Question

The ratio of acceleration of blocks $A$ placed on smooth incline with block $B$ placed on rough incline is $2:1$. The coefficient of kinetic friction between block $B$ and incline is:-

• A. $0.5$
• B. $0.75$
• C. $0.57$
• D. None of these

Answer 1

The correct option is A $0.5$

$\sum F_y=0$

$\Rightarrow N=10g\cos \theta /m_A=10kg$

$\sum F_x=m_A.a_A$

$\Rightarrow 10g.\sin \theta =10.a_A$

$\Rightarrow a_A=g\sin \theta ---\left(i\right)$

$\sum F_y=0$

$N=20g\cos \theta ----\left(ii\right)/m_B=20kg$

$\sum F_x=m_B{a}_B$

$\Rightarrow 20g\sin \theta -fk=20a_B$

$\Rightarrow 20g\sin \theta -{\mu }_k.20g\cos \theta =20a_{B}g$

$f_k={\mu }_k.N$

$={\mu }_k.20g\cos \theta$

from $\left(ii\right)$

$\Rightarrow \sin \theta -{\mu }_k.\cos \theta =\frac{a_B}{g}$

$\Rightarrow a_B=g(\sin \theta -m{u}_k\cos \theta )---\left(iii\right)$

$\left(i\right)÷\left(iii\right):$

$\frac{a_A}{a_B}=\frac{g\sin \theta }{g(\sin \theta -{\mu }_k\cos \theta )}=\frac{\sin \theta }{\sin \theta (1-{\mu }_k.\cot \theta )}$

$\Rightarrow \frac{2}{1}=\frac{1}{1-{\mu }_k}\Rightarrow (1-{\mu }_k)2=1$

$\theta ={45}^o$

$\cot {45}^o=1$

$\Rightarrow 2-2{\mu }_k=1\Rightarrow {\mu }_k=0.5$

Answer $A$