Question

Find the acceleration of the $3m$ block when the system is released from rest. All other surfaces are smooth except between $m$ and $2m$. Take $g=10m/s^2$

• A. $a=3m/s^2$
• B. $a=3.5m/s^2$
• C. $a=5m/s^2$
• D. $a=5.4m/s^2$

Answer 1

The correct option is D $a=5.4m/s^2$

The FBDs of the blocks are as shown


From the FBDs, we have
$N_1=mg$
$f_{max}=\mu N_1=0.3mg$ and
$T=3mg$

Assuming the system to move together with common acceleration, then it is given by
$a=\frac{\text{(Supporting Force-Opposing Force)}}{\text{Total mass}}$
$=\frac{3mg}{6m}=\frac{g}{2}$
(since friction is an internal force)

From the FBD of block $m$ we get
$f=m{a}_m=\frac{mg}{2}=0.5mg$

Since, $f>f_{max}$, our assumption is wrong.
The two blocks will separate and move with different accelerations.
So, acceleration of $m$ is $a_m=\frac{f_{max}}{m}=\frac{0.3mg}{m}=0.3g=3m/s^2$

The acceleration of $2m$ and $3m$ will be same. So, from their FBDs, we have
$T-f_{max}=2m{a}_1\Rightarrow T-0.3mg=2m{a}_1$
$3mg-T=3m{a}_1$

From above two equations, we get
$2.7mg=5m{a}_1\Rightarrow a_1=\frac{2.7g}{5}$
$\Rightarrow a_1=\frac{27}{5}=5.4m/s^2$
Hence, the acceleration of $3m$ is $5.4m/s^2$