Block `A` of mass `m` and block `B` of mass `2m` are placed on a fixed triangular wedge by means of a light and inextensible string and a frictionless pulley as shown in fig . The wedge is inclined at `45^(@)` to the horizontal on both sides . The coefficient of friction between the block `A`and the wedge is `2//3` and that between the block `B` and the wedge is `1//3` .If the system of `A and B` is released from rest then find .
a. the acceleration of `A`
b. tension in the string
c.the magnitude and direction of the frictional force acting on `A`

Options

A. `zero`
B. `(2m^(2))/(3)g`
C. `(4m^(2))/(3)g`
D. `(m^(2))/(sqrt2)g`

Answer

a

Explanation

Effective pulling force on masses along the plane when mass `2m` moves down and mass m moves up the plane is
`F_(eff) = 2mg sin 45^(@) - mg sin 45^(@)`
`= mg//sqrt2=sqrt2mg//2`
Effective force of friction on two masses is
`F_("friction") = f_(1) + f_(2) = mu_(1) mg cos theta + mu_(2) 2mg cos theta`
`=(2)/(3)mg 1/sqrt2 + (1)/(3)xx2mgxx(1)/sqrt2`
`=mg[sqrt2/(3) + sqrt2/(3)] =(2sqrt2)/(3) mg`
As `F_("eff") lt F_("friction")` so the masses will not move Hence the acceleration of the system will be zero
.