Linear Algebra/Topic: Dimensional Analysis/Solutions

Problem 1

Consider a projectile, launched with initial velocity ${\displaystyle v_{0}}$, at an angle ${\displaystyle \theta }$. An investigation of this motion might start with the guess that these are the relevant quantities. (de Mestre 1990)

quantity	dimensional   formula
horizontal position ${\displaystyle x}$	${\displaystyle L^{1}M^{0}T^{0}}$
vertical position ${\displaystyle y}$	${\displaystyle L^{1}M^{0}T^{0}}$
initial speed ${\displaystyle v_{0}}$	${\displaystyle L^{1}M^{0}T^{-1}}$
angle of launch ${\displaystyle \theta }$	${\displaystyle L^{0}M^{0}T^{0}}$
acceleration due to gravity ${\displaystyle g}$	${\displaystyle L^{1}M^{0}T^{-2}}$
time ${\displaystyle t}$	${\displaystyle L^{0}M^{0}T^{1}}$

1. Show that ${\displaystyle \{gt/v_{0},gx/v_{0}^{2},gy/v_{0}^{2},\theta \}}$ 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: ${\displaystyle x=v_{0}\cos(\theta )t}$ and ${\displaystyle y=v_{0}\sin(\theta )t-(g/2)t^{2}}$. Manipulate each to rewrite it as a relationship among the dimensionless products of the prior item.

Answer

1. This relationship

   ${\displaystyle (L^{1}M^{0}T^{0})^{p_{1}}(L^{1}M^{0}T^{0})^{p_{2}}(L^{1}M^{0}T^{-1})^{p_{3}}(L^{0}M^{0}T^{0})^{p_{4}}(L^{1}M^{0}T^{-2})^{p_{5}}(L^{0}M^{0}T^{1})^{p_{6}}=L^{0}M^{0}T^{0}}$

   gives rise to this linear system

   ${\displaystyle {\begin{array}{*{6}{rc}r}p_{1}&+&p_{2}&+&p_{3}&\ &\ &+&p_{5}&&&=&0\\&&&&&&&&&&0&=&0\\&&&&-p_{3}&&&-&2p_{5}&+&p_{6}&=&0\end{array}}}$

   (note that there is no restriction on ${\displaystyle p_{4}}$). The natural parametrization uses the free variables to give ${\displaystyle p_{3}=-2p_{5}+p_{6}}$ and ${\displaystyle p_{1}=-p_{2}+p_{5}-p_{6}}$. The resulting description of the solution set

   ${\displaystyle \{{\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\\p_{4}\\p_{5}\\p_{6}\end{pmatrix}}=p_{2}{\begin{pmatrix}-1\\1\\0\\0\\0\\0\end{pmatrix}}+p_{4}{\begin{pmatrix}0\\0\\0\\1\\0\\0\end{pmatrix}}+p_{5}{\begin{pmatrix}1\\0\\-2\\0\\1\\0\end{pmatrix}}+p_{6}{\begin{pmatrix}-1\\0\\1\\0\\0\\1\end{pmatrix}}\,{\big |}\,p_{2},p_{4},p_{5},p_{6}\in \mathbb {R} \}}$

   gives ${\displaystyle \{y/x,\theta ,xt/{v_{0}}^{2},v_{0}t/x\}}$ 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 ${\displaystyle gt/v_{0}}$, ${\displaystyle gx/v_{0}^{2}}$, ${\displaystyle gy/v_{0}^{2}}$, and ${\displaystyle \theta }$ into vectors gives this description (note the ?'s where the parameters will go).

