Question

Consider the figure shown below. If all the surfaces are smooth except that between mass $m$ and $2m$ then the acceleration (in ${\text{m/s}}^2$) of the mass $m$ is

• A. $\frac{2g}{5}$
• B. $\frac{g}{5}$
• C. $\frac{3g}{5}$
• D. $\frac{4g}{5}$

Answer 1

The correct option is A $\frac{2g}{5}$

The FBDs of the blocks are


From the FBDs, we have

$N_1=mg$

Also, $f_{\text{1 max}}=\mu N_1=\mu mg=\frac{mg}{2}$

Assuming the three masses as a system. Let the system move together with common acceleration $a$,

$a=\frac{\text{(Supporting Force - Opposing Force)}}{\text{Total mass}}$

$a=\frac{2mg}{5m}=\frac{2g}{5}$

(because friction $f_1$ and tension $T$ are internal forces for the system, it is not considered)

For mass $m$:

$f_1=ma=\frac{2mg}{5}$

$\because f_1<f_{\text{1 max}}$, our assumption is right.
Hence, they will move together with acceleration $a=\frac{2g}{5}$