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Question Number 46180 by ajfour last updated on 22/Oct/18

Commented by ajfour last updated on 22/Oct/18

If the stick is released in the position as shown in diagram, end A slides on yz wall then loses contact with it at P. Find coordinates of P. Assume all surfaces frictionless.

$${If}\:{the}\:{stick}\:{is}\:{released}\:{in}\:{the} \\ $$$${position}\:{as}\:{shown}\:{in}\:{diagram}, \\ $$$${end}\:{A}\:\:{slides}\:{on}\:{yz}\:{wall}\:{then} \\ $$$${loses}\:{contact}\:{with}\:{it}\:{at}\:{P}.\:{Find} \\ $$$${coordinates}\:{of}\:{P}.\:{Assume}\:{all} \\ $$$${surfaces}\:{frictionless}. \\ $$

Commented by ajfour last updated on 22/Oct/18

Commented by ajfour last updated on 22/Oct/18

Let A(0,y,z) ; B(x,0,0) N_1 = m((d^2 (x/2))/dt^2 ) = ((ma_1 )/2) ....(i) N_2 = ((ma_2 )/2) ; N_3 −mg = ((ma_3 )/2) ... (ii)&(iii) x= lcos θcos φ ; y= lcos θsin φ ; z = lsin θ ....(iv),(v),(vi) (N_1 sin φ−N_2 cos φ)(l/2)cos θ = (d/dt)(((ml^2 ω_φ cos^2 θ)/(12))) ....(vii) (N_1 cos φ+N_2 sin φ)(l/2)sin θ −N_3 ((l/2)cos θ)= (((ml^2 )/(12)))(d^2 θ/dt^2 ) ..(viii) unknowns N_1 , N_2 , N_3 , x, y, z, θ, φ .

$${Let}\:\:\:\:{A}\left(\mathrm{0},{y},{z}\right)\:\:;\:\:{B}\left({x},\mathrm{0},\mathrm{0}\right) \\ $$$${N}_{\mathrm{1}} =\:{m}\frac{{d}^{\mathrm{2}} \left({x}/\mathrm{2}\right)}{{dt}^{\mathrm{2}} }\:=\:\frac{{ma}_{\mathrm{1}} }{\mathrm{2}}\:\:\:\:\:\:....\left({i}\right) \\ $$$${N}_{\mathrm{2}} =\:\frac{{ma}_{\mathrm{2}} }{\mathrm{2}}\:\:;\:{N}_{\mathrm{3}} −{mg}\:=\:\frac{{ma}_{\mathrm{3}} }{\mathrm{2}}\:\:...\:\left({ii}\right)\&\left({iii}\right) \\ $$$$\:\:\:\:{x}=\:{l}\mathrm{cos}\:\theta\mathrm{cos}\:\phi\:; \\ $$$$\:\:\:\:{y}=\:{l}\mathrm{cos}\:\theta\mathrm{sin}\:\phi\:\:;\:\:{z}\:=\:{l}\mathrm{sin}\:\theta \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\left({iv}\right),\left({v}\right),\left({vi}\right) \\ $$$$\:\:\left({N}_{\mathrm{1}} \mathrm{sin}\:\phi−{N}_{\mathrm{2}} \mathrm{cos}\:\phi\right)\frac{{l}}{\mathrm{2}}\mathrm{cos}\:\theta \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\frac{{d}}{{dt}}\left(\frac{{ml}^{\mathrm{2}} \omega_{\phi} \mathrm{cos}\:^{\mathrm{2}} \theta}{\mathrm{12}}\right)\:\:\:\:....\left({vii}\right) \\ $$$$\left({N}_{\mathrm{1}} \mathrm{cos}\:\phi+{N}_{\mathrm{2}} \mathrm{sin}\:\phi\right)\frac{{l}}{\mathrm{2}}\mathrm{sin}\:\theta \\ $$$$\:\:−{N}_{\mathrm{3}} \left(\frac{{l}}{\mathrm{2}}\mathrm{cos}\:\theta\right)=\:\left(\frac{{ml}^{\mathrm{2}} }{\mathrm{12}}\right)\frac{{d}^{\mathrm{2}} \theta}{{dt}^{\mathrm{2}} }\:\:\:\:\:..\left({viii}\right) \\ $$$${unknowns}\:\:{N}_{\mathrm{1}} ,\:{N}_{\mathrm{2}} ,\:{N}_{\mathrm{3}} ,\:{x},\:{y},\:{z}, \\ $$$$\:\:\:\theta,\:\phi\:. \\ $$

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