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) gt 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 down the plane

Answer

B, D

Explanation

Let `m_(1)` moves up the 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 theta`
For the system `(m_(1) + m_(2))` to move up
`m_(2) g - (m_(1) g sin theta + f) gt 0`
`implies " " m_(2)g - (m_(1) g sin theta + mu m_(1) g cos theta ) gt 0 `
`implies " " m_(2) gt m_(1) ( sin theta + mu cos theta)`

Hence , option (b) is correct
Let the body moves down the plane , in this case f acts up the plane .
Hence , `" " m_(1) g sin theta - f gt m_(2) g `
`implies " " m_(1) g sin theta - mu m_(1) g cos theta gt m_(2) g `
`implies " " m_(1) (sin theta - mu cos theta ) gt m_(2)`
`implies " " m_(2) lt m_(1) (sin theta - mu cos theta )`
Hence , option (d) is correct