Question Number 102241 by ajfour last updated on 07/Jul/20
Commented by ajfour last updated on 07/Jul/20
$${If}\:{the}\:{solid}\:{cylinder}\:{and}\:{sphere}\:{have} \\ $$$${the}\:{same}\:{density}\:,\:{find}\:{maximum} \\ $$$${R}\:{such}\:{that}\:{cylinder}\:{doesn}'{t}\:{topple}. \\ $$$${center}\:{of}\:{sphere}\:{is}\:{at}\:{a}\:{level}\:{of}\:{half} \\ $$$${the}\:{cylinder}\:{height}\:{h}.\:{radius}\:{of} \\ $$$${cylinder}\:{is}\:{r}. \\ $$
Commented by mr W last updated on 07/Jul/20
$${i}\:{think}\:{it}\:{doesn}'{t}\:{matter}\:{at}\:{which} \\ $$$${level}\:{the}\:{center}\:{of}\:{the}\:{sphere}\:{is}. \\ $$
Commented by ajfour last updated on 07/Jul/20
$${true},\:{i}\:{understand}. \\ $$
Answered by mr W last updated on 07/Jul/20
$${W}_{{C}} =\pi{r}^{\mathrm{2}} {h}\rho{g} \\ $$$${W}_{{S}} =\frac{\mathrm{4}\pi{R}^{\mathrm{3}} }{\mathrm{3}}\rho{g} \\ $$$${W}_{{C}} {r}\geqslant{W}_{{S}} {R} \\ $$$$\pi{r}^{\mathrm{2}} {h}\rho{gr}\geqslant\frac{\mathrm{4}\pi{R}^{\mathrm{3}} }{\mathrm{3}}\rho{gR} \\ $$$${R}\leqslant{r}\sqrt[{\mathrm{4}}]{\frac{\mathrm{3}{h}}{\mathrm{4}{r}}} \\ $$
Commented by ajfour last updated on 07/Jul/20
$${Great}\:{Sir},\:{well}\:{presented},\:{give}\:{me} \\ $$$${a}\:{hint}\:{sir},\:{what}\:{questions}\:{would}\:{you} \\ $$$${like}\:{from}\:{me}?\:{I}\:{am}\:{not}\:{in}\:{good} \\ $$$${health},\:{its}\:{been}\:{months}\:{but}\:{these} \\ $$$${days}\:{progress}\:{is}\:{good},\:{and}\:{it}\:{feels} \\ $$$${great}\:{discussing}\:{with}\:{you},\:\:{but}\:{in} \\ $$$${some}\:{questions}\:{i}\:{lack}\:{the}\:{patience} \\ $$$${that}\:{you}\:{put}\:{in},\:{say}\:{the}\:{minimum} \\ $$$${velocity}\:{question},\:{i}\:{dint}\:{review}\:{it}.. \\ $$
Commented by mr W last updated on 07/Jul/20
$${sorry}\:{to}\:{know}\:{that}\:{you}\:{are}\:{not}\:{in} \\ $$$${good}\:{health}\:{sir}!\:{i}\:{hope}\:{sincerely}\:{that} \\ $$$${you}\:{will}\:{get}\:{well}\:{as}\:{soon}\:{as}\:{possible}. \\ $$$${in}\:{fact}\:{i}\:{find}\:{all}\:{questions}\:{from}\:{you} \\ $$$${creative}\:{and}\:{interesting}.\:{it}'{s}\:{a}\:{big} \\ $$$${pleasure}\:{to}\:{discuss}\:{with}\:{you}\:{and}\:{to} \\ $$$${see}\:{how}\:{you}\:{solve}\:{the}\:{problems}.\:{but} \\ $$$${at}\:{the}\:{moment}\:{i}\:{don}'{t}\:{wish}\:{that}\:{you} \\ $$$${give}\:{or}\:{solve}\:{any}\:{questions},\:{but}\:{that} \\ $$$${you}\:{take}\:{more}\:{care}\:{of}\:{your}\:{health}! \\ $$
Commented by ajfour last updated on 07/Jul/20
$${thanks}\:{for}\:{all}\:{your}\:{concern}\:{Sir},\:{but} \\ $$$${i}\:{am}\:{comfortable}\:{now}\:{with}\:{posting} \\ $$$${and}\:{answering}\:{few}\:{questions}\:{at}\:{the} \\ $$$${forum}. \\ $$
Commented by PRITHWISH SEN 2 last updated on 07/Jul/20
$$\mathrm{get}\:\mathrm{well}\:\mathrm{soon}\:\mathrm{sir}. \\ $$
Commented by $@y@m last updated on 08/Jul/20
https://tse2.mm.bing.net/th?id=OGC.035af8113ed0cbc479f25d449b737284&pid=Api&rurl=https%3a%2f%2fmedia.giphy.com%2fmedia%2fl2JJEZL5S5BAsFY5y%2fgiphy.gif&ehk=FsBG2ECL%2bvUZA84z1bKZjhzZWE%2fTO7f8leFAoQpAE9s%3d
Answered by 1549442205 last updated on 08/Jul/20
$$ \\ $$