An E. coli bacterium can be modeled as a 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 micrometers/s?

Video transcript

Hi everyone in this practice problem, we're being asked to calculate a living organism's terminal speed. We will have a microscopical living organism with a radius of one micrometer moving in blood. So the living organism will produce a horizontal force of 1.5 times 10 to the power of negative 12 Newton. And at 37 degrees Celsius, the viscosity of blood is going to be three multiplied by 10 to the power of negative three pascal seconds. We're being asked to determine the living organisms terminal speed. And the options given are a 1.5 micrometer per seconds. B 4.5 micrometer per second, C 27 micrometers per second. And lastly, D 162 micrometer per second. So what we want to actually notice is that the size of our living organism that we're talking about in this practice problem is going to be very, very small or micro metric. We are expecting a linear force because of this. And due to the really low Reynolds number, because our flow is then going to actually be a linear or laminar flow. So Reynolds number is given by our E equals to row multiplied by V multiplied by L all of that divided by mu. So row is the density of the fluid or the blood V is the velocity of the moving object. Or in this case, our living organism, L is going to be a measure of the object or the living organism's radius, which in this case is going to be very, very small because the radius is given to be one micrometer. So L will actually equals to 10 to the power of negative six m. And our mule is going to be the viscosity of the blood, which is going to be three times 10 to the power of negative three pascal seconds. So um without actually calculating the Reynolds number and knowing this two parameter, we can determine that the Loreal number is going to be very, very low and smaller than one because of this, we can actually use or we will consider the motion along the horizontal positive axis to be a linear. So the linear drag force F D is going to be given by the equation of six pi mu R V X where in this case, six pi is a constant mu is the phys of our fluid or the blood R is the radius of our living organism. And V X is the velocity of our living organism. When the living organism reaches its terminal speed, our um acceleration and the X direction is going to equals to zero m. Per second squared. So at V T A X will equals to zero m per second squared. And the way we wanna calculate our terminal speed is by applying Newton Second law. So according to Newton second law, in the X direction, our sigma FX will equals to m multiplied by A X. In this case, the sigma FX is going to be the force of the living organism that is being produced by the living organism itself minus F D, which is going to actually be acting upon the opposite direction of the F of the living organism. So then our Sigma FX is going to be a living organism or F live orc minus F D equals to M multiplied by A X. We gonna substitute our F D equation that we have recall for linear drag force so that our Newton Second law equation will then be f living organism minus six P U R V X equals to M multiplied by A X. What we wanna do next is to employ our condition at terminal velocity into our Newton Second Law equation by employing V X equals to P T and A X equals to zero m per second squared. To then get the equation of F living organism minus six P U R P equals to zero. We can then rearrange this so that we get an equation for P T which is the terminal velocity that we are interested at will then equals to F living or divided by six P U R. And that will then be the equation that we're going to use to calculate our final terminal velocity. So in this case, we can then substitute all of our known information that is given in the problem statement to finally calculate the terminal velocity. So F living orc is given in the problem statement to be 51.5 times 10 to the power of negative 12 Newton. So I'm going to input that 1.5 times 10 to the power of negative 12 Newton. And then we wanna divide that with six pi multiplied by mu which is going to be three times 10 to the power of negative three pascal seconds. And multiply that with our R which is going to be the radius, which is one micrometer which is just 10 to the power of negative six m. And calculating this, we will then get our terminal velocity P T to be 2. times 10 to the power of negative five m per second or putting it in micrometers per seconds. That will correspond to 27 micrometer per seconds. So 27 micrometer per seconds is then going to be the terminal speed or the terminal velocity of our living organism. And that will correspond to option C in our answer choices. So option C is going to be the answer to distract this problem and that will be it for this video if you guys still have any sort of confusion, please make sure to check out our other lesson videos on similar topics and that will be it for this one. Thank you.