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Question Number 72036 by mr W last updated on 23/Oct/19

Commented by mr W last updated on 23/Oct/19

the cup has the shape of hemisphere  with radius R. a stick of mass m and  length l (2R<l<4R) is released at  blue position. there is no friction  between cup and stick.  find the time t in terms of θ.  find θ_(max) .

$${the}\:{cup}\:{has}\:{the}\:{shape}\:{of}\:{hemisphere} \\ $$$${with}\:{radius}\:{R}.\:{a}\:{stick}\:{of}\:{mass}\:{m}\:{and} \\ $$$${length}\:{l}\:\left(\mathrm{2}{R}<{l}<\mathrm{4}{R}\right)\:{is}\:{released}\:{at} \\ $$$${blue}\:{position}.\:{there}\:{is}\:{no}\:{friction} \\ $$$${between}\:{cup}\:{and}\:{stick}. \\ $$$${find}\:{the}\:{time}\:{t}\:{in}\:{terms}\:{of}\:\theta. \\ $$$${find}\:\theta_{{max}} . \\ $$

Commented by ajfour last updated on 23/Oct/19

cup fixed, i guess ?

$${cup}\:{fixed},\:{i}\:{guess}\:? \\ $$

Commented by mr W last updated on 23/Oct/19

yes sir.

$${yes}\:{sir}. \\ $$

Answered by mr W last updated on 24/Oct/19

Commented by mr W last updated on 24/Oct/19

ϕ=(θ/2)  ω=(dϕ/dt)=(1/2)×(dθ/dt)  α=(dω/dt)=ω(dw/dϕ)=2ω(dω/dθ)  OS=R  let λ=(l/(4R)) <1  e=2R cos θ−(l/2)cos ϕ=2R(cos θ−(l/(4R)) cos (θ/2))  =2R(cos θ−λ cos (θ/2))  SM^2 =(2R)^2 +((l/2))^2 −2×2R×(l/2) cos ϕ  =4R^2 (1+λ^2 −2λ cos (θ/2))  I_S =((ml^2 )/(12))+m×SM^2   =4mR^2 (1+((4λ^2 )/3)−2λ cos (θ/2))    (1/2)I_S ω^2 =mg(R sin θ−(l/2)sin (θ/2))  2R(1+((4λ^2 )/3)−2λ cos (θ/2))ω^2 =g(sin θ−2λ sin (θ/2))  ⇒ω=(dθ/(2dt))=(√(g/(2R)))(√((sin θ−2λ sin (θ/2))/(1+((4λ^2 )/3)−2λ cos (θ/2))))  ⇒t=(√(R/(2g))) ∫_0 ^θ (√((1+((4λ^2 )/3)−2λ cos (θ/2))/(sin θ−2λ sin (θ/2))))dθ

$$\varphi=\frac{\theta}{\mathrm{2}} \\ $$$$\omega=\frac{{d}\varphi}{{dt}}=\frac{\mathrm{1}}{\mathrm{2}}×\frac{{d}\theta}{{dt}} \\ $$$$\alpha=\frac{{d}\omega}{{dt}}=\omega\frac{{dw}}{{d}\varphi}=\mathrm{2}\omega\frac{{d}\omega}{{d}\theta} \\ $$$${OS}={R} \\ $$$${let}\:\lambda=\frac{{l}}{\mathrm{4}{R}}\:<\mathrm{1} \\ $$$${e}=\mathrm{2}{R}\:\mathrm{cos}\:\theta−\frac{{l}}{\mathrm{2}}\mathrm{cos}\:\varphi=\mathrm{2}{R}\left(\mathrm{cos}\:\theta−\frac{{l}}{\mathrm{4}{R}}\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}\right) \\ $$$$=\mathrm{2}{R}\left(\mathrm{cos}\:\theta−\lambda\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}\right) \\ $$$${SM}^{\mathrm{2}} =\left(\mathrm{2}{R}\right)^{\mathrm{2}} +\left(\frac{{l}}{\mathrm{2}}\right)^{\mathrm{2}} −\mathrm{2}×\mathrm{2}{R}×\frac{{l}}{\mathrm{2}}\:\mathrm{cos}\:\varphi \\ $$$$=\mathrm{4}{R}^{\mathrm{2}} \left(\mathrm{1}+\lambda^{\mathrm{2}} −\mathrm{2}\lambda\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}\right) \\ $$$${I}_{{S}} =\frac{{ml}^{\mathrm{2}} }{\mathrm{12}}+{m}×{SM}^{\mathrm{2}} \\ $$$$=\mathrm{4}{mR}^{\mathrm{2}} \left(\mathrm{1}+\frac{\mathrm{4}\lambda^{\mathrm{2}} }{\mathrm{3}}−\mathrm{2}\lambda\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}\right) \\ $$$$ \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{I}_{{S}} \omega^{\mathrm{2}} ={mg}\left({R}\:\mathrm{sin}\:\theta−\frac{{l}}{\mathrm{2}}\mathrm{sin}\:\frac{\theta}{\mathrm{2}}\right) \\ $$$$\mathrm{2}{R}\left(\mathrm{1}+\frac{\mathrm{4}\lambda^{\mathrm{2}} }{\mathrm{3}}−\mathrm{2}\lambda\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}\right)\omega^{\mathrm{2}} ={g}\left(\mathrm{sin}\:\theta−\mathrm{2}\lambda\:\mathrm{sin}\:\frac{\theta}{\mathrm{2}}\right) \\ $$$$\Rightarrow\omega=\frac{{d}\theta}{\mathrm{2}{dt}}=\sqrt{\frac{{g}}{\mathrm{2}{R}}}\sqrt{\frac{\mathrm{sin}\:\theta−\mathrm{2}\lambda\:\mathrm{sin}\:\frac{\theta}{\mathrm{2}}}{\mathrm{1}+\frac{\mathrm{4}\lambda^{\mathrm{2}} }{\mathrm{3}}−\mathrm{2}\lambda\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}}} \\ $$$$\Rightarrow{t}=\sqrt{\frac{{R}}{\mathrm{2}{g}}}\:\int_{\mathrm{0}} ^{\theta} \sqrt{\frac{\mathrm{1}+\frac{\mathrm{4}\lambda^{\mathrm{2}} }{\mathrm{3}}−\mathrm{2}\lambda\:\mathrm{cos}\:\frac{\theta}{\mathrm{2}}}{\mathrm{sin}\:\theta−\mathrm{2}\lambda\:\mathrm{sin}\:\frac{\theta}{\mathrm{2}}}}{d}\theta \\ $$