   ${\displaystyle \{{\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\\p_{4}\\p_{5}\\p_{6}\end{pmatrix}}={\underline {\ {\textit {?}}\ }}{\begin{pmatrix}0\\0\\-1\\0\\1\\1\end{pmatrix}}+{\underline {\ {\textit {?}}\ }}{\begin{pmatrix}1\\0\\-2\\0\\1\\0\end{pmatrix}}+{\underline {\ {\textit {?}}\ }}{\begin{pmatrix}0\\1\\-2\\0\\1\\0\end{pmatrix}}+p_{4}{\begin{pmatrix}0\\0\\0\\1\\0\\0\end{pmatrix}}\,{\big |}\,p_{2},p_{4},p_{5},p_{6}\in \mathbb {R} \}}$

   The ${\displaystyle p_{4}}$ is already in place. Examining the rows shows that we can also put in place ${\displaystyle p_{6}}$, ${\displaystyle p_{1}}$, and ${\displaystyle p_{2}}$. 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

   ${\displaystyle {\frac {gx}{{v_{0}}^{2}}}={\frac {gt}{v_{0}}}\cos \theta }$

   so that Buckingham's function is ${\displaystyle f_{1}(\Pi _{1},\Pi _{2},\Pi _{3},\Pi _{4})=\Pi _{2}-\Pi _{1}\cos(\Pi _{4})}$. The second equation can be rewritten

   ${\displaystyle {\frac {gy}{{v_{0}}^{2}}}={\frac {gt}{v_{0}}}\sin \theta -{\frac {1}{2}}\left({\frac {gt}{v_{0}}}\right)^{2}}$

   and Buckingham's function here is ${\displaystyle f_{2}(\Pi _{1},\Pi _{2},\Pi _{3},\Pi _{4})=\Pi _{3}-\Pi _{1}\sin(\Pi _{4})+(1/2){\Pi _{1}}^{2}}$.

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 ${\displaystyle \nu }$	${\displaystyle L^{0}M^{0}T^{-1}}$
compressibility ${\displaystyle k}$	${\displaystyle L^{1}M^{-1}T^{2}}$
number of atoms per cubic cm ${\displaystyle N}$	${\displaystyle L^{-3}M^{0}T^{0}}$
mass of an atom ${\displaystyle m}$	${\displaystyle L^{0}M^{1}T^{0}}$

Show that there is one dimensionless product. Conclude that, in any complete relationship among quantities with these dimensional formulas, ${\displaystyle k}$ is a constant times ${\displaystyle \nu ^{-2}N^{-1/3}m^{-1}}$. This conclusion played an important role in the early study of quantum phenomena.

Answer

We consider

${\displaystyle (L^{0}M^{0}T^{-1})^{p_{1}}(L^{1}M^{-1}T^{2})^{p_{2}}(L^{-3}M^{0}T^{0})^{p_{3}}(L^{0}M^{1}T^{0})^{p_{4}}=(L^{0}M^{0}T^{0})}$

which gives these relations among the powers.

${\displaystyle {\begin{array}{*{4}{rc}r}&&p_{2}&-&3p_{3}&&&=&0\\&&-p_{2}&&&+&p_{4}&=&0\\-p_{1}&+&2p_{2}&&&&&=&0\end{array}}\;{\xrightarrow[{}]{\rho _{1}\leftrightarrow \rho _{3}}}\;{\xrightarrow[{}]{\rho _{2}+\rho _{3}}}\;{\begin{array}{*{4}{rc}r}-p_{1}&+&2p_{2}&&&&&=&0\\&&-p_{2}&&&+&p_{4}&=&0\\&&&\ &-3p_{3}&+&p_{4}&=&0\end{array}}}$

This is the solution space (because we wish to express ${\displaystyle k}$ as a function of the other quantities, ${\displaystyle p_{2}}$ is taken as the parameter).

${\displaystyle \{{\begin{pmatrix}2\\1\\1/3\\1\end{pmatrix}}p_{2}\,{\big |}\,p_{2}\in \mathbb {R} \}}$

Thus, ${\displaystyle \Pi _{1}=\nu ^{2}kN^{1/3}m}$ is the dimensionless combination, and we have that ${\displaystyle k}$ equals ${\displaystyle \nu ^{-2}N^{-1/3}m^{-1}}$ times a constant (the function ${\displaystyle {\hat {f}}}$ is constant since it has no arguments).

