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Question Number 68342 by Tony Lin last updated on 09/Sep/19

Commented by Tony Lin last updated on 09/Sep/19

$$whenmslidetothebottom$$$$ifthereisnofraction$$$$find{v}_m\mathrm{\&}{v}_M$$$$find{a}_m\mathrm{\&}{a}_M$$

Answered by mr W last updated on 09/Sep/19

Commented by mr W last updated on 10/Sep/19

$$METHOD1$$$$positionofblockM:x$$$$a_M=\frac{d^{2}x}{{dt}^2}$$$$positionofblockm:$$$$x_m=x+s\cos \theta$$$$y_m=s\sin \theta$$$$a_{m,x}=a_M+a\cos \theta witha=\frac{d^{2}s}{{dt}^2}$$$$a_{m,y}=a\sin \theta$$$$$$$${Ma}_M=N\sin \theta$$$$\Rightarrow N=\frac{{Ma}_M}{\sin \theta }$$$${ma}_{m,x}=-N\sin \theta$$$$m(a_M+a\cos \theta )=-N\sin \theta$$$${Ma}_M=N\sin \theta$$$$\Rightarrow (\frac{M}{m}+1)a_M+a\cos \theta =0...\left(i\right)$$$$$$$${ma}_{m,y}=-mg+N\cos \theta$$$$ma\sin \theta =-mg+N\cos \theta$$$$\Rightarrow \frac{M}{m}\times \frac{a_M}{\tan \theta }-a\sin \theta =g...\left(ii\right)$$$$$$$$\Rightarrow a_M=\frac{g\sin \theta \cos \theta }{\frac{M}{m}+{\sin }^2\theta }$$$$\Rightarrow a=-\frac{g(\frac{M}{m}+1)\sin \theta }{\frac{M}{m}+{\sin }^2\theta }$$$$\Rightarrow a_{m,x}=-\frac{g\frac{M}{m}\sin \theta \cos \theta }{\frac{M}{m}+{\sin }^2\theta }$$$$\Rightarrow a_{m,y}=-\frac{g(\frac{M}{m}+1){\sin }^2\theta }{\frac{M}{m}+{\sin }^2\theta }$$$$v_{m,y}=\sqrt{2{ha}_{m,y}}$$$$\Rightarrow v_{m,y}=\sin \theta \sqrt{\frac{2hg(\frac{M}{m}+1)}{\frac{M}{m}+{\sin }^2\theta }}$$$$$$$$v_{m,x}=\sqrt{\frac{2{ha}_{m,x}}{\tan \theta }}$$$$\Rightarrow v_{m,x}=\cos \theta \sqrt{\frac{2hg\frac{M}{m}}{\frac{M}{m}+{\sin }^2\theta }}$$

Commented by Tony Lin last updated on 10/Sep/19

$$thankssir$$

Answered by mr W last updated on 09/Sep/19

Commented by mr W last updated on 09/Sep/19

$$METHOD2$$$${Ma}_M=N\sin \theta$$$$\Rightarrow N=\frac{{Ma}_M}{\sin \theta }$$$$$$$$N+{ma}_M\sin \theta =mg\cos \theta$$$$\frac{{Ma}_M}{\sin \theta }+{ma}_M\sin \theta =mg\cos \theta$$$$(\frac{M}{m}+{\sin }^2\theta )a_M=g\sin \theta \cos \theta$$$$\Rightarrow a_M=\frac{g\sin \theta \cos \theta }{\frac{M}{m}+{\sin }^2\theta }$$$$$$$${ma}_M\cos \theta +mg\sin \theta =ma$$$$a=a_M\cos \theta +g\sin \theta$$$$\Rightarrow a=\frac{g(\frac{M}{m}+1)\sin \theta }{\frac{M}{m}+{\sin }^2\theta }$$$$a_{m,x}=a\cos \theta -a_M=\frac{g(\frac{M}{m}+1)\sin \theta \cos \theta }{\frac{M}{m}+{\sin }^2\theta }-\frac{g\sin \theta \cos \theta }{\frac{M}{m}+{\sin }^2\theta }$$$$\Rightarrow a_{m,x}=\frac{\frac{M}{m}g\sin \theta \cos \theta }{\frac{M}{m}+{\sin }^2\theta }$$$$a_{m,y}=a\sin \theta$$$$\Rightarrow a_{m,y}=\frac{g(\frac{M}{m}+1){\sin }^2\theta }{\frac{M}{m}+{\sin }^2\theta }$$

Commented by Tony Lin last updated on 10/Sep/19

$$thankssir$$

Answered by Tanmay chaudhury last updated on 09/Sep/19

Commented by Tanmay chaudhury last updated on 09/Sep/19

$$Mg+Ncos\theta =N_G$$$$MA=Nsin\theta$$$$form$$$$mgsin\theta +mAcos\theta =ma$$$$mgcos\theta =N+mAsin\theta$$$$MA=sin\theta (mgcos\theta -mAsin\theta )$$$$A(M+{msin}^2\theta )=mgsin\theta cos\theta$$$$A=\frac{mgsin\theta cos\theta }{M+{msin}^2\theta }$$$$A=accofwedge$$$$a=gsin\theta +\left(\frac{mgsin\theta cos\theta }{M+{msin}^2\theta }\right)cos\theta$$$$a=\frac{Mgsin\theta +{mgsin}^3\theta +mgsin\theta {cos}^2\theta }{M+{msin}^2\theta }$$$$a=\frac{Mgsin\theta +mgsin\theta ({sin}^2\theta +{cos}^2\theta )}{M+{msin}^2\theta }$$$$a=\frac{(M+m)gsin\theta }{M+{msin}^2\theta }$$$$$$$$$$

Commented by Tony Lin last updated on 10/Sep/19

$$thankssir$$

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