Commented by ajfour last updated on 24/Oct/19

αI_S =mge  please check sir, i have little  doubt..   (dL_S /dt)= I_S α+(dI_S /dt)ω=mge

$$\alpha{I}_{{S}} ={mge} \\ $$$${please}\:{check}\:{sir},\:{i}\:{have}\:{little} \\ $$$${doubt}..\:\:\:\frac{{dL}_{{S}} }{{dt}}=\:{I}_{{S}} \alpha+\frac{{dI}_{{S}} }{{dt}}\omega={mge} \\ $$

Commented by mr W last updated on 24/Oct/19

you are very right sir! it is a   system with variable MoI.  but  (1/2)I_S ω^2 =mgΔh  should be ok.

$${you}\:{are}\:{very}\:{right}\:{sir}!\:{it}\:{is}\:{a}\: \\ $$$${system}\:{with}\:{variable}\:{MoI}. \\ $$$${but} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{I}_{{S}} \omega^{\mathrm{2}} ={mg}\Delta{h} \\ $$$${should}\:{be}\:{ok}. \\ $$

Commented by ajfour last updated on 24/Oct/19

yes i too think the other one okay.

$${yes}\:{i}\:{too}\:{think}\:{the}\:{other}\:{one}\:{okay}. \\ $$

Commented by mr W last updated on 24/Oct/19

thanks alot sir!