Problem 3

The torque produced by an engine has dimensional formula ${\displaystyle L^{2}M^{1}T^{-2}}$. We may first guess that it depends on the engine's rotation rate (with dimensional formula ${\displaystyle L^{0}M^{0}T^{-1}}$), and the volume of air displaced (with dimensional formula ${\displaystyle L^{3}M^{0}T^{0}}$) (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 ${\displaystyle L^{-3}M^{1}T^{0}}$). Now find a complete set of dimensionless products.

Answer

1. Setting

   ${\displaystyle (L^{2}M^{1}T^{-2})^{p_{1}}(L^{0}M^{0}T^{-1})^{p_{2}}(L^{3}M^{0}T^{0})^{p_{3}}=(L^{0}M^{0}T^{0})}$

   gives this

   ${\displaystyle {\begin{array}{*{3}{rc}r}2p_{1}&&&+&3p_{3}&=&0\\p_{1}&&&&&=&0\\-2p_{1}&-&p_{2}&&&=&0\end{array}}}$

   which implies that ${\displaystyle p_{1}=p_{2}=p_{3}=0}$. That is, among quantities with these dimensional formulas, the only dimensionless product is the trivial one.
2. Setting

   ${\displaystyle (L^{2}M^{1}T^{-2})^{p_{1}}(L^{0}M^{0}T^{-1})^{p_{2}}(L^{3}M^{0}T^{0})^{p_{3}}(L^{-3}M^{1}T^{0})^{p_{4}}=(L^{0}M^{0}T^{0})}$

   gives this.

   ${\displaystyle {\begin{array}{*{4}{rc}r}2p_{1}&&&+&3p_{3}&-&3p_{4}&=&0\\p_{1}&&&&&+&p_{4}&=&0\\-2p_{1}&-&p_{2}&&&&&=&0\end{array}}\;{\xrightarrow[{\rho _{1}+\rho _{3}}]{(-1/2)\rho _{1}+\rho _{2}}}\;{\xrightarrow[{}]{\rho _{2}\leftrightarrow \rho _{3}}}{\begin{array}{*{4}{rc}r}2p_{1}&&&+&3p_{3}&-&3p_{4}&=&0\\&&-p_{2}&+&3p_{3}&-&3p_{4}&=&0\\&\ &&&(-3/2)p_{3}&+&(5/2)p_{4}&=&0\end{array}}}$

   Taking ${\displaystyle p_{1}}$ as parameter to express the torque gives this description of the solution set.

   ${\displaystyle \{{\begin{pmatrix}1\\-2\\-5/3\\-1\end{pmatrix}}p_{1}\,{\big |}\,p_{1}\in \mathbb {R} \}}$

   Denoting the torque by ${\displaystyle \tau }$, the rotation rate by ${\displaystyle r}$, the volume of air by ${\displaystyle V}$, and the density of air by ${\displaystyle d}$ we have that ${\displaystyle \Pi _{1}=\tau r^{-2}V^{-5/3}d^{-1}}$, and so the torque is ${\displaystyle r^{2}V^{5/3}d}$ times a constant.

Problem 4

Dominoes falling make a wave. We may conjecture that the wave speed ${\displaystyle v}$ depends on the spacing ${\displaystyle d}$ between the dominoes, the height ${\displaystyle h}$ of each domino, and the acceleration due to gravity ${\displaystyle g}$. (Tilley)

1. Find the dimensional formula for each of the four quantities.
2. Show that ${\displaystyle \{\Pi _{1}=h/d,\Pi _{2}=dg/v^{2}\}}$ is a complete set of dimensionless products.
3. Show that if ${\displaystyle h/d}$ is fixed then the propagation speed is proportional to the square root of ${\displaystyle d}$.

