An E. coli bacterium can be modeled as a 0.50 μm diameter sphere that has the density of water. Rotating flagella propel a bacterium through 40°C water with a force of 65 fN, where 1 fN = 1femtonewton = 10^-15 N. What is the bacterium's speed in μm/s?

Verified step by step guidance

1. Model the bacterium as a sphere and calculate its volume using the formula for the volume of a sphere: $V_{\mathrm{sphere}}=\frac{4}{3}\pi r^3$, where $r$ is the radius of the bacterium.
2. Determine the mass of the bacterium using the density of water, which is approximately $1000 \mathrm{kg}/m^3$. The mass is given by $m=\rho V$, where $\rho$ is the density and $V$ is the volume.
3. Apply Stokes' law to calculate the drag force acting on the bacterium as it moves through the water. Stokes' law is given by $F$_drag = 6πηrv, where $\eta$ is the dynamic viscosity of water at 40°C (approximately $6.92\times {10}^{-} \mathrm{Pa} s$), $r$ is the radius of the bacterium, and $v$ is the speed.
4. Set the drag force equal to the propelling force exerted by the flagella, $F$_drag = 65 fN, and solve for the speed $v$. Rearrange Stokes' law to isolate $v$: $v=\frac{F}{}$_drag/(6πηr).
5. Convert the calculated speed from meters per second to micrometers per second by multiplying by $106$, since $1 \mu m=10-6 m$.

Key Concepts

Density and Volume

Density is defined as mass per unit volume and is crucial for understanding how the mass of the E. coli bacterium relates to its size. Since the bacterium is modeled as a sphere, its volume can be calculated using the formula V = (4/3)πr³, where r is the radius. Knowing the density allows us to determine the mass of the bacterium, which is essential for calculating its motion in a fluid.

Force and Motion

Newton's second law states that force equals mass times acceleration (F = ma). In this scenario, the force exerted by the rotating flagella propels the bacterium through water. By rearranging this equation, we can find the acceleration of the bacterium, which is necessary to determine its speed as it moves through the fluid medium.

Speed Calculation

Speed is defined as the distance traveled per unit of time. In this context, once we have the acceleration from the force and mass, we can calculate the speed of the bacterium. The final speed can be expressed in micrometers per second, requiring conversion from standard units, which is important for understanding the bacterium's movement in a biological context.