$[10$ points$]$ Problem $4$: Dimensional Analysis

It's Thanksgiving and time to cook a turkey! You went to the grocery store and bought a

turkey that weighs $7kg.$ Last year you perfected your turkey recipe, cooking the turkey

in the oven for $150$ minutes. Unfortunately, the turkey you purchased last year was only

$5kg$ in weight. You wonder, how much longer should you cook this year's turkey?

Dimensional analysis can help!

Figure $4\mathrm{.}1$: Dimensional analysis of a spherical turkey

Consider the cooking time of the turkey, $t.$ The time it takes to cook the turkey depends

on the time it takes for heat to travel from the outer most part of the turkey to the inner

most part of the turkey. Let's assume that the turkey is so plump that it can be assumed

to be a sphere of radius, $R,$ with mass, $m.$ Additionally, the heat transfer will depend on

the specific heat capacity, $c,$ and the heat conductivity, $k.$

The specific heat capacity, $c,$ measures the amount of energy required to heat a

unit mass of turkey by one unit of temperature; the SI units of $c$ are $\frac{J}{kg-K}$

The heat conductivity, $k,$ measures the rate of energy transfer per unit length per

unit of temperature change; the $SI$ units of $k$ are $\frac{W}{m-K}$

J$-$joule; W $-$ watt; $m-$ meter; kg$-$kilogram; and K$-$Kelvin.

a$)[6$ points$]$ Express the relation $t=f(R,m,c,k)$ in dimensionless form.

b$)[2$ points$]$ Which of the physical properties describing the turkey are intensive

properties, and which are extensive properties?

c$)[2$ points$]$ If the size of the turkey and the mass of the turkey are not independent,

but instead are related by $R=m^{\frac{1}{3}},$ where $$ is a proportionality constant, what

can you say about the relation between $t$ and $m?$ Estimate the amount of time it

will take you to cook this year's turkey to perfection.