I have learned about the commutators, and read this: $$[A, f(B)] = f'[A,B]+\frac{1}{2}f''([A,B]B+B[A,B])+\frac{1}{3!}f'''([A,B]B^2+B[A,B]+B^2[A,B])+...$$ then Simplified to $$[A, f(B)] = [A,B](f'+f''B+\frac{1}{2}f'''B^2+...)=[A,B]\frac{df}{dB}$$ I do understand the first two equations, only don't...

The goal is to transform the following coordinates: $$x(t)= R(\Phi-\sin\Phi)$$ and $$z(t)=R(2 +\cos\Phi)$$ with the substitution: $u=\cos\left(\Phi/2\right)$ in order to get: $$x(t)=2R(\arccos(u)-u\sqrt{1-u^2})$$ and $$z(t)=R(1+2u^2)$$ How do I go about solving this problem? I already tried using...

I made a classical experiment to demonstrate centrifugal force. The experimental setup is made to stand vertically and spun along the vertical axis. The balls that initially rest at the bottom, on spinning, move up to the chamber. Can someone please explain in layman terms as to why this happens?...