A homogeneous shaft PQR with fixed supports at both ends is subjected to a torsional moment T at point Q, as shown in the figure. The polar moments of inertia of the portions PQ and QR of the shaft with circular cross-sections are $J_1$ and $J_2$ , respectively. The torsional moment reactions at the supports P and R are $T_p$ and $T_R$ , respectively. If $\frac {T_p}{T_R} = 4$ and $\frac {J_1}{J_2} = 2$ , the ratio of the lengths $\frac {L_1}{L_2}$ is

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A homogeneous shaft PQR with fixed supports at both ends is subjected to a torsional moment T at point Q, as shown in the figure. The polar moments of inertia of the portions PQ and QR of the shaft with circular cross-sections are $J_1$  and  $J_2$ , respectively. The torsional moment reactions at the supports P and R are  $T_p$  and  $T_R$ , respectively. If  $\frac {T_p}{T_R} = 4$  and  $\frac {J_1}{J_2} = 2$ , the ratio of the lengths  $\frac {L_1}{L_2}$  is

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