Linear Algebra/Topic: Dimensional Analysis/Solutions

SolutionsEdit

Problem 1

Consider a projectile, launched with initial velocity  , at an angle  . An investigation of this motion might start with the guess that these are the relevant quantities. (de Mestre 1990)

quantity     dimensional

  formula

horizontal position        
vertical position        
initial speed        
angle of launch        
acceleration due to gravity        
time        
  1. Show that   is a complete set of dimensionless products. (Hint. This can be done by finding the appropriate free variables in the linear system that arises, but there is a shortcut that uses the properties of a basis.)
  2. These two equations of motion for projectiles are familiar:   and  . Manipulate each to rewrite it as a relationship among the dimensionless products of the prior item.
Answer
  1. This relationship
     
    gives rise to this linear system
     
    (note that there is no restriction on  ). The natural parametrization uses the free variables to give   and  . The resulting description of the solution set
     
    gives   as a complete set of dimensionless products (recall that "complete" in this context does not mean that there are no other dimensionless products; it simply means that the set is a basis). This is, however, not the set of dimensionless products that the question asks for. There are two ways to proceed. The first is to fiddle with the choice of parameters, hoping to hit on the right set. For that, we can do the prior paragraph in reverse. Converting the given dimensionless products  ,  ,  , and   into vectors gives this description (note the ?'s where the parameters will go).
     
    The   is already in place. Examining the rows shows that we can also put in place  ,  , and  . The second way to proceed, following the hint, is to note that the given set is of size four in a four-dimensional vector space and so we need only show that it is linearly independent. That is easily done by inspection, by considering the sixth, first, second, and fourth components of the vectors.
  2. The first equation can be rewritten
     
    so that Buckingham's function is  . The second equation can be rewritten
     
    and Buckingham's function here is  .
Problem 2
Einstein (Einstein 1911) conjectured that the infrared characteristic frequencies of a solid may be determined by the same forces between atoms as determine the solid's ordanary elastic behavior. The relevant quantities are
quantity     dimensional

  formula

characteristic frequency        
compressibility        
number of atoms per cubic cm        
mass of an atom        

Show that there is one dimensionless product. Conclude that, in any complete relationship among quantities with these dimensional formulas,   is a constant times  . This conclusion played an important role in the early study of quantum phenomena.

Answer

We consider

 

which gives these relations among the powers.

 

This is the solution space (because we wish to express   as a function of the other quantities,   is taken as the parameter).

 

Thus,   is the dimensionless combination, and we have that   equals   times a constant (the function   is constant since it has no arguments).

Problem 3

The torque produced by an engine has dimensional formula  . We may first guess that it depends on the engine's rotation rate (with dimensional formula  ), and the volume of air displaced (with dimensional formula  ) (Giordano, Wells & Wilde 1987).

  1. Try to find a complete set of dimensionless products. What goes wrong?
  2. Adjust the guess by adding the density of the air (with dimensional formula  ). Now find a complete set of dimensionless products.
Answer
  1. Setting
     
    gives this
     
    which implies that  . That is, among quantities with these dimensional formulas, the only dimensionless product is the trivial one.
  2. Setting
     
    gives this.
     
    Taking   as parameter to express the torque gives this description of the solution set.
     
    Denoting the torque by  , the rotation rate by  , the volume of air by  , and the density of air by   we have that  , and so the torque is   times a constant.
Problem 4

Dominoes falling make a wave. We may conjecture that the wave speed   depends on the spacing   between the dominoes, the height   of each domino, and the acceleration due to gravity  . (Tilley)

  1. Find the dimensional formula for each of the four quantities.
  2. Show that   is a complete set of dimensionless products.
  3. Show that if   is fixed then the propagation speed is proportional to the square root of  .
Answer
  1. These are the dimensional formulas.
    quantity     dimensional

      formula

    speed of the wave        
    separation of the dominoes        
    height of the dominoes        
    acceleration due to gravity        
  2. The relationship
     
    gives this linear system.
     
    Taking   and   as parameters, the solution set is described in this way.
     
    That gives   as a complete set.
  3. Buckingham's Theorem says that  , and so, since   is a constant, if   is fixed then   is proportional to  .
Problem 5

Prove that the dimensionless products form a vector space under the   operation of multiplying two such products and the   operation of raising such the product to the power of the scalar. (The vector arrows are a precaution against confusion.) That is, prove that, for any particular homogeneous system, this set of products of powers of  , ...,  

 

is a vector space under:

 

and

 


(assume that all variables represent real numbers).

Answer

Checking the conditions in the definition of a vector space is routine.

Problem 6

The advice about apples and oranges is not right. Consider the familiar equations for a circle   and  .

  1. Check that   and   have different dimensional formulas.
  2. Produce an equation that is not dimensionally homogeneous (i.e., it adds apples and oranges) but is nonetheless true of any circle.
  3. The prior item asks for an equation that is complete but not dimensionally homogeneous. Produce an equation that is dimensionally homogeneous but not complete.

(Just because the old saying isn't strictly right, doesn't keep it from being a useful strategy. Dimensional homogeneity is often used as a check on the plausibility of equations used in models. For an argument that any complete equation can easily be made dimensionally homogeneous, see (Bridgman 1931, Chapter I, especially page 15.)

Answer
  1. The dimensional formula of the circumference is  , that is,  . The dimensional formula of the area is  .
  2. One is  .
  3. One example is this formula relating the length of arc subtended by an angle to the radius and the angle measure in radians:  . Both terms in that formula have dimensional formula  . The relationship holds for some unit systems (inches and radians, for instance) but not for all unit systems (inches and degrees, for instance).

ReferencesEdit

  • Bridgman, P. W. (1931), Dimensional Analysis, Yale University Press .
  • de Mestre, Neville (1990), The Mathematics of Projectiles in sport, Cambridge University Press .
  • Giordano, R.; Wells, M.; Wilde, C. (1987), "Dimensional Analysis", UMAP Modules (COMAP) (526) .
  • Einstein, A. (1911), Annals of Physics 35: 686 .
  • Tilley, Burt, Private Communication .