Mechanics & Wave Motion Problem Set 6

Y&F Chapter 1, section 1.10 on the scalar product.  We'll do vector products in a couple of weeks.  Y&F Chapter 6 on Work and Energy, particularly section 6.4 that discusses power.  You may want to read ahead in Chapter 7 (at least section 7.1) since most of these problems can also be done (more easily, IMHO) using potential energy.

1. Assignment 6 on our MasteringPhysics course site.
2. Y&F end of chapter problems.
   1. 1.55.
   2. 1.93.  Solve this using the dot product, not by using geometry.
   3. 6.3.
   4. 6.50.
   5. 6.63.  The results of problems 6.63 and 6.64 are actually very useful.  In Physics 17 you will encounter similar rules for putting resistors in series or in parallel.
   6. 6.64.
   7. 6.77.  This is easier to do using the potential energy of a spring.
   8. 6.82.  Solve this using energy methods.  It is possible to do it using Newton's laws, but harder.
   9. 6.102.  Simple if you think about it right, except that you have to find the non-trivial minimum of some functions.  Break out your calculus and, if you're unsure, plot the function to check you've located the minimum correctly. 
3. 
   The plot to the right (click on it to enlarge / save to disk), which was taken from Steven Vogel's book Comparative Biomechanics, shows the energy (J) expended by a person per mass (kg^-1) of the human per distance walked (m^-1) for a person walking up or down slopes of various steepness (aka grade).   
   1. Derive a (simple) formula for the energy required to push a 1 kg object 1 m up an incline at angle q, assuming the object is on perfectly frictionless wheels.  Going down the incline (i.e. with q negative), you could in principle gain energy: treat this is a negative energy cost.  This is the minimum physically possible energy (per kg per m) for perfectly efficient transport of any physical object, including a human body. 
   2. Print Vogel's plot of observed efficiencies and add a line showing the minimum energy you constructed in part 1.  You will need to extend the graph into negative energies/kg/m.  Note that the secondary axis at the top of the plot gives the slope in degrees.  Your line will always lie below Vogel's because our bodies are not perfectly efficient.  The difference between Vogel's data and your line could plausibly be called the physiological inefficiency of the human body.
      
   3. Graphically, find the part of Vogel's walking graph with the least physiological inefficiency.  Note that this close to, but not exactly the same as, the minimum of the walking cost curve.
      
      1. Describe how to locate (graphically) the point with the smallest inefficiency.
      2. What grade (in degrees) is least inefficient?
      3. What is the physiological inefficiency (in J/kg.m) on that optimum grade?  This is the premium that we pay for the privilege of having legs. 
             In general, our bodies do not operate in the most energy efficient manner, presumably because we have evolved in a high-food environment where the primary threats were things like predators, war and disease rather than starvation.