# Dynamic equilibrium equations

The system represented in the figure consists of a 2 kg pulley to which two springs and a rigid square with 10 kg/m are connected, which in turn is articulated at point B (System is in the xz plane).

Dynamic equilibrium equations:

Can someone explain these 2 terms in yellow? It's making me a bit of confusion since I thought it should be "+" and not "-" and both terms should be equal to each other and not symmetrical

this was the free body diagramm that i draw

In other exercice that i done, they consider JTeta1 and JTeta2 in the opposite direction

• Could you please mark the variables $\theta_1$ and $\theta_2$ in the figure ? Preferably in such a way that the rotation corresponding to the positive value of the variables are clear. The confusion between + and - may go away if the positive sense of the rotations are clearly marked in the figure.
– AJN
Mar 6 at 2:41
• Already update it Mar 6 at 8:34
• can someone give me a help? Mar 6 at 12:34

1. The total term which includes the highlighted portion represents the torque exerted by the vertical spring.
2. The spring torque depends on spring force which in turn depends on the spring extension / compression.
3. If both ends of the spring move by equal amount, the spring doesn't get more compressed or extended and hence no change in the spring force. So the difference of the movement between the two ends of the spring determines the extension / compression of the spring.
4. The highlighted term gives the extension / compression of the spring. The following diagram shows the movement of the each side of the spring. The movement of the top end of the spring is $$r_2 \theta_1$$ positive in the downward direction due to the choice of the direction of positive $$\theta_1$$. The movement of the bottom end of the spring is $$2L \theta_2$$ which is also positive in the downward direction due to the choice of the direction of positive $$\theta_2$$. Hence the extension / compression of the spring is given by $$\mp(r_2 \theta_1 - 2L\theta_2)$$.
5. The term is the same in both equations (except for an overall sign as seen in your example) since the extension / compression of the spring is not dependent on which equation we are considering at any instant.

• could you explain why my free body diagramm was wrong? Mar 7 at 17:24
• I do not know what is wrong in the free body diagram; but a few things stand out. The $J \ddot\theta_1$ and $J \theta_2$ are marked in the opposite direction of the blue arrows representing $\theta_1$ and $\theta_2$. I don't know if that is a correct practice. Also why are there two forces marked at the top of the spring? $K x_2$ and $K 2 L \theta_2$ ? Wouldn't there be only a single spring force? How did you decide the direction of the arrows there?
– AJN
Mar 8 at 1:42
• Only one force (tension on the rope) needs to be marked in the location on the top of the spring. that tension depends on the compression/expansion of the spring. Expression for that is the yellow highlighted portion which you asked in the question
– AJN
Mar 8 at 1:47
• there is another exercice (i put in the original post) that they consider JTeta1 and JTeta2 in the opposite direction, that is why i done that. So that is is a wrong practice? Mar 9 at 6:51
• I am not sure. Irrespective of that, the expression for the compression/ extension of the spring is what is highlighted in yellow
– AJN
Mar 9 at 11:25