$${thanks}\:{alot}\:{sir}! \\ $$

Answered by ajfour last updated on 23/Oct/19

Commented by ajfour last updated on 24/Oct/19

AP =2Rcos (θ/2)     ;  x=l−2Rcos (θ/2)  let φ=(θ/2)  τ_P =−(Nsin φ)(l−x)+(mgcos φ)((l/2)−x)                                                    .....(i)  Also along rod   Ncos φ−mgsin φ=m(d^2 x/dt^2 )+mω^2 ((l/2)−x)                                            ........(ii)  L=I_P  ω   let  ω=(dφ/dt)=((d(θ/2))/dt)  I_P =((ml^2 )/(12))+m((l/2)−x)^2   (dL/dt)=2ωm((l/2)−x)(−(dx/dt))+          [((ml^2 )/2)+m((l/2)−x)^2 ](dω/dt)   ..(iii)    x=l−2Rcos φ  ⇒ −(dx/dt)=−ωRsin φ       (d^2 x/dt^2 )=ω^2 Rcos φ  Now from (ii)   Ncos φ−mgsin φ=m(d^2 x/dt^2 )+mω^2 ((l/2)−x)  ⇒ N=mgtan φ+mω^2 R+mω^2 ((l/2)−x)sec φ  using this and (i), (iii) in     (dL/dt)=τ_P    ⇒  2ωm((l/2)−x)(−(dx/dt))+[((ml^2 )/(12))+m((l/2)−x)^2 ](dω/dt)  =−(Nsin φ)(l−x)+(mgcos φ)((l/2)−x)  ⇒  2ω(2Rcos φ−(l/2))(−ωRsin φ)  +[(l^2 /(12))+(2Rcos φ)^2 ]((ωdω)/dφ) =    −{gtan φ+ω^2 R+ω^2 (2Rcos φ−(l/2))sec φ}sin φ(2Rcos φ)      +gcos φ(2Rcos φ−(l/2))  ⇒  ((l^2 /(12))+4R^2 cos^2 φ)((ωdω)/dφ)    +2ω^2 R^2 sin φcos φ =       2gR(cos^2 φ−sin^2 φ)−((glcos φ)/2)  let  ω^2 =z  ⇒  (dz/dφ)+(((4R^2 sin φcos φ)/((l^2 /(12))+4R^2 cos^2 φ)))z          = ((4gR(cos^2 φ−sin^2 φ)−glcos φ)/(l^2 /12+4R^2 cos^2 φ))  e^(∫Pdx) =e^(∫((4R^2 sin φcos φdφ)/(l^2 /12+4R^2 cos^2 φ)))       =−(1/(√(l^2 /12+4R^2 cos^2 φ)))  ⇒ −(z/(√(l^2 /12+4R^2 cos^2 φ)))      =∫(([4gR(cos^2 φ−sin^2 φ)−glcos φ]dφ)/((l^2 /12+4R^2 cos^2 φ)^(3/2) ))  .....

