Question

The friction coefficient between the two blocks shown in figure (6−E9) is μ but the floor is smooth. (a) What maximum horizontal force F can be applied without disturbing the equilibrium of the system? (b) Suppose the horizontal force applied is double of that found in part (a). Find the accelerations of the two masses.
Figure

Answer 1

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(a) According to the diagram of the two blocks, we have

$R_{1}=m g \quad \ldots \text { equation (i) }$

$F=\mu R_{1}+T \quad \ldots \text { equation (ii) }$

$T-\mu R_{1}=0 \quad . . \text { equation (iii) }$

We obtain from equations (i) and (ii),  

$T=F-\mu m g$        …(ii)

We obtain from equations (i) and (iii),  

$T=\mu m g$

In equation (ii) by putting $T=\mu m g$ we get

$F=\mu m g+\mu m g=2 \mu m g$

(b) From the upper block’s free body diagram, we get

$2 F-\mu m g-T=m a$      ….(i)

From the lower block’s free body diagram, we get

$T=\mu m g+M a$

In equation (i), putting the value of T, we obtain

$m a=2 F-\mu m g-M a-\mu m g$

We obtain by putting $F=2 \mu m g$

$\Rightarrow a(M+m)=4 \mu m g-2 \mu m g$

$\Rightarrow a=m+2 \mu m g M$ in opposite directions.

Answer 2

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