Question

The mass of wedge, shown in $Fig.6.157$, is M and that of the block is m. Neglecting friction at all the places and mass of the pulley, calculate the acceleration of wedge. Thread is inextensible.

Answer 1

Constraint relation:

It is clear from above discussion,

$a_x=A(1-\cos \theta )$ and $a_y=A\sin \theta$

Equations of motion

Considering the motion of wedge in horizontal direction.

For wedge :

$T-T\cos \theta +N\sin \theta =MA$........(i)

Considering the motion of block parallel and perpendicular to the sloping side in horizontal direction.

For block : In the direction parallel to inclined surface,

$mg\sin \theta -T={ma}_x=MA(1-\cos \theta )$

or $T=mg\sin \theta =Ma(1-\cos \theta )$.......(ii)

In the direction perpendicular to inclined plane,

$mg\cos \theta -N={ma}_y=mA\sin \theta$

$N=mg\cos \theta =mA\sin \theta$.......(iii)

Substituting the value of N from (ii) and (iii) in (i), we get,

$A=\frac{mg\sin \theta }{M+2m(1-\cos \theta )}$