In the co-efficinet of friction between the floor and the body `B` is 0.1. The co-efficient of friction beteen the bodies `B` and `A` is 0.2 A fore `F` is applied as shown `B` The mass of `A` is `m//2` and of `B` is m Which of the following statements are ture ?
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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 be rest if F = `0.1` mg
D. The maximum value of F for which the two bodies will move together is `0.45` mg

Answer

A, B, D

Explanation

Consider the adjacent diagram . Frictional force on `B(f_(1))` and frictional force on A `(f_(2))` will be as shown.
Let A and B are moving together `a_("common") = (F-f_(1))/(m_(A) + m_(B)) = (F - f_(1))/((m//2) + m) = (2(F- f_(1)))/( 3m)`
Pseudo force on A = `(m_(A)) xx a_("common")`
=` m_(A) xx (2(F- f_(1)))/(3m) = (m)/(2) xx (2(F-f_(1)))/(3m) = ((F - f_(1)))/(3)`

The force (F) will be maximum when
Pseudo force on A = Frictional force on A
`implies " " (F_("max") - f_(1))/(3) = mu m_(A) g `
= `0.2 xx (m)/(2) xx g = 0.1` mg
`implies " " F_("max") = 0.3 mg + f_(1)`
=`0.3` mg + `(0.1) (3)/(2) mg = 0.45` mg
`implies` Hence , maximum force upto which bodies will move together is `F_("max") = 0.45` mg
(a) Hence for F = `0.25` mg `lt F_("max")` bodies will move together .
(b) For F = `0.5 mg gt F_("max") ` , body A will slip with respect to B .
(c) For F = `0.5` mg `gt F_("max")` , bodies slip
`(f_(1))_("max") = mu M_( B) g = (0.1) xx (3)/(2) m xx g = 0.15` mg
`(f_(2))_("max") = mu m_(A) g = (0.2) ((m)/(2)) (g) = 0.1` mg
Hence , minimum force required for movement of the system (A + B)
`F_("min") = (f_(1))_("max") + (f_(2))_("max") `
= `0.15` mg + `0.1` mg = `0.25` mg
(d) Given , force F = `0.1` mg `lt F_("min")`
Hence , the bodies will be at rest .
(e) Maximum force for combined movement `F_("max") = 0.45` mg .