Question

Two blocks of masses 5 $kg$ and 3 $kg$ are placed in contact over an inclined surface of angle ${37}^{\circ }$ as shown in figure. ${\mu }_1$ is friction coefficient between 5 $kg$ block and the surface of the incline and similarly, ${\mu }_2$ is friction coefficient between the 3 $kg$ block and the surface of the incline. After the release of the blocks from the inclined surface,

• A. if ${\mu }_1=0.5$ and ${\mu }_2=0.3$, then 5 $kg$ block exerts 3 $N$ force on the 3 $kg$ block.
• B. if ${\mu }_1=0.5$ and ${\mu }_2=0.3$, then 5 $kg$ block exerts 8 $N$ force on the 3 $kg$ block.
• C. if ${\mu }_1=0.3$ and ${\mu }_2=0.5$, then 5 $kg$ block exerts 1 $N$ force on the 3 $kg$ block.
• D. if ${\mu }_1=0.3$ and ${\mu }_2=0.5$, then 5 $kg$ block exerts no force on the 3 $kg$ block.

Answer 1

Answer: (A), (D)

The correct options are
A if ${\mu }_1=0.5$ and ${\mu }_2=0.3$, then 5 $kg$ block exerts 3 $N$ force on the 3 $kg$ block.
D if ${\mu }_1=0.3$ and ${\mu }_2=0.5$, then 5 $kg$ block exerts no force on the 3 $kg$ block.

Case I : ${\mu }_1=0.5,{\mu }_2=0.3$
In this case,
Along the incline, acceleration of 5 $kg$ block will be less than the acceleration of 3 $kg$ block provided they move alone on the incline. The reason is, greater friction coefficient of 5 kg block. The acceleration along the incline is $gsin\theta -\mu gcos\theta$.
Hence, 5 $kg$ block will apply normal force on 3 $kg$ block. both blocks will move together.
In this case, FBDs of both blocks are shown.
For 5 $kg$ block,

$m_{1}gsin\theta +N-{\mu }_1{m}_{1}gcos\theta =m_{1}a$

For $3kg$ block,
$m_{2}gsin\theta -N-{\mu }_2{m}_{2}gcos\theta =m_{2}a$

In this case, FBDs of both blocks are shown.
For 5 $kg$ block,

$m_{1}gsin\theta +N-{\mu }_1{m}_{1}gcos\theta =m_{1}a$

For $3kg$ block,
$m_{2}gsin\theta -N-{\mu }_2{m}_{2}gcos\theta =m_{2}a$

On solving we get, N $=3N$.

Case II: ${\mu }_1=0.3,{\mu }_2=0.5$
In this case, friction on of 5 $kg$ is lesser than the block of 3 $kg$. So, the acceleration of block of 5 $kg$ will be greater as compari to the 3 $kg$ block. So, they will lose contact of each other and the value of normal force will be zero.