A constant force F pushes the block `m` till the wedge M starts sliding. If the stiffness of the light spring connecting M and m is K, coefficient of friction between block and wedge is `mu_1` and between the wedge and ground is `mu_2`, find the minimum value of the force F

Answer

`(mu_2^2(M+m)^2g^2-mu_1^2m^2g^2)/(2k)`

Explanation

Let `x_0` is the compression required so that wedge M just starts moving.

`mu_1mg+kx_0=mu_2(M+m)g`
`impliesx_0=(mu_2(M+m)g-mu_1mg)/(k)`
Now for block: `W_(t otal)=Deltak=k_2-k_1`
`impliesFx_0-mu_1mgx_0-1/2kx_0^2=0`
`impliesW_F=Fx_0=mu_1mgx_0+(kx_0^2)/(2)`
`impliesW_F=(mu_1mg+(kx_0)/(2))x_0=(mu_2^2(M+m)^2g^2-mu_1^2m^2g^2)/(2k)`