A block `A` of mass ` m` which is placed a rough inclined face of a wedge of same mass being pulled through light string and with force `F` as shown in figure The coefficient of friction between inclined face and block `A` is `mu ` while there is no friction between the ground and wedge it the whole system moves with same acceleration then find the value of `F`

Explanation

If the whole system moves together, than the acceleration of system,
`a = (F)/(2m)`
`T + mg sin theta = f + ma cos theta`

But `T = F `. Therefore
`f = F + mg sin theta - ma cos theta`
for no relative sliding `f lt mu N`
` F + mg sin theta - ma cos theta lt mu (mg cos theta + ma sin theta)`
`F + mg sin theta - m.((F)/(2m)) cos theta lt mu (mg cos theta + m((F)/(2m)) sin theta)`
` F lt (2mg (mu cos theta - sin theta))/((2 - cos theta - mu sin theta))`