$${AP}\:=\mathrm{2}{R}\mathrm{cos}\:\frac{\theta}{\mathrm{2}}\:\:\:\:\:;\:\:{x}={l}−\mathrm{2}{R}\mathrm{cos}\:\frac{\theta}{\mathrm{2}} \\ $$$${let}\:\phi=\frac{\theta}{\mathrm{2}} \\ $$$$\tau_{{P}} =−\left({N}\mathrm{sin}\:\phi\right)\left({l}−{x}\right)+\left({mg}\mathrm{cos}\:\phi\right)\left(\frac{{l}}{\mathrm{2}}−{x}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....\left({i}\right) \\ $$$${Also}\:{along}\:{rod} \\ $$$$\:{N}\mathrm{cos}\:\phi−{mg}\mathrm{sin}\:\phi={m}\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }+{m}\omega^{\mathrm{2}} \left(\frac{{l}}{\mathrm{2}}−{x}\right)\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:........\left({ii}\right) \\ $$$${L}={I}_{{P}} \:\omega\:\:\:{let}\:\:\omega=\frac{{d}\phi}{{dt}}=\frac{{d}\left(\theta/\mathrm{2}\right)}{{dt}} \\ $$$${I}_{{P}} =\frac{{ml}^{\mathrm{2}} }{\mathrm{12}}+{m}\left(\frac{{l}}{\mathrm{2}}−{x}\right)^{\mathrm{2}} \\ $$$$\frac{{dL}}{{dt}}=\mathrm{2}\omega{m}\left(\frac{{l}}{\mathrm{2}}−{x}\right)\left(−\frac{{dx}}{{dt}}\right)+ \\ $$$$\:\:\:\:\:\:\:\:\left[\frac{{ml}^{\mathrm{2}} }{\mathrm{2}}+{m}\left(\frac{{l}}{\mathrm{2}}−{x}\right)^{\mathrm{2}} \right]\frac{{d}\omega}{{dt}}\:\:\:..\left({iii}\right) \\ $$$$\:\:{x}={l}−\mathrm{2}{R}\mathrm{cos}\:\phi \\ $$$$\Rightarrow\:−\frac{{dx}}{{dt}}=−\omega{R}\mathrm{sin}\:\phi \\ $$$$\:\:\:\:\:\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }=\omega^{\mathrm{2}} {R}\mathrm{cos}\:\phi \\ $$$${Now}\:{from}\:\left({ii}\right) \\ $$$$\:{N}\mathrm{cos}\:\phi−{mg}\mathrm{sin}\:\phi={m}\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }+{m}\omega^{\mathrm{2}} \left(\frac{{l}}{\mathrm{2}}−{x}\right) \\ $$$$\Rightarrow\:{N}={mg}\mathrm{tan}\:\phi+{m}\omega^{\mathrm{2}} {R}+{m}\omega^{\mathrm{2}} \left(\frac{{l}}{\mathrm{2}}−{x}\right)\mathrm{sec}\:\phi \\ $$$${using}\:{this}\:{and}\:\left({i}\right),\:\left({iii}\right)\:{in} \\ $$$$\:\:\:\frac{{dL}}{{dt}}=\tau_{{P}} \:\:\:\Rightarrow \\ $$$$\mathrm{2}\omega{m}\left(\frac{{l}}{\mathrm{2}}−{x}\right)\left(−\frac{{dx}}{{dt}}\right)+\left[\frac{{ml}^{\mathrm{2}} }{\mathrm{12}}+{m}\left(\frac{{l}}{\mathrm{2}}−{x}\right)^{\mathrm{2}} \right]\frac{{d}\omega}{{dt}} \\ $$$$=−\left({N}\mathrm{sin}\:\phi\right)\left({l}−{x}\right)+\left({mg}\mathrm{cos}\:\phi\right)\left(\frac{{l}}{\mathrm{2}}−{x}\right) \\ $$$$\Rightarrow \\ $$$$\mathrm{2}\omega\left(\mathrm{2}{R}\mathrm{cos}\:\phi−\frac{{l}}{\mathrm{2}}\right)\left(−\omega{R}\mathrm{sin}\:\phi\right) \\ $$$$+\left[\frac{{l}^{\mathrm{2}} }{\mathrm{12}}+\left(\mathrm{2}{R}\mathrm{cos}\:\phi\right)^{\mathrm{2}} \right]\frac{\omega{d}\omega}{{d}\phi}\:= \\ $$$$\:\:−\left\{{g}\mathrm{tan}\:\phi+\omega^{\mathrm{2}} {R}+\omega^{\mathrm{2}} \left(\mathrm{2}{R}\mathrm{cos}\:\phi−\frac{{l}}{\mathrm{2}}\right)\mathrm{sec}\:\phi\right\}\mathrm{sin}\:\phi\left(\mathrm{2}{R}\mathrm{cos}\:\phi\right) \\ $$$$\:\:\:\:+{g}\mathrm{cos}\:\phi\left(\mathrm{2}{R}\mathrm{cos}\:\phi−\frac{{l}}{\mathrm{2}}\right) \\ $$$$\Rightarrow \\ $$$$\left(\frac{{l}^{\mathrm{2}} }{\mathrm{12}}+\mathrm{4}{R}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \phi\right)\frac{\omega{d}\omega}{{d}\phi} \\ $$$$\:\:+\mathrm{2}\omega^{\mathrm{2}} {R}^{\mathrm{2}} \mathrm{sin}\:\phi\mathrm{cos}\:\phi\:= \\ $$$$\:\:\:\:\:\mathrm{2}{gR}\left(\mathrm{cos}\:^{\mathrm{2}} \phi−\mathrm{sin}\:^{\mathrm{2}} \phi\right)−\frac{{gl}\mathrm{cos}\:\phi}{\mathrm{2}} \\ $$$${let}\:\:\omega^{\mathrm{2}} ={z}\:\:\Rightarrow \\ $$$$\frac{{dz}}{{d}\phi}+\left(\frac{\mathrm{4}{R}^{\mathrm{2}} \mathrm{sin}\:\phi\mathrm{cos}\:\phi}{\frac{{l}^{\mathrm{2}} }{\mathrm{12}}+\mathrm{4}{R}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \phi}\right){z} \\ $$$$\:\:\:\:\:\:\:\:=\:\frac{\mathrm{4}{gR}\left(\mathrm{cos}\:^{\mathrm{2}} \phi−\mathrm{sin}\:^{\mathrm{2}} \phi\right)−{gl}\mathrm{cos}\:\phi}{{l}^{\mathrm{2}} /\mathrm{12}+\mathrm{4}{R}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \phi} \\ $$$${e}^{\int{Pdx}} ={e}^{\int\frac{\mathrm{4}{R}^{\mathrm{2}} \mathrm{sin}\:\phi\mathrm{cos}\:\phi{d}\phi}{{l}^{\mathrm{2}} /\mathrm{12}+\mathrm{4}{R}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \phi}} \\ $$$$\:\:\:\:=−\frac{\mathrm{1}}{\sqrt{{l}^{\mathrm{2}} /\mathrm{12}+\mathrm{4}{R}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \phi}} \\ $$$$\Rightarrow\:−\frac{{z}}{\sqrt{{l}^{\mathrm{2}} /\mathrm{12}+\mathrm{4}{R}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \phi}} \\ $$$$\:\:\:\:=\int\frac{\left[\mathrm{4}{gR}\left(\mathrm{cos}\:^{\mathrm{2}} \phi−\mathrm{sin}\:^{\mathrm{2}} \phi\right)−{gl}\mathrm{cos}\:\phi\right]{d}\phi}{\left({l}^{\mathrm{2}} /\mathrm{12}+\mathrm{4}{R}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \phi\right)^{\mathrm{3}/\mathrm{2}} } \\ $$$$..... \\ $$

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