= Solution
Measure each <generalized coordinate> $\theta_i$ from an unstressed configuration. The diagram has one <torsional spring> joining the frame to object 1, one joining objects 1 and 2, and one joining objects 2 and 3. The <rotational kinetic energy> and <elastic potential energy> are
$$
T=\frac12\sum_{i=1}^3I_i\dot\theta_i^2,\qquad
U=\frac12k_1\theta_1^2+\frac12k_2(\theta_2-\theta_1)^2+\frac12k_3(\theta_3-\theta_2)^2.
$$
Each spring contribution follows by integrating its restoring <torque>: $\int_0^{\Delta\theta}k_j s\,ds=k_j(\Delta\theta)^2/2$. Thus
$$
\boxed{L=T-U.}
$$
The <Euler-Lagrange equations> give
$$
\boxed{\begin{aligned}
I_1\ddot\theta_1+(k_1+k_2)\theta_1-k_2\theta_2&=0,\\
I_2\ddot\theta_2-k_2\theta_1+(k_2+k_3)\theta_2-k_3\theta_3&=0,\\
I_3\ddot\theta_3-k_3\theta_2+k_3\theta_3&=0.
\end{aligned}}
$$
These hold for either interpretation of the square's <moment of inertia>; the geometry-based value from (a) is used first below.
Back to article page
