Propelling a paramecium
Unicellular organisms such as bacteria and protists are small objects that live in
dense fluids. Theresistive force they feel is large and their masses are small, so
their motion looks very different from motion in a medium with little resistance.
Paramecia move by pushing their cilia (little hairs on their surface) against the fluid.
The fluid pushes back on them (according to Newton's third law), and this results in
a force that moves the paramecium forward. For simplicity, we will consider the
forces on all the cilia as a single combined force on the paramecium -- this is the
applied force, $F_{app}$, and we will assume that it is constant and in the direction that
the paramecium is moving.
A paramecium with mass $m$ is initially at rest, but then starts moving its cilia to
move forward with an acceleration $a$. What is the applied force $F_{app}$ equal to in
this case?
$F_v - ma$
$F_v + ma$
$ma - F_v$
$F_v$
$-F_v$
$ma$
The paramecium also feels a resistive viscous force from the fluid. If we
approximate the paramecium as a sphere with radius $R$, the magnitude of the
viscous force is given by the equation $F_v = 6\pi\mu Rv$. The viscosity of the fluid is $\mu$,
and $v$ is the paramecium's speed. The viscous force is opposite the direction of
motion.
Assume the paramecium is moving in the positive direction.
8
1 point
As the speed of the paramecium increases, the viscous force it experiences grows
rapidly. The paramecium quickly reaches a terminal speed, $v = v_T$. What is true
about this terminal speed? If there is more than one true statement, you must
select all of them.
The terminal speed is directly proportional to the applied force.
The terminal speed is inversely proportional to the applied force.
The terminal speed is directly proportional to the mass.
The terminal speed is inversely proportional to the mass.
The terminal speed is directly proportional to the fluid viscosity.
The terminal speed is inversely proportional to the fluid viscosity.
The terminal speed is directly proportional to the radius of the paramecium.
The terminal speed is inversely proportional to the radius of the
paramecium.
