Mass `m_(1)` moves on a slope making an angle `theta` with the horizontal and is attached to mass `m_(2)` by a string passing over a frictionless pulley as shown in The co-efficient of friction between `m_(1)` and the slopping surface is `mu` Which of the following statements are true ? .

Options

A. If `m_(2)gt m_(1) sin theta`, the body will move up the plane .
B. If `m_(2)lt m_(1) (sin theta+mu cos theta)` the body will move up the plane .
C. If `m_(2)lt m_(1) (sin theta-mu cos theta)`, the body will move up the plane .
D. If `m_(2) lt m_(1) (sin theta - mu cos theta) ` the body will move up the plane .

Answer

B, D

Explanation

Let `m_(1)` moves up plane Different forces involved are shown in the diagram
`N =`Normal reaction `f =` Frictional force `T =` Tension in the string
`f = mu N = mu m_(1) g cos thata`
For the system `(m_(1)+m_(2))` to move up `m_(2)g -(m_(1) g "sin" theta +f) gt 0`
`rArr m_(2) g - (m_(1) f "sin" theta + mu m_(1) g "cos" theta)gt0`
`rArr m_(2)lt m_(1) ("sin" theta + mu "cos" theta)`
Hence, option (b) is corrected
Let the body moves down the plane in this case of acts up the plane.
Hence `m_(1) g "sin" theta - f gt m_(2)g`
`rArr m_(1)g "sin" theta -mu m_(1) g "cos" theta lt m_(2)g`
`rArr m_(1) ("sin" theta - mu "cos" theta) gtm_(2)`
`rArr m_(2)lt m_(1) ("sin" theta - mu "cos" theta)`
Hence option (d) is correct
.