# Motion of two bodies one resting on the other – 2nd Condition

Motion of two bodies one resting on the other – 2nd 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 aA = 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 Fl = 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 Fl = μsmg.

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

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

Where, F’l = μsmg = Limiting friction between body A and B, both the body will move with different acceleration. Here force of kinetic friction μkmg 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 = μs (m + M) g = Limiting friction between body B and surface

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