Dimensional Analysis

Consider a projectile, launched with initial velocity $v_{0}$, at an angle $\theta$. To study its motion we may guess that these are the relevant quantities.

• Show that $\{\hskip 1.00006ptgt/v_{0},gx/v_{0}^{2},gy/v_{0}^{2},\theta\hskip 1.00006pt\}$ is a complete set of dimensionless products. (Hint. One way to go is to find the appropriate free variables in the linear system that arises but there is a shortcut that uses the properties of a basis.)

• These two equations of motion for…

conjectured that the infrared characteristic frequencies of a solid might be determined by the same forces between atoms as determine the solid’s ordinary elastic behavior. The relevant quantities are these.

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

The torque produced by an engine has dimensional formula $L^{2}M^{1}T^{-2}$. We may first guess that it depends on the engine’s rotation rate (with dimensional formula $L^{0}M^{0}T^{-1}$), and the volume of air displaced (with dimensional formula $L^{3}M^{0}T^{0}$).

• Try to find a complete set of dimensionless products. What goes wrong?

• Adjust the guess by adding the density of the air (with dimensional formula $L^{-3}M^{1}T^{0}$). Now find a complete set of…

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

• Find the dimensional formula for each of the four quantities.

• Show that $\{\hskip 1.00006pt\Pi_{1}=h/d,\Pi_{2}=dg/v^{2}\hskip 1.00006pt\}$ is a complete set of dimensionless products.

• Show that if $h/d$ is fixed then the propagation speed is proportional to the square root of

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

$\{\hskip 1.00006ptm_{1}^{p_{1}}\dots m_{k}^{p_{k}}\mid\text{$p_{1}$, \ldots, $p_{k}$ satisfy the system}\hskip 1.00006pt\}$

is a vector space under:

and

(assume that all var…

The advice about apples and oranges is not right. Consider the familiar equations for a circle $C=2\pi r$ and $A=\pi r^{2}$.

• Check that $C$ and $A$ have different dimensional formulas.

• Produce an equation that is not dimensionally homogeneous (i.e., it adds apples and oranges) but is nonetheless true of any circle.

• The prior item asks for an equation that is complete but not dimensionally homogeneous. Produce an equ…