Pendulums in inertial and non-inertial frames

Question

If I have a non-inertial frame accelerating with acceleration $A$ with respect to an inertial frame and if I have two pendulums, one in the inertial frame and the other in the non inertial frame, then how will the motion of the pendulum differ when seen from each frame? The pendulum in the inertial frame should oscillate about the vertical and the pendulum in the non-inertial frame should oscillate about an angle $\phi = \arctan\left(\frac{A}{g}\right)$ with the vertical. I think both observers should agree with that? But if I represent the coordinates in the inertial frame as unprimed coordinates and the coordinates in the non-inertial frame as primed coordinates then I have $$a' = a - A$$ which means that the two observers should disagree about the angular acceleration $\ddot{\theta}$ which depends on the tangential acceleration of the pendulums. But do they really disagree? Or am I missing something?

Answer 1

The EOM's

with $x'=v(\tau)\,\tau+x$ where $v(\tau)$ is the velocity between $x'$ and $x$

you obtain the kinetic energy and the potential energy of a pendulum that move in the prime system.

Pendulum position vector

$$\vec{R}=\left[ \begin {array}{c} v \left( \tau \right) \tau+L\sin \left( \varphi \right) \\L\cos \left( \varphi \right) \end {array} \right] $$

thus:

Kinetic energy

$$T=\frac{m}{2}\vec{\dot{R}}\cdot\vec{\dot{R}} $$

Potential Energy

$$U=-m\,g\,L\cos(\phi)$$

and the the equation of motion

$$\ddot{\varphi}+{\frac {g\sin \left( \varphi \right) }{L}}+{\frac {\cos \left( \varphi \right) \left( {\frac {d^{2}}{d{\tau}^{2}}}v \left( \tau \right) \right) \tau+2\,\cos \left( \varphi \right) {\frac {d}{d \tau}}v \left( \tau \right) }{L}}=0 \tag 1$$

with $v(\tau)=-v_0-\frac{1}{2}\,A\,\tau\quad$ in equation (1)

$${\frac {d^{2}}{d{\tau}^{2}}}\varphi \left( \tau \right) +{\frac {\sin \left( \varphi \left( \tau \right) \right) g}{L}}-\,{\frac {\cos \left( \varphi \left( \tau \right) \right) A}{L}} =0\tag 2$$

Simulation

Thus:

The acceleration parameter $A$ dose't affected the frequency ,$\omega=\sqrt{g/L}$ of the pendulum, which is the same for the $x,y$ space. the amplitude of $\varphi(\tau)$ is proportional to the acceleration parameter $A$.