Question

Two touching blocks $1$ and $2$ are placed on an inclined plane forming an angle $60^{\circ}$ with the horizontal. The masses are $m_{1}$ and $m_{2}$ and the coefficient of friction between the inclined plane and the two blocks are $1.5\, \mu$ and $1.0 \,\mu$, respectively. The force of reaction between the blocks during the motion is $(g=$ acceleration due to gravity)

• A. $\left(m_{2}-m_{1}\right) \mu g$
• B. $\left(m_{2}+m_{1}\right) \mu g$
• C. $\frac{1}{2} \frac{m_{1} m_{2}}{m_{1}+m_{2}} \mu g$
• D. $\frac{1}{4} \frac{m_{1} m_{2}}{m_{1}+m_{2}} \mu g$

Answer 1

Answer: (D)

According to the question, there are two blocks of masses $m_{1}$ and $m_{2}$ which are kept in contact on inclined plane of inclination $60^{\circ}$ as shown in the figure,

Coefficient of friction between the surface of plane and surface of first block, $\mu_{1}=1.5 \mu$ Coefficient of friction between the surface of plane and surface of second block, $\mu_{2}=1 \mu$
Now, free body diagram for the first block,

If $a$ be the common acceleration, then
$m_{1}\, g \,\sin\, 60^{\circ}-\mu_{1} \cdot R_{1}+R=m_{1} a$
$m_{1} \,g \frac{\sqrt{3}}{2}-1.5 \,\mu \,m_{1} g \,\cos\, 60^{\circ}+R=m_{1}\, a$
$m_{1}\, g \frac{\sqrt{3}}{2}-\frac{1.5 \,\mu\, m_{1}\, g}{2}+R=m_{1}\, a\,\,\,...(i)$
Similarly, for second body,

$m_{2} \,g \,\sin\, 60^{\circ}-\mu_{2} \,R_{2}-R=m_{2} a$
$m_{2} \,g \frac{\sqrt{3}}{2}-\mu\, m_{2} \,g \,\cos \,60^{\circ}-R=m_{2} a$
$m_{2} \,g \frac{\sqrt{3}}{2}-\frac{\mu \,m_{2} g}{2}-R=m_{2} a\,\,\,...(ii)$
From Eqs. (i) and (ii), we get
$\frac{m_{1} \,g \frac{\sqrt{3}}{2}-\frac{1.5 \,\mu \,m_{1}\, g}{2}+R}{m_{2} \,g \frac{\sqrt{3}}{2}-\frac{\mu\, m_{2} g}{2}-R}=\frac{m_{1} a}{m_{2} a}$
$\Rightarrow \frac{\sqrt{3}}{2} m_{1} \,m_{2} \,g-0.75 \,\mu\, m_{1} \,m_{2} \,g+m_{2} \,R $
$=\frac{\sqrt{3}}{2} m_{1}\, m_{2}\, g-0.5 \,\mu \,m_{1} \,m_{2}\, g-m_{1}\, R $
$ \Rightarrow \left(m_{1}+m_{2}\right) R =0.25 \,\mu \,m_{1} \,m_{2} \,g $
$\Rightarrow R=\frac{\mu\, m_{1} \,m_{2}\, g}{4\left(m_{1}+m_{2}\right)} $
Hence, the force of reaction between the blocks doing the motion is, $R=\frac{1}{4} \frac{m_{1}\, m_{2}}{\left(m_{1}+m_{2}\right)} \mu \,g$.