# Problem

Say I have the following equation.

$$y=f(\theta)$$

Where

$$\theta = \int\int\alpha$$

Is it possible to express the equation in terms of $$\alpha$$ and not the double integral of it?

# Origin

This problem originates from the problem shown in the image below.

Problem.jpg

Where the equations $$F_y=M_ysin(\frac{\pi}{2}-\theta)$$ and $$F_x=M_ycos(\frac{\pi}{2}-\theta)$$ both depend on the angle $$\theta$$ but where only the angular acceleration $$\alpha$$ is known.

• Very confusing question. Could you type in the original problem? Is it a pendulum? Is the force constant? What is exactly the question? – Andrei Apr 8 at 18:21
• @Andrei No, it's not confusing. How can $F_y=M_ysin(\frac{\pi}{2}-\theta)$ and the other equation be expressed in terms of $\alpha$ and not $\theta$. Simply put, remove the double integral. – Full_Nitrous Apr 8 at 20:43
• I meant that the problem in the image makes no sense to me. You have some formulas there, and some pictures, but I do not know what they represent – Andrei Apr 8 at 20:45
• @Andrei Oh, yes ok. The force $F$ is the force of a gimballed rocket engine with the angle $v$ mounted on a rocket body which has a orientation angle of $\theta$. The forces $F_y$ and $F_x$ are the translative forces resulting from the force $F$ disregarding that the rotation itself creates a force. – Full_Nitrous Apr 8 at 20:47

With the additional comments, the angular acceleration $$\alpha$$ is a constant (not depending of $$\theta$$). You can then write $$\theta=\theta_0+\omega_0t+\frac 12\alpha t^2$$ Here $$\theta_0$$ is the original angle of the rocket, and $$\omega_0$$ is the original angular velocity. $$t$$ is time.
• Using $t^2$ means the same as the integral. This does not solve the problem. – Full_Nitrous Apr 8 at 21:07
• Then the answer is no. Since $\theta$ depends on the time dependent value of $\alpha$, there is no other way to express $\theta$. – Andrei Apr 8 at 21:10
• The question is not how to express $\theta$ rather how to change the entire equation so that $\alpha$ can be used instead. – Full_Nitrous Apr 8 at 21:11
• Probably a good idea, but let me just try to clarify it now. For the equations of motion (in the differential form) you will write $\alpha$ as a function of $\theta$. That is $$\alpha=\ddot\theta=f(\theta)$$ – Andrei Apr 8 at 21:21