Differential equation of spring-mass system attached to one end of seesaw

[Tahir Mushtaq]

please help me to find differential equation of spring-mass system attached to one end of seesaw.

case 1: Seesaw is balanced with its fulcrum point or pivotal point. At one end of seesaw, the spring (with spring constant K1) is attached. Now the seesaw has total mass M which is attached to spring, form a spring-mass system. I confuse on this point that the oscillation of seesaw will be around pivotal point. Then what is differential equation of spring-mass system.

case 2: The same case with both ends of Seesaw is attached with springs having spring constant K1 and K2. Then what is differential equation of this system.

 

[kyiydnlm]

$$\frac{L^2}{4} \theta K+J\ddot{\theta} = 0$$
$$J = \frac{1}{12}ML^2$$
case 1: K = K1
case 2: K = K1+K2

Assuming uniform seesaw, pivot is center of mass (also geometry center). If otherwise seesaw is not uniform, J will change accordingly. If pivot is not geometry center, the equation will be slightly different.