Reading

Y&F Chapter 9.  I won't make you do moment of inertia integrals, so you can skip section 9.6.

Questions

Due in class on Wednesday, April 14.
  1. Assignment 9 on our MasteringPhysics course site.
  2. Y&F end of chapter problems. 
    1. 9.46.  This is a rotational version of problem 7.75.
    2. 9.47
    3. 9.86.  Note that the pulley is not massless.
    4. 9.89
    5. 9.98.  (This is a hard-core physics problem; typically these are completely confusing until you figure out how to do them, at which point they're completely obvious.)  Hint:  write the kinetic energy of the neutron star in terms of its period T, then calculate the power by carefully taking a time derivative (remembering that everything except T is constant) using the chain rule; don't forget that you'll generate a term dT/dt when you do this.  As a spot check: part (a) should yield I = 1.09 x 1038 kg m2.  
  3. (This is a hard-core biophysics problem.)  A car generates energy in spurts every time a piston fires; that energy is stored in a rotating flywheel.  A swimming bacterium also generates energy in spurts (though the mechanism is not completely understood), but cannot store the energy in any flywheel-like structure.  To see why not, evaluate the kinetic energy stored in a rotating bacterial flagellum:
    1. A bacterial flagellum contains about 40,000 flagellin proteins; each flagellin has a mass of 60 kDa.  (The Dalton is a crazy biologist's unit of mass equal to 1.6 x 10-27 kg).  Find the total mass of a flagellum.
    2. The flagellum is a helix of diameter 1 µm; the moment of inertia of a helix (for rotation about its centerline) is the same as the moment of inertia of a hoop with the same mass and radius.  (To see why, imagine looking straight down a helix as it's rotating).  Find the moment of inertia of a flagellum.
    3. A swimming E. coli rotates its flagellum at around 100 Hz.  Find the total kinetic energy stored in a rotating flagellum.
    4. You answer to part 3 is a very small number, but of course bacteria are themselves very small.  So is it small or large for a bacterium?  To answer this, calculate the energy released from hydrolysis of a single molecule of ATP (the typical energy currency of a cell), given that hydrolyzing 1 mole of ATP yields about 60 kJ of energy.  What fraction of an ATP energy is stored in the rotating flagellum?  If you've worked the numbers correctly you'll see why we typically ignore the kinetic energy of the flagellum when we're calculating the energy balance in a swimming cell.

The beauty of physics is that its concepts (here, the kinetic energy associated associated with a rotating object) apply over such a huge range: between the last two problems the moment of inertia changes by about 70 orders of magnitude, but the physics is still the same!