Solution - Two Swinging Ball

First of all, we have to find the position of center of mass (CM) of the balls. Because the balls are identical, their CM is at the middle of the rod. CM can be used for finding its potential energy and gravitational torque.

And then we need to find its angular velocity and angular acceleration when the CM forms an angle θ with horizontal. It's for determine tension of the strings.

We can use conservation of energy to find its angular velocity. (use pivot as relative point)






Because the energy is conserved,

And then we can use CM to find its angular acceleration relative to the pivot.

Angular acceleration relative to the pivot equals to angular acceleration relative to its CM. So we can say that total torque relative to CM will result the same angular acceleration.




By substituting α from equation (2), we can get

Now we use centripetal acceleration to get one more equation. We can substitute the two balls become one ball of mass 2m and located at the CM.

Substituting ω from equation (1) and solve the equation, we can get,

From equation (3) and (4), we can get,

Hence, the maximum value of the tensions are

That's my solution, if you have any question, correction, suggestion, or comment, please leave a comment below.
Thank you.