In the coefficient of friction between the floor and the body B is 0.1. The coefficient of friction between the bodies B and A is 0.2. A force F is applied as shown on B. The mass of m A is `1/2`and of B is m. Which of the following statements are true ?

Options

A. The bodies will move together if F = 0.25 mg
B. The body A will slip with respect to B if F = 0.5 mg
C. The bodies will move together if F = 0.5 mg
D. The bodies will be at rest if F = 0.1 mg

Answer

A, B, D

Explanation

Consider diagram shown in figure
Let frictional force on B be `F_(1)` and friction
Force on A be `F_(2)`
F= external force

Common acceleration of system be a
`a= (F-f_(1))/(M_(a) +m_(3)) = (F-F_(1))/((m)/(2)+m) = 2(F-f_(1))/(3m)`
Pseudo force on A
The force ( f) will be maximum when pseudo force on A= frictional force on A
`=0.2 ((m)/(2))g`
`F_(max) - F_(1) = 3 xx 0.1 mg = 0 .3 mg`
`=0.3 mg + (0-1) ((3)/(2)) mg`
Hence maximum force upto which bodies will move together is `F_(max) = 0.45 mg`
( a) Hence for F= 0.15 `lt F_(max)` hodies will moves together
(b ) For F = 0 .5 mg `gt F_(max)` body A will slip with respect to B
( d ) Given force F=- 0.1 mg `lt F`
System (both block ) will be at rest
(e ) Maximum force required tomove both block
`F_(max) = 0.45 mg`