question_answer

Find the force F to be applied on M, so that m remains stationary with respect to M?

A)  \[\left( M+m \right)g\]                

B)  \[\left( M + m \right) g\,sin\,\theta \]

C)  \[\left( M+m \right)g\,cos\,\theta \]           

D)  \[\left( M+m \right)g\,\tan \,\theta \]

Correct Answer: D

Solution :

If we work in frame of wedge the free body diagram for small block is as follows For block to remain stationary \[\operatorname{ma}\,cos \theta  =mg\,sin \theta \] \[\Rightarrow ~~a=\,\,g\,\,tan\,\theta \]                                        ... (i) Now, we assume whole system \[\left( M+m \right)\] is moving with acceleration a \[\operatorname{F}=\left( M+m \right)a\,\,\Rightarrow \,\, a=\,\,\frac{F}{(M+m)}\]              ... (ii) By Eqs. (i) and (ii), we get \[\operatorname{F}=\,\,\left( M+m \right)g\,tan\,\theta \]