Question

In figure both the pulleys are massless and frictionless. A force $F$ (of any possible magnitude) is applied in horizontal direction. There is no friction between $M$ and the ground. ${\mu }_1$ and ${\mu }_2$ are coefficients of friction as shown between the blocks. Column I gives the different relations between ${\mu }_1$ and ${\mu }_2$ and Column II is regarding the motion of $M$. Match the columns.


$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{i. If}{\mu }_1={\mu }_2=0& \text{a. may accelerate towards right}\\ \text{ii. If}{\mu }_1={\mu }_2\ne 0& \text{b. may accelerate towards left}\\ \text{iii. If}{\mu }_1>{\mu }_2& \text{c. does not accelerate}\\ \text{iv. If}{\mu }_1<{\mu }_2& \text{d. may or may not accelerate}\end{array}$

• A. $\text{(i- c; ii- c; iii- b, d; iv- a, d)}$
• B. $\text{(i-c; ii- b; iii- c, d; iv- a, d)}$
• C. $\text{(i- b,d; ii- c; iii- b, d; iv- a, d)}$
• D. $\text{(i- a,c; ii- c; iii- d; iv- d)}$

Answer 1

The correct option is A $\text{(i- c; ii- c; iii- b, d; iv- a, d)}$

Friction force and tension acting between blocks and string are shown below,

F.B.D of lower block is,
So the movement of lower block depend on the value of ${\mu }_1$ and ${\mu }_2$.
If ${\mu }_1={\mu }_2=0,$ then $f_1$ and $f_2$ are also zero. Hence the block does not accelerate.
If ${\mu }_1={\mu }_2,$ then $f_1=f_2$ . In this case also the block does not accelerate.
If ${\mu }_1>{\mu }_2$,
Both kinetic and static friction come into picture, and we can say,
(1) Limiting friction on block in L.H.S $\ge$ R.H.S.
(2) Limiting friction in R.H.S $\ge$ kinetic friction in L.H.S of block.
(3) Kinetic friction in R.H.S of block is greater than L.H.S side of block.
From here we can conclude that
Block may accelerate towards left or may or may not accelerate.
Similarly, if ${\mu }_1<{\mu }_2$ , we can conclude that block may accelerate towards right or may or may not accelerate.