Two blocks of masses `m_(1)=1 kg and m_(2)=2 kg` are connected by a string and side down a plane inclined at an angle `theta=45^(@)` with the horizontal. The coefficient of sliding friction between `m_(1)` and plane is `mu_(1)=0.4,` and that between `m_(2)` and plane is `mu_(2)=0.2.` Calculate the common acceleration of the two blocks and the tension in the string.

Explanation

As `mu_(2)ltmu_(1)`. Block `m_(2)` has greater acceleration than `m_(1)` if we separately consider the motion of blocks. But they are connected so they move together as a system with common acceleration.
So acceleration of the bloks: `a=((m_(1)+m_(2))g sin theta-mu_(1)m_(2)g cos theta-mu_(2)m_(2)g cos theta)/(m_(1)+m_(2))`
`=((1+2)(10)((1)/sqrt2)-0.4xx1xx10xx(1)/sqrt(2)-0.2xx2xx10xx(1)/sqrt(2))/(1+2)=(22)/(3sqrt(2))ms^(-2)~
For block `m_(2)`: `m_(2)g sin theta-mu_(2)m_(2)g cos theta-T=m_(2)arArrT=m_(2)g sin theta-mu_(2)gcos theta-m_(2)a`
`=2xx10xx(1)/sqrt(2)-0.2xx2xx10xx(1)/sqrt(2)-2xx(22)/(3sqrt(2))=(2)/(3sqrt(2))N`