Many bacteria exhibit \"run and tumble\" motion. They swim in straight lines the

Question

Many bacteria exhibit "run and tumble" motion. They swim in straight lines then stop themselves. They then rotate ("tumble") for a period of time before going off in a straight line in a new, random direction. This allows them to randomly seek out different directions. Eventually, they run into food. We are going to simplify the situation and analyze it from a physics perspective. Let an E. Coli bacterium be a cylinder. We'll actually be looking at the motion of two of them. E.coli#1 is 1 micron in diameter and 10 microns in length. E.coli#2 has a diameter of 1 micron and is 2 microns in length. In their "running" state, they move with a speed of 30 microns per second. Their density is 1.15 g/mL (1150 kg/m^3). What is the mass of E.coli#1 and the mass of E.coli#2? What is the moment of inertia of each? Assume they are rotating about their centers of mass in a "windmill"-like rotation. When they stop and rotate, assume all of the translational energy is transformed into rotational energy. What is the angular velocity of each bacteria during the "tumble" state?

Explanation / Answer

(a) This bacteria have shape just similar to cylindrical rod therfore first we calculate ther volume
Coli 1 (V1 - volume) = Pi*D2L where D and L is the diameter and length
V1 = Pi*12*10 *10-18 = 31.41*10-18 m3
Similarly Volume of coli2 = Pi*12*2 *10-18 = 6.283*10-18 m3
Now we know that
mass = denstiy*volume
Mass of coli 1= 1150*31.41*10-18 = 36.125*10-15 kg
Mass of coil2 = 1150*6.283*10-18 = 7.225*10-15 kg
(b) Moment of inertia of rod = ML2/12
Coli 1 I1 = 36.125*10-15*(10*10-6)2 / 12 = 3.01*10-25 kg-m2
Coli 2 I2 = 7.225*10-15 *(2*10-6)2 / 12 = 2.408 *10-27 kg -m2
(c) Kinetic energy of the coli 1 = rotational energy
(1/2)mV2 = (1/2)I1W12   where V = 30 microns/sec is the linear velocity and W is the angular velocity
on solving we get
W1 =10.39 rad/s
Similarly for coli 2
(1/2)m2*V2 = (1/2)I2W22
On solving we get
W2 = 56.878 rad/s

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