Energy Needed Per Second to Sustain a Velocity through a Liquid

In summary, the conversation discusses how bacteria use swimming to seek out food, and the smallest distance they need to swim in order to outrun diffusion and reach an area with higher food concentration. The question is then posed of how many ATP molecules the bacterium must consume per second in order to travel at a certain speed, assuming all energy usage goes towards overcoming fluid drag. Various equations and variables are mentioned, including the speed of the bacterium, diffusion constant of food molecules, propulsive force, and the relationship between power, force, and velocity. The question of how to incorporate theta into the calculations is also raised.
  • #1
Oijl
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Homework Statement


Bacteria use swimming to seek out food. Imagine that the bacterium is in a region of low food concentration. For the bacterium to profit from swimming to a region with more food, it has to reach there before diffusion of food molecules makes the concentrations in the two regions the same. Here we find the smallest distance that a bacterium needs to swim so it can outrun diffusion.

(a) ...
(b) ...
(c) Estimate the number of ATP molecules the bacterium must consume (hydrolyze) per second in order to travel at this speed [speed of bacterium], assuming that all of the energy usages goes into overcoming fluid drag. The amount of energy released from one ATP molecule is approximately 20 kT. Note that the bacterial flagellar motor is actually powered by a proton gradient and this estimate focuses on the ATP equivalents associated with overcoming fluid drag.


Homework Equations


Speed of bacterium = 30µm/s
Diffusion constant of food molecule = 500µm^2 / s

Propulsive force:
Fp = 2πnLvcos(theta)sin(theta)
where n is the viscosity of water, L is the length of the flagellum with L = 10µm, theta is the angle at which a small segment of the flagellum moves with respect to the direction of motion of the bacterium, v is the speed of a section of the flagellum perpendicular to the direction of motion of the bacterium, and
v = πDf,
where D = 0.5µm and f = 100Hz

Speed of bacterium:
V = vsin(theta)cos(theta)



Also,
tan(theta) = (πD)/P, where P = 2µm.


The Attempt at a Solution



I can't quite figure what to do with sin(theta)cos(theta)...

But really, I can't figure how to move from these equations to energy/time. I thought that maybe

F = -(gradient)U

could be helpful, but when I thought of how to apply that, I wasn't sure what to take F with respect to... theta seems to be the only value to change in the Fp equation, but even if I found the potential energy, how would I move to something that is energy per time?

Thanks!
 
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  • #2
There is an equation relating power, force and velocity. It should be in any introductory physics book, in the chapter that deals with work, energy, and power.

Hope that helps. Not sure what to do about theta, without further information.
 

1. How is energy needed per second calculated?

The energy needed per second to sustain a velocity through a liquid is calculated by multiplying the force exerted on an object by its velocity. This is known as the power equation (P = F x v).

2. What is the unit of measurement for energy needed per second?

The unit of measurement for energy needed per second is watt (W). This is equivalent to one joule per second.

3. How does the density of the liquid affect the energy needed per second?

The density of the liquid does not directly affect the energy needed per second. However, it does affect the force exerted on the object, which in turn affects the energy needed per second.

4. Can the energy needed per second be negative?

No, the energy needed per second cannot be negative. This is because energy is a scalar quantity and cannot have a negative value. However, the direction of the force and velocity can affect the sign of the power.

5. Are there any other factors that can affect the energy needed per second?

Yes, besides force and velocity, other factors that can affect the energy needed per second include the shape and size of the object, the viscosity of the liquid, and any external forces acting on the object.

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