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

The correct option is A i- a, c; ii- b, c; iii- b, c; iv- b, d
As 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=Nmcosθ2M+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, d

Physics

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