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