Question Number 70423 by ajfour last updated on 04/Oct/19
Commented by ajfour last updated on 04/Oct/19
$${Within}\:{a}\:{hollow}\:{sphere},\:{mass} \\ $$$${M}_{\mathrm{0}} \:,\:{radius}\:{R},\:{rests}\:{two}\:{smaller} \\ $$$${spheres}\:{masses},\:{M}\:{and}\:{m},\:{radii} \\ $$$${b},\:{a}\:,\:{respectively}.\:{Find}\:{equilibrium} \\ $$$${angles}\:\alpha,\:\beta. \\ $$
Answered by ajfour last updated on 04/Oct/19
Commented by ajfour last updated on 04/Oct/19
$${Let}\:{the}\:{normal}\:{reaction}\:{between} \\ $$$${the}\:{two}\:{smaller}\:{spheres}\:{be}\:{F}. \\ $$$${R}\mathrm{sin}\:\gamma=\left({R}−{b}\right)\mathrm{cos}\:\alpha−\left({R}−{a}\right)\mathrm{cos}\:\beta \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….\left({i}\right) \\ $$$${R}\mathrm{cos}\:\gamma=\left({R}−{b}\right)\mathrm{sin}\:\alpha+\left({R}−{a}\right)\mathrm{sin}\:\beta \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….\left({ii}\right) \\ $$$$\:{From}\:{concurrency}\:{of}\:{forces} \\ $$$$\:{F}\:=\:\frac{{Mg}\mathrm{sin}\:\alpha}{\mathrm{cos}\:\left(\gamma+\alpha\right)}=\frac{{mg}\mathrm{sin}\:\beta}{\mathrm{cos}\:\left(\beta−\gamma\right)} \\ $$$$\Rightarrow\:{M}\left(\frac{\mathrm{1}}{\mathrm{tan}\:\beta}+\:\mathrm{tan}\:\gamma\right) \\ $$$$\:\:\:\:\:\:\:\:\:=\:{m}\left(\frac{\mathrm{1}}{\mathrm{tan}\:\alpha}−\mathrm{tan}\:\gamma\right) \\ $$$$\Rightarrow\:\:\mathrm{tan}\:\gamma\:=\:\frac{{m}\mathrm{cot}\:\alpha−{M}\mathrm{cot}\:\beta}{{m}+{M}}\:\:..\left({I}\right) \\ $$$${Also}\:{from}\:\left({i}\right)\:\&\:\left({ii}\right) \\ $$$$\mathrm{tan}\:\gamma\:=\:\frac{\left({R}−{b}\right)\mathrm{cos}\:\alpha−\left({R}−{a}\right)\mathrm{cos}\:\beta}{\left({R}−{b}\right)\mathrm{sin}\:\alpha+\left({R}−{a}\right)\mathrm{sin}\:\beta} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:……\left({II}\right) \\ $$$${let}\:{R}−{a}={p}\:,\:{R}−{b}={q}\:,\:{M}=\mu{m}, \\ $$$$\:\:\:\:\:\:{r}={a}+{b} \\ $$$$\:\:\:\:{from}\:{cosine}\:{rule}\:{in}\:\bigtriangleup{ACB} \\ $$$$\:\:\:\mathrm{cos}\:\left(\alpha+\beta\right)=\frac{{p}^{\mathrm{2}} +{q}^{\mathrm{2}} −{r}^{\mathrm{2}} }{\mathrm{2}{pq}}\:=\:\boldsymbol{{k}}\:\:\:..\left(\mathrm{1}\right) \\ $$$${And}\:{from}\:\left({I}\right)\:\&\:\left({II}\right) \\ $$$$\frac{\mathrm{cot}\:\alpha−\mu\mathrm{cot}\:\beta}{\mathrm{1}+\mu}=\frac{{q}\mathrm{cos}\:\alpha−{p}\mathrm{cos}\:\beta}{{q}\mathrm{sin}\:\alpha+{p}\mathrm{sin}\:\beta} \\ $$$$\Rightarrow\:\:\left({q}\mathrm{sin}\:\alpha+{p}\mathrm{sin}\:\beta\right)\left(\mathrm{sin}\:\beta\mathrm{cos}\:\alpha−\mu\mathrm{cos}\:\beta\mathrm{sin}\:\alpha\right) \\ $$$$\:\:\:=\left(\mathrm{1}+\mu\right)\left({q}\mathrm{cos}\:\alpha−{p}\mathrm{cos}\:\beta\right)\left(\mathrm{sin}\:\alpha\mathrm{sin}\:\beta\right) \\ $$$$\Rightarrow \\ $$$$\:{q}\mathrm{sin}\:\alpha\mathrm{cos}\:\alpha\mathrm{sin}\:\beta−\mu{q}\mathrm{sin}\:^{\mathrm{2}} \alpha\mathrm{cos}\:\beta \\ $$$$+{p}\mathrm{cos}\:\alpha\mathrm{sin}\:^{\mathrm{2}} \beta−\mu{p}\mathrm{sin}\:\alpha\mathrm{sin}\:\beta\mathrm{cos}\:\beta \\ $$$$=\left(\mathrm{1}+\mu\right){q}\mathrm{sin}\:\alpha\mathrm{cos}\:\alpha\mathrm{sin}\:\beta \\ $$$$\:\:\:\:\:\:\:\:−\left(\mathrm{1}+\mu\right){p}\mathrm{sin}\:\alpha\mathrm{sin}\:\beta\mathrm{cos}\:\beta \\ $$$$\Rightarrow \\ $$$$\mu{q}\mathrm{sin}\:\alpha\mathrm{cos}\:\alpha\mathrm{sin}\:\beta−{p}\mathrm{sin}\:\alpha\mathrm{sin}\:\beta\mathrm{cos}\:\beta \\ $$$$\:\:\:\:={p}\mathrm{cos}\:\alpha\mathrm{sin}\:^{\mathrm{2}} \beta−\mu{q}\mathrm{sin}\:^{\mathrm{2}} \alpha\mathrm{cos}\:\beta \\ $$$$\Rightarrow \\ $$$$\:\:\left(\mu{q}\mathrm{sin}\:\alpha\right)\mathrm{sin}\:\left(\alpha+\beta\right) \\ $$$$\:\:\:\:\:\:\:\:={p}\mathrm{sin}\:\beta\:\mathrm{sin}\:\left(\alpha+\beta\right) \\ $$$$\Rightarrow\:\:\frac{\mathrm{sin}\:\alpha}{\mathrm{sin}\:\beta}\:=\:\frac{{p}}{\mu{q}}\:\:\:\:\left({has}\:{to}\:{be}\:{a}\:{short}\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{way}\:{to}\:{this}\right) \\ $$$${Now}\:{using}\:\left(\mathrm{1}\right) \\ $$$$\:\:\mathrm{cos}\:\alpha\sqrt{\mathrm{1}−\mathrm{sin}\:^{\mathrm{2}} \beta}−\mathrm{sin}\:\alpha\mathrm{sin}\:\beta\:={k} \\ $$$$\Rightarrow \\ $$$$\:\:\:\mathrm{cos}\:\alpha\sqrt{\mathrm{1}−\frac{\mu^{\mathrm{2}} {q}^{\mathrm{2}} }{{p}^{\mathrm{2}} }\mathrm{sin}\:^{\mathrm{2}} \alpha} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\frac{\mu{q}}{{p}}\:\mathrm{sin}\:^{\mathrm{2}} \alpha\:=\:{k} \\ $$$$\:\:\left(\mathrm{1}−\mathrm{sin}\:^{\mathrm{2}} \alpha\right)\left(\mathrm{1}−\frac{\mu^{\mathrm{2}} {q}^{\mathrm{2}} }{{p}^{\mathrm{2}} }\mathrm{sin}\:^{\mathrm{2}} \alpha\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\left({k}+\frac{\mu{q}}{{p}}\mathrm{sin}\:^{\mathrm{2}} \alpha\right)^{\mathrm{2}} \\ $$$$\:{lets}\:{call}\:\:\mathrm{sin}\:\alpha={x}\:,\:\:\frac{\mu{q}}{{p}}={c} \\ $$$$\Rightarrow\:\:\mathrm{1}−\left(\mathrm{1}+{c}^{\mathrm{2}} \right){x}^{\mathrm{2}} ={k}^{\mathrm{2}} +\mathrm{2}{ckx}^{\mathrm{2}} \\ $$$$\Rightarrow\:\:\:\:{x}^{\mathrm{2}} =\frac{\mathrm{1}−{k}^{\mathrm{2}} }{\left({c}^{\mathrm{2}} +\mathrm{2}{ck}+\mathrm{1}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:{y}^{\mathrm{2}} =\:{c}^{\mathrm{2}} {x}^{\mathrm{2}} \:=\:\frac{{c}^{\mathrm{2}} \left(\mathrm{1}−{k}^{\mathrm{2}} \right)}{\left({c}^{\mathrm{2}} +\mathrm{2}{ck}+\mathrm{1}\right)} \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$$\Rightarrow\:\:\boldsymbol{\alpha}=\mathrm{sin}^{−\mathrm{1}} \sqrt{\frac{\mathrm{1}−{k}^{\mathrm{2}} }{{c}^{\mathrm{2}} +\mathrm{2}{ck}+\mathrm{1}}} \\ $$$$\:\:\:\:\:\:\boldsymbol{\beta}=\mathrm{sin}^{−\mathrm{1}} \sqrt{\frac{{c}^{\mathrm{2}} \left(\mathrm{1}−{k}^{\mathrm{2}} \right)}{{c}^{\mathrm{2}} +\mathrm{2}{ck}+\mathrm{1}}} \\ $$$$\:\:\:\:\:{c}\:=\:\frac{{M}\left({R}−{b}\right)}{{m}\left({R}−{a}\right)}\: \\ $$$$\:\:\:\:\:{k}=\frac{\left({R}−{a}\right)^{\mathrm{2}} +\left({R}−{b}\right)^{\mathrm{2}} −\left({a}+{b}\right)^{\mathrm{2}} }{\mathrm{2}\left({R}−{a}\right)\left({R}−{b}\right)}\:\:\:\: \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\blacksquare \\ $$$${If}\:\:{as}\:{an}\:{example} \\ $$$$\:\:\:\mu=\mathrm{8}\:,\:{b}=\mathrm{2}{a}\:=\:\frac{{R}}{\mathrm{2}}\:\:\:\:\left({same}\:{densities}\right) \\ $$$$\:\:\:{c}=\frac{\mathrm{16}}{\mathrm{3}}\:,\:{k}=\frac{\frac{\mathrm{9}}{\mathrm{16}}+\frac{\mathrm{4}}{\mathrm{16}}−\frac{\mathrm{9}}{\mathrm{16}}}{\frac{\mathrm{12}}{\mathrm{16}}}=\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\:\:\mathrm{sin}\:\alpha=\sqrt{\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{9}}}{\frac{\mathrm{256}}{\mathrm{9}}+\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{3}}\right)\left(\frac{\mathrm{16}}{\mathrm{3}}\right)+\mathrm{1}}} \\ $$$$\:\:\alpha\:=\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{297}}}\right)\approx\:\mathrm{9}.\mathrm{446}° \\ $$$$\:\:\beta\:=\:\mathrm{sin}^{−\mathrm{1}} \left(\sqrt{\frac{\mathrm{2048}}{\mathrm{2673}}}\right)\:\approx\:\mathrm{61}.\mathrm{082}°\:. \\ $$
Answered by mr W last updated on 04/Oct/19
Commented by mr W last updated on 04/Oct/19
$${p}={R}−{a} \\ $$$${q}={R}−{b} \\ $$$${r}={a}+{b} \\ $$$${M}=\mu{m} \\ $$$$\mathrm{cos}\:\theta=\frac{{q}^{\mathrm{2}} +{r}^{\mathrm{2}} −{p}^{\mathrm{2}} }{\mathrm{2}{qr}}=\frac{{q}^{\mathrm{2}} +\frac{{r}^{\mathrm{2}} }{\left(\mathrm{1}+\mu\right)^{\mathrm{2}} }−{h}^{\mathrm{2}} }{\mathrm{2}{q}\frac{{r}}{\mathrm{1}+\mu}} \\ $$$$\frac{{q}^{\mathrm{2}} +{r}^{\mathrm{2}} −{p}^{\mathrm{2}} }{\mathrm{1}+\mu}={q}^{\mathrm{2}} +\frac{{r}^{\mathrm{2}} }{\left(\mathrm{1}+\mu\right)^{\mathrm{2}} }−{h}^{\mathrm{2}} \\ $$$$\Rightarrow{h}^{\mathrm{2}} =\frac{\mu{q}^{\mathrm{2}} }{\mathrm{1}+\mu}+\frac{{p}^{\mathrm{2}} }{\mathrm{1}+\mu}−\frac{\mu{r}^{\mathrm{2}} }{\left(\mathrm{1}+\mu\right)^{\mathrm{2}} } \\ $$$$\Rightarrow{h}^{\mathrm{2}} =\frac{\mathrm{1}}{\left(\mathrm{1}+\mu\right)^{\mathrm{2}} }\left[\mu\left(\mathrm{1}+\mu\right){q}^{\mathrm{2}} +\left(\mathrm{1}+\mu\right){p}^{\mathrm{2}} −\mu{r}^{\mathrm{2}} \right] \\ $$$$\mathrm{cos}\:\alpha=\frac{{q}^{\mathrm{2}} +{h}^{\mathrm{2}} −\frac{{r}^{\mathrm{2}} }{\left(\mathrm{1}+\mu\right)^{\mathrm{2}} }}{\mathrm{2}{qh}} \\ $$$$=\frac{\left(\mathrm{1}+\mathrm{2}\mu\right){q}^{\mathrm{2}} +{p}^{\mathrm{2}} −{r}^{\mathrm{2}} }{\mathrm{2}{q}\sqrt{\mu\left(\mathrm{1}+\mu\right){q}^{\mathrm{2}} +\left(\mathrm{1}+\mu\right){p}^{\mathrm{2}} −\mu{r}^{\mathrm{2}} }} \\ $$$$\Rightarrow\alpha=\mathrm{cos}^{−\mathrm{1}} \frac{\left(\mathrm{1}+\mathrm{2}\mu\right){q}^{\mathrm{2}} +{p}^{\mathrm{2}} −{r}^{\mathrm{2}} }{\mathrm{2}{q}\sqrt{\mu\left(\mathrm{1}+\mu\right){q}^{\mathrm{2}} +\left(\mathrm{1}+\mu\right){p}^{\mathrm{2}} −\mu{r}^{\mathrm{2}} }} \\ $$$$\mathrm{cos}\:\beta=\frac{{p}^{\mathrm{2}} +{h}^{\mathrm{2}} −\frac{\mu^{\mathrm{2}} {r}^{\mathrm{2}} }{\left(\mathrm{1}+\mu\right)^{\mathrm{2}} }}{\mathrm{2}{ph}} \\ $$$$=\frac{\mu{q}^{\mathrm{2}} +\left(\mathrm{2}+\mu\right){p}^{\mathrm{2}} −\mu{r}^{\mathrm{2}} }{\mathrm{2}{p}\sqrt{\mu\left(\mathrm{1}+\mu\right){q}^{\mathrm{2}} +\left(\mathrm{1}+\mu\right){p}^{\mathrm{2}} −\mu{r}^{\mathrm{2}} }} \\ $$$$\Rightarrow\beta=\mathrm{cos}^{−\mathrm{1}} \frac{\mu{q}^{\mathrm{2}} +\left(\mathrm{2}+\mu\right){p}^{\mathrm{2}} −\mu{r}^{\mathrm{2}} }{\mathrm{2}{p}\sqrt{\mu\left(\mathrm{1}+\mu\right){q}^{\mathrm{2}} +\left(\mathrm{1}+\mu\right){p}^{\mathrm{2}} −\mu{r}^{\mathrm{2}} }} \\ $$$$ \\ $$$${example}: \\ $$$$\mu=\mathrm{8},\:{a}=\frac{{R}}{\mathrm{4}},\:{b}=\frac{{R}}{\mathrm{2}} \\ $$$${p}={R}−{a}=\frac{\mathrm{3}{R}}{\mathrm{4}},\:{q}={R}−{b}=\frac{\mathrm{2}{R}}{\mathrm{4}},\:{r}=\frac{\mathrm{3}{R}}{\mathrm{4}} \\ $$$$\mathrm{cos}\:\alpha=\frac{\mathrm{17}}{\mathrm{3}\sqrt{\mathrm{33}}}\:\Rightarrow\mathrm{sin}\:\alpha=\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}\sqrt{\mathrm{33}}}\:\Rightarrow\mathrm{9}.\mathrm{446}° \\ $$$$\mathrm{cos}\:\beta=\frac{\mathrm{25}}{\mathrm{9}\sqrt{\mathrm{33}}}\:\Rightarrow\mathrm{sin}\:\beta=\frac{\mathrm{32}\sqrt{\mathrm{2}}}{\mathrm{9}\sqrt{\mathrm{33}}}\:\Rightarrow\mathrm{61}.\mathrm{083}° \\ $$
Commented by ajfour last updated on 04/Oct/19
$${sir}\:{please}\:{consider}\:{the}\:{example} \\ $$$${if}\:{have}\:{chosen}\:{and}\:{bring}\:{out} \\ $$$$\:{values}\:{for}\:\alpha,\:\beta\:. \\ $$
Commented by mr W last updated on 04/Oct/19
$${the}\:{results}\:{are}\:{the}\:{same}\:{sir}. \\ $$
Commented by ajfour last updated on 04/Oct/19
$${Great},\:{i}\:{had}\:{done}\:{blunders}, \\ $$$${edited},\:{corrected}.\:{Your}\:{solution} \\ $$$${is}\:{shorter}.\:{Thank}\:{you}\:{Sir}. \\ $$
Commented by mr W last updated on 04/Oct/19
$${i}\:{treated}\:{the}\:{two}\:{balls}\:{as}\:{a}\:{single}\: \\ $$$${object}\:{whose}\:{center}\:{of}\:{mass}\:{must} \\ $$$${be}\:{vertically}\:{beneath}\:{the}\:{center}\:{of} \\ $$$${hollow}\:{sphere}\:{when}\:{in}\:{equilibrium}. \\ $$