Can you explain the homework problem about maximizing the mass of sand?

This was problem 6-21. The idea is to use Newton's Laws to find some equations that relate the mass of the sand to the angle at which the string is just barely not breaking (for which the tension is $T_{\rm max}$). You apply the ``force how-to'' to set up the equations, then solve for $m$ as a function of $\theta$ (with all the other quantities known, and $T=T_{\rm max}$). Then to find to find the maximum $m$ as a function of $\theta$ (and the value of $\theta=\theta_{\rm max}$ which maximizes $m(\theta)$), you differentiate $m(\theta)$ with respect to $\theta$ and set it equal to zero, i.e., $\frac{dm}{d\theta}=0$. Then you solve for $\theta_{\max}$ and plug it back in to $m(\theta)$ to find $m_{\max}$.

I hope this procedure for finding extremum values of functions is familiar to you from calculus-- it's extremely useful and we'll be using it again.