Part(a)

Equation of motion for the man on the platform is T + n - Mg =Ma

Equation of motion for the platform is T - n - mg =ma

add the above two equations to eliminate n then we will get

\(\displaystyle{T}={\frac{{{\left({M}+{m}\right)}{\left({a}+{g}\right)}}}{{{2}}}}\)

Part(b)

acceleration of the rope = -2 x acceleration of theplatform

\(\displaystyle{a}_{{{r}{o}{p}{e}}}=-{2}\times{1.80}=-{3.6}\frac{{m}}{{s}^{{2}}}\)

\(\displaystyle{a}_{{{r}{o}{p}{e}}}\) with respect to man is \(\displaystyle=-{3.6}-{1.8}=-{5.4}\frac{{m}}{{s}^{{2}}}\)