Many bacteria have flagella or cilia, tiny little waving appendages, which they

Question

Many bacteria have flagella or cilia, tiny little waving appendages, which they use to propel themselves. These were always believed to just wave around to move the bacteria, but it turns out that some of them actually act as tiny propellers. However the smallest natural propeller is part of an ATPase molecule. An ATPasc is an enzyme which either breaks down or builds up an ATP (Adenosine tri-phosphate) molecule. ATP is the energy currency of cells: energy is liberated by breaking one of the phosphate bonds, or stored by attaching a phosphate. The FI-ATPasc molecule has seven sub-units, six of which form a ring around the seventh sub-unit, as shown below. This middle piece actually spins around like the rotor of an electric motor, but it was only by attaching another molecule like a propeller blade that it was possible to observe this movement. If each rotation takes 100 ms. what is the angular velocity of the attached actin molecule? If the actin molecule is I pm long and has a mass of 2 Times 10^-22kg. what is the moment of inertia of the propeller? What is the angular momentum of the actin at this velocity? Assuming constant angular acceleration, if it takes 100 ms to perform a rotation starting from rest, what is the angular acceleration of the actin? I_rod = 1/3 ml^2.

Explanation / Answer

(a)

w = 2pi/T = 2pi/ 100* 10^-3 s = 62.8 rad/s

(b)

I = ml^2/3 = 2 * 10^-22 kg ( 1 * 10^-6)^2/3

I = 7 * 10^-35 kg m^2

(c)

L = I w = 7 * 10^-35 kg m^2 (62.8 rad/s) =4.39 * 10^-33 kg m^2/s

(d)

alpha = dw/dt = 62.8 rad/s - 0/0.100 s = 628 rad/s^2

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