Question

Two containers of sand are arranged like the block as shown in figure. The containers alone have negligible mass; the sand in these containers has a total mass $M_{tot}$; the sand in the hanging container $H$ has mass $m$.
To measure the magnitude of the acceleration of the system, a large number of experiments have been carried out where $m$ varies from experiment to experiment but $M_{tot}$ does not; that is, sand is shifted between the containers before each trial.

Which of the curves in the graph correctly gives the acceleration magnitude as a function of the ratio $m/M_{tot}$ (vertical axis is for acceleration)?

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

For calculating the equation of graph, we need the relation between acceleration of blocks and $\frac{m}{M_{tot}}$.
Direction and magnitude of acceleration are shown below.


F.B.D of blocks are shown below.


For $M$ mass block,
Using Newton's second law,
$T=Ma$ ----------(1)

Similarly for $m$ mass block we can write,
$mg-T=ma$

Replce the expression of $T$ from equation (1),

$mg-Ma=ma$
$⟹a=g-\frac{Ma}{m}$
$a[1+\frac{M}{m}]=g$
$⟹a=\frac{g}{1+\frac{M}{m}}=\frac{mg}{M+m}$

From question,
$M_{tot}=m+M$
Hence,
$a=\frac{mg}{M_{tot}}$--------(2)

We can treat $a$ as dependent variable ($y$) and $\frac{m}{M_{tot}}$ as independent variable ($x$).
$\therefore$ equation (2) is a equation of straight line in the form of $y=mx$. Hence graph number 3 is the correct answer to this question.