Masses `M_(1)` , `M_(2)` and `M_(3)` are connected by strings of negligible mass which pass over massless and frictionless pulleys `P_(1)` and `P_(2)` as shown in the fig. The masses move such that the portion of the string between `P_(1)` and `P_(2)` is parallel to the incline and the portion of the string between `P_(2)` and `M_(3)` is horizontal. The masses `M_(3)` and `M_(2)` are `4.0kg` each and the coefficient of kinetic friction between the masses and the surface is `0.25`. The inclined plane makes an angle of `37^(@)` with the horizontal. Find the mass `M_(1)`.

Explanation

Let the tension between the masses `M_(1)` and `M_(2)` be `T_(1)` and that between `M_(2)` and `M_(3)` be `T_(2)`. The kinetic friction between the masses and the surface is the `mu_(k)`. The equations of motion for different masses are,
`T_(1)-T_(2)-mu_(k)M_(2)gcostheta=M_(2)gsintheta`
`T_(1)=M_(1)g`
`M_(1)g-T_(2)-mu_(k)M_(2)gcostheta-M_(2)gsintheta=0`
`M_(1)xx9.8-9.8-0.25xx4xx9.8xxcos37-4xx9.8xxsin37=0`
`9.8M_(1)-9.8-7.82-23.59=0`
`9.8M_(1)=41.21`
`M_(1)=4.205kg`