The correct option is
B gsin2θ(1+sin2θ)As mass
m can slide freely on the wedge. Hence, the block slides downwards with an acceleration
ar relative to the wedge and
(arcosθ−a) with respect to the ground in the horizontal direction.
Since, except gravity as there is no external force, hence to keep the COM of the system to be at rest, the wedge has to move leftward with an acceleration
a.
Since, net horizontal force on the system is zero. therefore, horizontal component of acceleration of COM of the system is zero.
Since there is no net force in horizontal direction, so
Fnet,horizontal=0
m(arcosθ−a)−ma=0
2ma=marcosθ⇒arcosθ=2a
a=arcosθ2 ........(1)
From, FBD of the wedge we get,
Fx=ma
Nsinθ=ma
N=masinθ ........(2)
From FBD of the block we get,
N+masinθ=mgcosθ ......(3)
Putting
(2) in
(3) we get,
masinθ+masinθ=mgcosθ
a(1+sin2θsinθ)=gcosθ
Using eq.
(1) in the above equation we get,
arcosθ2(1+sin2θsinθ)=gcosθ
ar=2gsinθ1+sin2θ
Therefore, acceleration of the block vertically downwards is,
ay,b=arsinθ
ay,b=2gsin2θ1+sin2θ
Since net force in horizontal direction or
x− direction is zero. So acceleration of centre of mass in
x− direction will be
∴ax,com=0
Using relation for acceleration of the centre of mass in
y− direction
ay,com=mway,w+mbay,bmw+mb [∵ ay,w=0]
ay,com=may,b2m=ay,b2
Thus, acceleration of COM of the system will be,
acom=ay,b2=gsin2θ(1+sin2θ)
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Hence, option
(B) is the correct answer.