Momentum- Propelling a Paramecium

[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.

 

[paranormal]

a. Newton's second law states that the net force on an object is equal to its mass multiplied by its acceleration. In the case of a paramecium feeling two forces, the equation would be F net = ma = F - F viscous, where F is the applied force and F viscous is the viscous force. This equation illustrates how the two forces act in opposite directions, with the viscous force slowing down the paramecium's motion while the applied force propels it forward.

b. If the mass of the paramecium is small enough, the "ma" term becomes negligible compared to the two forces. In this case, the equation for Newton's second law would become F net = F - F viscous. This means that the net force on the paramecium is equal to the difference between the applied force and the viscous force. As a result, the paramecium's motion would appear more fluid and effortless compared to a low or no resistance example, as it would not require a significant amount of force to maintain its constant velocity.

c. When the paramecium starts from rest and begins to move, the three terms in the full N2 equation behave in different ways. The applied force, F, initially increases as the paramecium uses its cilia to propel itself forward. The viscous force, F viscous, also increases as the paramecium moves through the fluid and experiences more resistance. However, as the paramecium reaches a constant velocity, both forces become equal and opposite, resulting in a net force of zero and a constant velocity. This can be seen in the graph of F net, where the line levels off at zero. The acceleration, a, initially increases as the paramecium gains speed, but eventually levels off when it reaches a constant velocity. This can be seen in the graph of a, which starts at a high value and then becomes zero. The velocity, v, also increases at first and then levels off at a constant value, as shown in the graph of v.