Question Number 24186 by Tinkutara last updated on 14/Nov/17
$$\left({n}\:−\:\mathrm{1}\right)\:\mathrm{equal}\:\mathrm{point}\:\mathrm{masses}\:\mathrm{each}\:\mathrm{of}\:\mathrm{mass} \\ $$$${m}\:\mathrm{are}\:\mathrm{placed}\:\mathrm{at}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{regular} \\ $$$${n}-\mathrm{polygon}.\:\mathrm{The}\:\mathrm{vacant}\:\mathrm{vertex}\:\mathrm{has}\:\mathrm{a} \\ $$$$\mathrm{position}\:\mathrm{vector}\:{a}\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{centre}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polygon}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{position} \\ $$$$\mathrm{vector}\:\mathrm{of}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{mass}. \\ $$
Commented by mrW1 last updated on 14/Nov/17
$$−\frac{\mathrm{1}}{{n}−\mathrm{1}}\boldsymbol{{a}} \\ $$
Commented by Tinkutara last updated on 14/Nov/17
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$
Commented by mrW1 last updated on 14/Nov/17
$${let}'{s}\:{say}\:{the}\:{center}\:{of}\:\left({n}−\mathrm{1}\right){m}\:{is}\:{x} \\ $$$${from}\:{the}\:{center}\:{of}\:{polygon}.\:{we}\:{know} \\ $$$${if}\:{we}\:{also}\:{put}\:{a}\:{mass}\:{m}\:{at}\:{the}\:{vacant}\:{vertex}, \\ $$$${then}\:{we}\:{have}\:{n}\:{mass}\:{m},\:{and}\:{the} \\ $$$${center}\:{of}\:{nm}\:{is}\:{the}\:{center}\:{of}\:{polygon}. \\ $$$$\left({n}−\mathrm{1}\right){mg}\mid{x}\mid={mg}\mid{a}\mid \\ $$$$\Rightarrow\mid{x}\mid=\frac{\mid{a}\mid}{{n}−\mathrm{1}} \\ $$$${x}\:{is}\:{opposite}\:{to}\:{a} \\ $$$$\Rightarrow\boldsymbol{{x}}=−\frac{\mathrm{1}}{{n}−\mathrm{1}}\boldsymbol{{a}} \\ $$
Commented by mrW1 last updated on 14/Nov/17
