First slide

Two block problem

Question

A block of mass m is placed on another blocks of mass M which itself is lying on a horizontal surface. The coefficient of friction between the two blocks is
µ₁ and that between the block of mass M and horizontal surface is pr. What maximum horizontal force can be applied to the lower block so that the two blocks move without separation ?

Moderate

A

$(M + m) (μ_{2} - μ_{1}) g$
B

$(M - m) (μ_{2} - μ_{1}) g$
C

$(M - m) (μ_{2} + μ_{1}) g$
D

$(M + m) (μ_{2} + μ_{1}) g$

Solution

Here the force applied should be such that the force on block of mass m should not exceed from the force of friction (µ₁ m g). Let the system move with acceleration
a. Then
$F - μ_{2} (M + m) g = (M + m) a$ …(1)
For block of mass m,
$F_{1} = ma = μ_{1} mg or a = μ_{1} g$ …(2)
From eqs. (1) and (2), we get
$\begin{aligned} F & = μ_{2} (M + m) g + (M + m) μ_{1} g \\ = (M + m) g (μ_{1} + μ_{2}) \end{aligned}$

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