Question $2$

$($a$)$ The ice in an ice rink is maintained at $\backslash (-1\mathrm{.}5^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The contact surface of an ice skate is $\backslash (0\mathrm{.}1\backslash$mathrm$\{$~mm$\}\backslash$times $175\backslash$mathrm$\{$~mm$\}\backslash ).$ Assume that, for the most part, the weight of the figure skater is transferred to the ice via one skate only. What is the minimum weight of the person required to melt the ice locally under the skate, thus enabling low frictional resistance to the motion? Comment on your answer.

You will need the following data on water to answer the question :

$[15$ marks$]$

$($b$)$ In order to save on energy costs, the management are considering extra refrigeration at the overnight electricity rates, leading to an ice temperature of $\backslash (-8^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ at $06$:$00.$ If the ice subsequently heats up at a uniform rate during the day of $\backslash (0\mathrm{.}5^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ per hour, at what time could the average child $($weighing $40$ kg $)$ and the average adult $($weighing $80$ kg $)$ start skating in optimum conditions $?$

Comment on your answer.