Answer

1. These are the dimensional formulas.

   quantity	dimensional   formula
   speed of the wave ${\displaystyle v}$	${\displaystyle L^{1}M^{0}T^{-1}}$
   separation of the dominoes ${\displaystyle d}$	${\displaystyle L^{1}M^{0}T^{0}}$
   height of the dominoes ${\displaystyle h}$	${\displaystyle L^{1}M^{0}T^{0}}$
   acceleration due to gravity ${\displaystyle g}$	${\displaystyle L^{1}M^{0}T^{-2}}$

2. The relationship

   ${\displaystyle (L^{1}M^{0}T^{-1})^{p_{1}}(L^{1}M^{0}T^{0})^{p_{2}}(L^{1}M^{0}T^{0})^{p_{3}}(L^{1}M^{0}T^{-2})^{p_{4}}=(L^{0}M^{0}T^{0})}$

   gives this linear system.

   ${\displaystyle {\begin{array}{*{4}{rc}r}p_{1}&+&p_{2}&+&p_{3}&+&p_{4}&=&0\\&&&&&&0&=&0\\-p_{1}&&&&&-&2p_{4}&=&0\end{array}}\;{\xrightarrow[{}]{\rho _{1}+\rho _{4}}}\;{\begin{array}{*{4}{rc}r}p_{1}&+&p_{2}&+&p_{3}&+&p_{4}&=&0\\&&p_{2}&+&p_{3}&-&p_{4}&=&0\end{array}}}$

   Taking ${\displaystyle p_{3}}$ and ${\displaystyle p_{4}}$ as parameters, the solution set is described in this way.

   ${\displaystyle \{{\begin{pmatrix}0\\-1\\1\\0\end{pmatrix}}p_{3}+{\begin{pmatrix}-2\\1\\0\\1\end{pmatrix}}p_{4}\,{\big |}\,p_{3},p_{4}\in \mathbb {R} \}}$

   That gives ${\displaystyle \{\Pi _{1}=h/d,\Pi _{2}=dg/v^{2}\}}$ as a complete set.
3. Buckingham's Theorem says that ${\displaystyle v^{2}=dg\cdot {\hat {f}}(h/d)}$, and so, since ${\displaystyle g}$ is a constant, if ${\displaystyle h/d}$ is fixed then ${\displaystyle v}$ is proportional to ${\displaystyle {\sqrt {d\,}}}$.

Problem 5

Prove that the dimensionless products form a vector space under the ${\displaystyle {\vec {+}}}$ operation of multiplying two such products and the ${\displaystyle {\vec {\cdot }}}$ 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 ${\displaystyle m_{1}}$, ..., ${\displaystyle m_{k}}$

${\displaystyle \{m_{1}^{p_{1}}\dots m_{k}^{p_{k}}\,{\big |}\,p_{1},\ldots ,p_{k}{\text{ satisfy the system}}\}}$

is a vector space under:

${\displaystyle m_{1}^{p_{1}}\dots m_{k}^{p_{k}}{\vec {+}}m_{1}^{q_{1}}\dots m_{k}^{q_{k}}=m_{1}^{p_{1}+q_{1}}\dots m_{k}^{p_{k}+q_{k}}}$

and

${\displaystyle r{\vec {\cdot }}(m_{1}^{p_{1}}\dots m_{k}^{p_{k}})=m_{1}^{rp_{1}}\dots m_{k}^{rp_{k}}}$

(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 ${\displaystyle C=2\pi r}$ and ${\displaystyle A=\pi r^{2}}$.

1. Check that ${\displaystyle C}$ and ${\displaystyle A}$ 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 ${\displaystyle L}$, that is, ${\displaystyle L^{1}M^{0}T^{0}}$. The dimensional formula of the area is ${\displaystyle L^{2}}$.
2. One is ${\displaystyle C+A=2\pi r+\pi r^{2}}$.
3. One example is this formula relating the length of arc subtended by an angle to the radius and the angle measure in radians: ${\displaystyle \ell -r\theta =0}$. Both terms in that formula have dimensional formula ${\displaystyle L^{1}}$. The relationship holds for some unit systems (inches and radians, for instance) but not for all unit systems (inches and degrees, for instance).