\(\sum F_y=0\)

\(\Rightarrow N =10 g \cos \theta / m _{ A }=10\,kg\)

\(\sum F_x=m_A \cdot a_A\)

\(\Rightarrow 10 g \cdot \sin \theta=10 . a _{ A }\)

\(\Rightarrow a_A=g \sin \theta---\) (i)

\(\sum F_y=0\)

\(N =20 g \cos \theta----\) (ii) \(/ m _{ B }=20\,kg\)

\(\sum F _{ x }= m _{ B } a _{ B }\)

\(\Rightarrow 20\, g \sin \theta- fk =20 a _{ B }\)

\(\Rightarrow 20\, g \sin \theta-\mu_{ k } \cdot 20 g \cos \theta=20 a _{ B } g\)

\(f _{ k }=\mu_{ k } \cdot N\)

\(=\mu_{ k } \cdot 20\, g \cos \theta\)

from \((ii)\)

\(\Rightarrow \sin \theta-\mu_{ k } \cdot \cos \theta=\frac{a_{ B }}{g}\)

\(\Rightarrow a _{ B }= g \left(\sin \theta- mu _{ k } \cos \theta\right)---\) (iii)

\((i) \div (iii):\)

\(\frac{a_A}{a_B}=\frac{g \sin \theta}{g\left(\sin \theta-\mu_k \cos \theta\right)}=\frac{\sin \theta}{\sin \theta\left(1-\mu_{ K } \cdot \cot \theta\right)}\)

\(\Rightarrow \frac{2}{1}=\frac{1}{1-\mu_{ k }} \Rightarrow\left(1-\mu_{ k }\right) 2=1\)

\(\theta=45^{\circ}\)

\(\cot 45^{\circ}=1\)

\(\Rightarrow 2-2 \mu_{ k }=1 \Rightarrow \mu_{ k }=0.5\)