Question-224144

Question Number 224144 by mr W last updated on 23/Aug/25

Commented by mr W last updated on 23/Aug/25

find the normal force N between two solid cylinders with masses m_1 , m_2 and radii r_1 , r_2 respectively which rest inside an other thin walled hollow cylinder with radius R as shown. all contact surfaces are frictionless.

$${find}\:{the}\:{normal}\:{force}\:\boldsymbol{{N}}\:{between}\: \\ $$$${two}\:{solid}\:{cylinders}\:{with}\:{masses} \\ $$$$\boldsymbol{{m}}_{\mathrm{1}} ,\:\boldsymbol{{m}}_{\mathrm{2}} \:{and}\:{radii}\:\boldsymbol{{r}}_{\mathrm{1}} ,\:\boldsymbol{{r}}_{\mathrm{2}} \:{respectively} \\ $$$${which}\:{rest}\:{inside}\:{an}\:{other}\:{thin}\: \\ $$$${walled}\:{hollow}\:{cylinder}\:{with}\:{radius} \\ $$$$\boldsymbol{{R}}\:{as}\:{shown}.\: \\ $$$${all}\:{contact}\:{surfaces}\:{are}\:{frictionless}. \\ $$

Answered by mr W last updated on 24/Aug/25

Commented by mr W last updated on 24/Aug/25

say μ_1 =(m_1 /(m_1 +m_2 )), μ_2 =(m_2 /(m_1 +m_2 )) AD=((m_2 ×AB)/(m_1 +m_2 ))=μ_2 (r_1 +r_2 ) DB=((m_1 ×AB)/(m_1 +m_2 ))=μ_1 (r_1 +r_2 ) μ_2 (R−r_2 )^2 +μ_1 (R−r_1 )^2 =CD^2 +μ_1 μ_2 (r_1 +r_2 )^2 ⇒CD=(√(μ_1 (R−r_1 )^2 +μ_2 (R−r_2 )^2 −μ_1 μ_2 (r_1 +r_2 )^2 )) ((sin θ)/(DB))=((sin β)/(CD)) ⇒((sin θ)/(sin β))=((μ_1 (r_1 +r_2 ))/(CD)) (N/(sin θ))=((m_2 g)/(sin β)) ⇒N=((m_2 g sin θ)/(sin β)) (N/((m_1 +m_2 )g))=((μ_2 sin θ)/(sin β)) =((μ_1 μ_2 (r_1 +r_2 ))/(CD)) =((μ_1 μ_2 (r_1 +r_2 ))/( (√(μ_1 (R−r_1 )^2 +μ_2 (R−r_2 )^2 −μ_1 μ_2 (r_1 +r_2 )^2 )))) =((μ_1 μ_2 )/( (√(μ_1 (((R−r_1 )/(r_1 +r_2 )))^2 +μ_2 (((R−r_2 )/(r_1 +r_2 )))^2 −μ_1 μ_2 )))) ⇒N=((μ_1 μ_2 (m_1 +m_2 )g)/( (√(μ_1 (((R−r_1 )/(r_1 +r_2 )))^2 +μ_2 (((R−r_2 )/(r_1 +r_2 )))^2 −μ_1 μ_2 )))) (N_1 /((m_1 +m_2 )g))=((sin θ)/(sin (α+β)))=((sin θ sin β)/(sin β sin (α+β))) =((μ_1 AB)/(CD))×((AC)/(AB))=((μ_1 AC)/(CD)) =((μ_1 (R−r_1 ))/( (√(μ_1 (R−r_1 )^2 +μ_2 (R−r_2 )^2 −μ_1 μ_2 (r_1 +r_2 )^2 )))) ⇒N_1 =(((R−r_1 )m_1 g)/( (√(μ_1 (R−r_1 )^2 +μ_2 (R−r_2 )^2 −μ_1 μ_2 (r_1 +r_2 )^2 )))) similarly ⇒N_2 =(((R−r_2 )m_2 g)/( (√(μ_1 (R−r_1 )^2 +μ_2 (R−r_2 )^2 −μ_1 μ_2 (r_1 +r_2 )^2 ))))

