**Friction
Acting on Multiple Surfaces**

**1. If Force Applied on Lower Block: **The maximum friction acting on m₂ is f₂ = µ₂m₂g. If the system moves with the common acceleration, then

F – µ₁ (m₁ + m₂) g = (m₁ + m₂) a and f₂ = m₂a

Maximum acceleration of m₂,

µ₂m₂g = m₂ a_{max}

∴
a_{max} = µ₂g.

**2. If Force Applied on Upper Block:**

f₂ = Limiting friction between m₁ and m₂.

f₁ = Limiting friction between the surface and m₁.

If F > f₂, then both block move with different acceleration and the maximum friction acts between the blocks.

F – f₂ = m₂a₂.

F – µ₂m₂g = m₂a₂ and N₂ = m₂g

N₁ = N₂ + m₁g = (m₁ + m₂) g

∴ f₁ = µ₁N₁ = µ₁ (m₁ + m₂) g and f₂ = µ₂m₂g

If f₂ < f₁ then m₁ remains at rest. If f₂ > f₁, then m₁ moves in the direction of f₂.

f₂ – f₁ = m₁ a₁.

If F < f₂ then no slipping is found between m₁ and m₂, i.e. m₁ and m₂ move together.

If F < f₁ then the system is in rest. If F > f₁, then the system moves with the common acceleration. In this case:

F – f₁ = (m₁ + m₂) g

F – µ₁ (m₁ + m₂) g = (m₁ + m₂) a.