  Question

There is no friction anywhere in the system shown in figure. The pulley is light. The wedge is free to move on the frictionless surface. A horizontal force F is applied on the system in such a way that m does not slide on M or both move together with some common acceleration. Given M>√2m. Match the entries of Column I with that of Column II. Column IColumn IIi.Pseudo-force acting on m as seen from the frame of M is a. equal tomFm+Mii.Pseudo-force acting on M as seen from the frame of m is b. greater thanmFm+Miii. Normal force (for θ=45∘)between m and M isc. less than mgsinθiv. Normal force between ground and M isd. greater than mgsinθ

A
i- a, c; ii- b, c; iii- b, c; iv- b, d  B
i- b, c; ii- a, c; iii- b, c; iv- b, d  C
i- b, c; ii- a, c; iii- b, d; iv- b, c  D
i- b, c; ii- b, d; iii- a, c; iv- b, c  Solution

The correct option is A i- a, c; ii- b, c; iii- b, c; iv- b, dAs the blocks move together with same acceleration: a=FM+m Let R be the reaction force on wedge by the ground. Here, R=(M+m)g The FBD of the wedge: The FBD of the block: For (i): Pseudo force acting on m: Fi=mFm+M Hence (a) matches. We can also say that, F=macosθ+mgsinθ F=Ficosθ+mgsinθ Fi(M+m)m=Ficosθ+mgsinθ By solving this, I can say that Fi<mgsinθ Hence (c) matches. For (ii): Pseudo force acting on M: Fii=Ma But, Ma>ma, so Fii>Fi Fii>mFm+M Fii=MFm+M Hence (b) matches. Moreover, lets take F=macosθ+mgsinθ F=mMacosθ/M+mgsinθ Fii(M+m)M=FiimcosθM+mgsinθ By solving this, we can say that Fii<mgsinθ Hence (c) matches. For (iii): (θ=45∘) If we balance forces in FBD of M, we get Nsinθ=Ma N/√2=Ma N=√2Ma N=√2Fii As, Fii>mFm+M N>mFm+M Hence (b) matches. Again, lets take previously derived equation Fii(M+m)M=FiimcosθM+mgsinθ But, Fii=N/√2 Hence, we get N(M+m)√2M=Nmcosθ√2M+mgsinθ By solving this, we can say that N<mgsinθ For (iv): We have R=(M+m)g R>mg R>mgsinθ Hence (d) matches. Moreover, we know that Fi<mgsinθ Hence, R>mgsinθ>Fi R>mFm+M Hence (b) matches. So combining all answers we can say option 'a' is correct. i- a, c; ii- b, c; iii- b, c; iv- b, dPhysics

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