica, Universidade do Minho, Braga, Portugal

Viscosity is internal friction in a fluid. Viscous forces oppose the motion of one portion of a fluid relative to another. Viscous effects are important in the flow of fluids in pipes, the flow of blood, the lubrication of engine parts, and many other situations.

The present work, the authors intend to use experiments for the oscillations of a physical pendulum, in the form of a long and light string that carries a ball at its lower end, immersed in water to measure the water viscosity. So we are in the present of viscously damped pendulum and we use theories describing the Drag Force approaches to a viscously damped oscillating sphere.The experimental part of the present work is based on a very simple and low cost image capturing process. A standard video digital camera recorder is used video analysis software Tracker to measure amplitude as a function of time. The experiment was developed inside a PVC cylinder container.

IntroductionThe topics of oscillations and harmonic motion are of fundamental importance to physics and engineering undergraduate courses and they are covered theoretical and experimentally in most of the undergraduate courses. The oscillating water columns, swinging pendulums, masses attached to springs,,and other methods, are the most used examples. The collection and the analysis of the experimental data is normally done with the use of potentiometers2, photocells, photosensors webcam5, and force sensors3,4.In the present work, a very simple and low cost image capturing process is used to analyze the motion of a simple pendulum in water in order to measure the water viscosity. A standard video digital camera recorder is used on the video analysis software Tracker to measure the amplitude of the oscillations as a function of time. The theories describing the drag force approaches to a viscously damped oscillating sphere are used to determine the water viscosity. Theories and ModelsA typical simple pendulum consists of wire by which a massive object, like a sphere, is suspended from one end of an unstretchable massless string that is fixed at the other end, as is shown in figure 1 . The forces acting on the object are the tension force T from the string, , FD is the drag force, Fg is the gravitacional force (weight) and B is the buoyancy, for a sphere immersed in a fluid.

There are three forces acting in the tangential direction, namely, the drag force, FD, and the projections of the weight and buoyancy forces on this direction. The magnitude of the weight and the buoyancy are given by: (1)

(2)where g is the gravity acceleration, is the fluid density, is the mass and the diameter of the sphere and is the sphere volume.

The motion equation by Newtons second law gives for the tangential direction: (3)

(3)

However, if the angle is very small, is very nearly equal to in radians, Fig.1. and like we can rewriting this equation one obtains:

(4)Now let's consider the drag force models.I-Stokes lawThe drag force for a sphere that is moving in a viscous fluid depends on how large the Reynolds number (Re) is, which is given by (5)

where is sphere radius, is the fluid viscosity and is the relative speed between fluid and object. For small Reynolds numbers , the drag force is linear with speed is given by Stokes law:

(6)

Where d is diameter sphere.Replacing Eqs. (6) in Eq. (4), the motion equation can be written as:

(7)And rewriting this equation one obtains:

(8)Its motion equation can be written as

(9)Where the constants are given by

, The case of underdamped simple harmonic motion, when , is achieved when the mass oscillates in a low viscosity medium such as distilled water, the motion is described by3,5,,

(10)where A, and are the amplitude, phase and frequency, respectively, of the oscillations.

Where the frequency is given by

It was found that the Stokes law for modelling the oscillatory system has limitations, because the fit of the experimental results with eq. (10) is not the best and the obtained viscosities are much greater than the real ones. Such limitations result because its application requires not only a small Reynolds number but also a stationary motion, which is not fulfilled in the damped oscillatory case.II- LandauLifshitzLandau and Lifshitz developed the viscous dragging force theory for a spherical body in oscillatory motion12. This theory does not require a small Reynolds number (in our case we have 50