**Motion of two bodies
one resting on the other – 2 ^{nd} Condition**

When a body A of mass m is resting on a body B of mass M then two conditions are possible:

**2) A force F is applied to the lower
body and then following four situations are possible**

**i) When there is no friction:**

a) B will move with acceleration while A will remain at rest as there is no pulling force on A.

\({{a}_{B}}=\left( \frac{F}{M} \right)\) and a_{A}
= 0.

b) As relative to B. A will move backwards with acceleration and so will fall from it in time t.

∴ \(t=\sqrt{\frac{2L}{a}}=\sqrt{\frac{2ML}{F}}\).

**ii) If friction is present between A
and B only and F < F’ _{l}:**

Where,
F’ = Pseudo force on body A and F_{l} = Limiting friction between body
A and B.

a) Both the body will move together with common acceleration, \(a=\frac{F}{M+m}\).

b)
Pseudo force on the body A, \({{F}^{‘}}=mq=\frac{mF}{m+M}\) and F_{l}
= μ_{s}mg.

c) \({{F}^{‘}}<{{F}_{l}}\Rightarrow \frac{mF}{m+M}<{{\mu }_{s}}mg\) ⇒ F < μ_{s} (m + M) g, so both bodies will move together with acceleration \({{a}_{A}}={{a}_{B}}=\frac{F}{m+M}\), if F < μ_{s} (m + M) g

**iii) If friction is present between A
and B only and F > F’ _{l}:**

Where,
F’_{l} = μ_{s}mg = Limiting friction between body A and B, both
the body will move with different acceleration. Here force of kinetic friction μ_{k}mg
will oppose the motion of B while will cause the motion of A.

Acceleration of body A relative to B will be: \(a={{a}_{A}}-{{a}_{B}}=-\left[ \frac{F-{{\mu }_{k}}g\left( m+M \right)}{M} \right]\), here -ve sign implies that relative to B, A will move backwards and will fall it after time.

\(t=\sqrt{\frac{2L}{a}}=\sqrt{\frac{2ML}{F-{{\mu }_{k}}g\left( m+M \right)}}\).

**iv) If there is friction between B and
floor and F > F’ _{l }:**

Where,
F’_{l} = μ_{s} (m + M) g = Limiting friction between body B and
surface

The
system will move only if F > F’_{l}then replacing F by F – F’_{l}. The entire case (iii) will
be valid. However, if F > F’_{l}the system will not move andfriction
between B and floor will be F while between A and B is zero.