b
\({v_2} = {v_1}\frac{{{\alpha ^2}}}{\beta }\) \( \Rightarrow \,\,[{L_2}T_2^{ - 1}]\, = \,[{L_1}T_1^{ - 1}]\,\frac{{{\alpha ^2}}}{\beta }\)......\((i)\)

\({a_2} = {a_1}\alpha \beta \) \( \Rightarrow \,\,\,[{L_2}T_2^{ - 2}]\, = [{L_1}T_1^{ - 2}]\,\alpha \beta \)......\((ii)\)

\({F_2} = \frac{{{F_1}}}{{\alpha \beta }}\) \( \Rightarrow \,\,\,[{M_2}{L_2}T_2^{ - 2}]\, = \,[{M_1}{L_1}T_1^{ - 2}]\, \times \frac{1}{{\alpha \beta }}\)......\((iii)\)

Dividing equation \((iii)\)\ by equation \((ii)\) we get \({M_2} = \frac{{{M_1}}}{{(\alpha \beta )\,\alpha \beta }}\) \( = \frac{{{M_1}}}{{{\alpha ^2}{B^2}}}\)

Squaring equation \((i)\) and dividing by equation \((ii)\) we get \({L_2} = {L_1}\frac{{{\alpha ^3}}}{{{\beta ^3}}}\)

Dividing equation \((i)\) by equation \((ii)\) we get \({T_2} = {T_1}\frac{\alpha }{{{\beta ^2}}}\)