Question $2.............................................................................10$ marks

Fig. $2$ shows a wooden crate of mass $\backslash (75\backslash$mathrm$\{$k$\}$ g $\backslash )$ resting on a brass trolly of weight $\backslash (17\mathrm{.}5\backslash$mathrm$\{$k$\}$ g $\backslash ),$ that are in frictional contact with each other where the crate is resting on the trolly as shown. Further, the brass trolly has wheels that are in frictional contact with the ground surface at $\backslash ($ B $\backslash )$ and $\backslash ($ A $\backslash ).$

You are told that the coefficient of static friction between the wheels and the ground surface is $\backslash (\backslash$mu$\_\{$w$\}=0\mathrm{.}35\backslash ),$ and the coefficient of static friction between the crate and the trolly is $\backslash (\backslash$mu$\_\{$t$\}=0\mathrm{.}5\backslash ).$

You are then told, that the wheels at $\backslash ($ B $\backslash )$ are free to roll, whilst the wheels at $\backslash ($ A $\backslash )$ are locked.

Now, as part of your design, you must do the following:

$($a$)$ Determine: The maximum force $\backslash (\backslash$mathbf$\{$P$\}\backslash )$ that may be applied without causing the crate to slip off the trolly.

$($b$)$ Derive: The equation of coefficient of static friction from first principles.