$${say}\:\mu_{\mathrm{1}} =\frac{{m}_{\mathrm{1}} }{{m}_{\mathrm{1}} +{m}_{\mathrm{2}} },\:\mu_{\mathrm{2}} =\frac{{m}_{\mathrm{2}} }{{m}_{\mathrm{1}} +{m}_{\mathrm{2}} } \\ $$$${AD}=\frac{{m}_{\mathrm{2}} ×{AB}}{{m}_{\mathrm{1}} +{m}_{\mathrm{2}} }=\mu_{\mathrm{2}} \left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right) \\ $$$${DB}=\frac{{m}_{\mathrm{1}} ×{AB}}{{m}_{\mathrm{1}} +{m}_{\mathrm{2}} }=\mu_{\mathrm{1}} \left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right) \\ $$$$\mu_{\mathrm{2}} \left({R}−{r}_{\mathrm{2}} \right)^{\mathrm{2}} +\mu_{\mathrm{1}} \left({R}−{r}_{\mathrm{1}} \right)^{\mathrm{2}} ={CD}^{\mathrm{2}} +\mu_{\mathrm{1}} \mu_{\mathrm{2}} \left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right)^{\mathrm{2}} \\ $$$$\Rightarrow{CD}=\sqrt{\mu_{\mathrm{1}} \left({R}−{r}_{\mathrm{1}} \right)^{\mathrm{2}} +\mu_{\mathrm{2}} \left({R}−{r}_{\mathrm{2}} \right)^{\mathrm{2}} −\mu_{\mathrm{1}} \mu_{\mathrm{2}} \left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$$$\frac{\mathrm{sin}\:\theta}{{DB}}=\frac{\mathrm{sin}\:\beta}{{CD}} \\ $$$$\Rightarrow\frac{\mathrm{sin}\:\theta}{\mathrm{sin}\:\beta}=\frac{\mu_{\mathrm{1}} \left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right)}{{CD}} \\ $$$$\frac{{N}}{\mathrm{sin}\:\theta}=\frac{{m}_{\mathrm{2}} {g}}{\mathrm{sin}\:\beta} \\ $$$$\Rightarrow{N}=\frac{{m}_{\mathrm{2}} {g}\:\mathrm{sin}\:\theta}{\mathrm{sin}\:\beta} \\ $$$$\frac{{N}}{\left({m}_{\mathrm{1}} +{m}_{\mathrm{2}} \right){g}}=\frac{\mu_{\mathrm{2}} \:\mathrm{sin}\:\theta}{\mathrm{sin}\:\beta} \\ $$$$\:\:\:\:=\frac{\mu_{\mathrm{1}} \mu_{\mathrm{2}} \left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right)}{{CD}} \\ $$$$\:\:\:\:=\frac{\mu_{\mathrm{1}} \mu_{\mathrm{2}} \left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right)}{\:\sqrt{\mu_{\mathrm{1}} \left({R}−{r}_{\mathrm{1}} \right)^{\mathrm{2}} +\mu_{\mathrm{2}} \left({R}−{r}_{\mathrm{2}} \right)^{\mathrm{2}} −\mu_{\mathrm{1}} \mu_{\mathrm{2}} \left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right)^{\mathrm{2}} }} \\ $$$$\:\:\:\:=\frac{\mu_{\mathrm{1}} \mu_{\mathrm{2}} }{\:\sqrt{\mu_{\mathrm{1}} \left(\frac{{R}−{r}_{\mathrm{1}} }{{r}_{\mathrm{1}} +{r}_{\mathrm{2}} }\right)^{\mathrm{2}} +\mu_{\mathrm{2}} \left(\frac{{R}−{r}_{\mathrm{2}} }{{r}_{\mathrm{1}} +{r}_{\mathrm{2}} }\right)^{\mathrm{2}} −\mu_{\mathrm{1}} \mu_{\mathrm{2}} }} \\ $$$$\Rightarrow{N}=\frac{\mu_{\mathrm{1}} \mu_{\mathrm{2}} \left({m}_{\mathrm{1}} +{m}_{\mathrm{2}} \right){g}}{\:\sqrt{\mu_{\mathrm{1}} \left(\frac{{R}−{r}_{\mathrm{1}} }{{r}_{\mathrm{1}} +{r}_{\mathrm{2}} }\right)^{\mathrm{2}} +\mu_{\mathrm{2}} \left(\frac{{R}−{r}_{\mathrm{2}} }{{r}_{\mathrm{1}} +{r}_{\mathrm{2}} }\right)^{\mathrm{2}} −\mu_{\mathrm{1}} \mu_{\mathrm{2}} }} \\ $$$$\frac{{N}_{\mathrm{1}} }{\left({m}_{\mathrm{1}} +{m}_{\mathrm{2}} \right){g}}=\frac{\mathrm{sin}\:\theta}{\mathrm{sin}\:\left(\alpha+\beta\right)}=\frac{\mathrm{sin}\:\theta\:\mathrm{sin}\:\beta}{\mathrm{sin}\:\beta\:\mathrm{sin}\:\left(\alpha+\beta\right)} \\ $$$$\:\:\:\:=\frac{\mu_{\mathrm{1}} {AB}}{{CD}}×\frac{{AC}}{{AB}}=\frac{\mu_{\mathrm{1}} {AC}}{{CD}} \\ $$$$\:\:\:\:=\frac{\mu_{\mathrm{1}} \left({R}−{r}_{\mathrm{1}} \right)}{\:\sqrt{\mu_{\mathrm{1}} \left({R}−{r}_{\mathrm{1}} \right)^{\mathrm{2}} +\mu_{\mathrm{2}} \left({R}−{r}_{\mathrm{2}} \right)^{\mathrm{2}} −\mu_{\mathrm{1}} \mu_{\mathrm{2}} \left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right)^{\mathrm{2}} }} \\ $$$$\Rightarrow{N}_{\mathrm{1}} =\frac{\left({R}−{r}_{\mathrm{1}} \right){m}_{\mathrm{1}} {g}}{\:\sqrt{\mu_{\mathrm{1}} \left({R}−{r}_{\mathrm{1}} \right)^{\mathrm{2}} +\mu_{\mathrm{2}} \left({R}−{r}_{\mathrm{2}} \right)^{\mathrm{2}} −\mu_{\mathrm{1}} \mu_{\mathrm{2}} \left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right)^{\mathrm{2}} }} \\ $$$${similarly} \\ $$$$\Rightarrow{N}_{\mathrm{2}} =\frac{\left({R}−{r}_{\mathrm{2}} \right){m}_{\mathrm{2}} {g}}{\:\sqrt{\mu_{\mathrm{1}} \left({R}−{r}_{\mathrm{1}} \right)^{\mathrm{2}} +\mu_{\mathrm{2}} \left({R}−{r}_{\mathrm{2}} \right)^{\mathrm{2}} −\mu_{\mathrm{1}} \mu_{\mathrm{2}} \left({r}_{\mathrm{1}} +{r}_{\mathrm{2}} \right)^{\mathrm{2}} }} \\ $$

Commented by mr W last updated on 24/Aug/25