How Does a Paramecium Move Through Viscous Fluid?

[jmwood15]

Unicellular organisms such as bacteria and protists are small objects that live in dense fluids. As a result, the resistive force they feel is large and viscous. Since their masses are small their motion looks very different from motion in a medium with little resistance. Paramecia move by pushing their cilia (little hairs on their surface) through the fluid. The fluid (of course) pushes back on them. We will call this back force of the fluid on the cilia of the paramecium "the applied force", F (since it wouldn't happen if the paramecium didn't try to move its cilia).

a. Write Newton's second law for a paramecium feeling two forces: the applied force and the viscous force. (Recall that the viscous force takes the form F viscous = - η v, proportional to the velocity and in the opposite direction.)


b. If the mass is small enough, for most of the time the term "ma" can be much smaller than the two forces, which are large and nearly cancel. Write what the equation for Newton's second law (N2) turns into if we ignore the "ma" term. Describe what the motion would be like and how it would appear different from a low or no resistance example.


c. Suppose the paramecium is starting from rest and starts to move, coming quickly to a constant velocity. Describe how the three terms in the full N2 equation behave, illustrating your discussion with graphs of x, v, a, F net, F, and Fviscous.

 

[fzero]

Either show some of your own work or clearly point out what it is that you don't understand. If you don't want to put any effort into solving the problem, no one else will either.

 

[jmwood15]

i don't understand how the viscous and applied force are acting together to create the net force. I believe that some form of a momentum equation has to be used for part B

 

[fzero]

For part a you should draw the force diagram for the paramecium. Using the correct directions of the forces with respect to the velocity, you can then write down Newton's 2nd law with the correct signs. For part b, you will need to use the relationship

\vec{v}(t) = \frac{ d\vec{x}(t)}{dt}

between velocity and displacement. The displacement \vec{x}(t) describes the motion.