\[F=ma\]
for each particle:Particle 1:
\[m_1 \omega^2 r_1=T_1-T_2\]
Particle 2:
\[m_2 \omega^2 r_2=T_2\]
From the second of these, Particle 1:
\[T_2=m_2 \omega^2 r_2\]
Then from the first Particle 1:
\[T_1=m_1 \omega^2 r_1+T_2=m_1 \omega^2 r_1+m_2 \omega^2 r_2=\omega^2(m_1r_1+m_2r_2)\]