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A block of mass m1 is attached to a cord of length L1, which is fixed at one end. The block moves in a horizontal circle on a frictionless tabletop. A second block of mass m2 is attached to the first by a cord of length L2 and also moves in a circle on the same frictionless tabletop, as shown below. If the period of the motion is T, find the tension in each cord in terms of the given symbols. (Use any variable or symbol stated above as necessary. Ignore the width of the blocks.)
T1 = (cord of length L1)

T2 = (cord of length L2)

Ben, are you writing a new physics textbook?  You have a lot of posts on this question and answer section.

Look at outerblock first.   T2 must counter the centifugal force on m2 so T2 = m2v2^2/(L1+L2).   How do we find V2?   Well we know that the period is T and that the mass must move in a circle with circumference (L1+L2)*2*pi.   So  v2 = (L1+L2)*2*pi/T.

Now let's look at the inner block.    The net Tension on the block must counter the centifugal force applied to the block of mass m1.   m1Vi^2/(L1) =  T1 - T2 or   T1 =   T2 + m1v1^2/(L1).   We can find v1 using the same technique used above   v1 = (2*p1*L1)/T.