Which strategy should I use to approach a problem? Sometimes I'll use $U= 1/2kx^2$ when I should use $F=-kx$ and vice versa. What are clues in a problem that would let me identify the correct approach to a given problem? And shouldn't both approaches give me the same answer in any scenario?
[Physics] Spring force vs Spring Potential Energy
newtonian-mechanicsspring
Related Solutions
If you solve for $t$ in Eq. (5.1), and plug that into equation (1.1), you'll see that the solution looks like $x_B \propto v_A^2 sin(\theta) cos(\theta)$. The function on the right is symmetric about $\pi/4$, thus, as long as $\theta$ doesn't equal $\pi/4$, there will be two solutions (symmetrically about $\pi/4$). Of course, in general, there could be one, two, or no solutions (see figure below).
While you have 2-equations and 2-unknowns, the periodic functions introduce some degeneracy into the problem---allowing multiple solutions.
As Farcher points out, the system is subject to a constant unbalanced force, so the resulting motion is constant acceleration. It is never in equilibrium : it is neither static nor moving with constant speed.
Your eqn 1 is correct : $K_1x_1=K_2x_2$. However, the tension in the springs is constant, so $x_1$ and $x_2$ are constants, and when you differentiate they vanish. $x_1$ and $x_2$ are not measures of the positions of the two masses, so differentiating them does not give you the linear acceleration of each mass.
Perhaps your difficulty is caused by the ideal conditions required for the two masses to move with constant acceleration. Unless the masses are released very carefully, and the pulley rotates very smoothly, any small jerk in the motion will be amplified by the springs : tension and extension will vary with time. The resulting motion of each mass will not be constant acceleration, even if the motion of the $CM$ is. The 'solution' of constant acceleration is an unstable one. I find it intuitive that the masses will oscillate as the heavier one descends.
Best Answer
Obviously both are equivalent, in terms of applicability (you get the second directly by taking the gradient of the first).
My "rule of thumb" is that for simple exercices,
Of course that's only a way to choose one way to start solving a given problem; most of the time both are perfectly usable.
When going to more complicated problems however this "rule" is not necessary true ; in any case you should choose the one that gives you as little unnecessary information as possible - while still answering the question.