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(C) For the first track of radius $R$ and speed $v$:Time period $T = \frac{2\pi R}{v} \Rightarrow v = \frac{2\pi R}{T}$.Centripetal acceleration $a_c

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$T/2$

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(C) For the first track of radius $R$ and speed $v$:Time period $T = \frac{2\pi R}{v} \Rightarrow v = \frac{2\pi R}{T}$.Centripetal acceleration $a_c = \frac{v^2}{R}$.For the second track of radius $R' = 2R$ and speed $v'$:New centripetal acceleration $a_c' = 8a_c = \frac{v'^2}{2R}$.Substituting $a_c = \frac{v^2}{R}$ into the equation: $\frac{v'^2}{2R} = 8 \left( \frac{v^2}{R} \right) \Rightarrow v'^2 = 16v^2 \Rightarrow v' = 4v$.The new time period $T'$ is given by:$T' = \frac{2\pi R'}{v'} = \frac{2\pi (2R)}{4v} = \frac{1}{2} \left( \frac{2\pi R}{v} \right) = \frac{T}{2}$.

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