What is meant by simple harmonic oscillation? Give examples and explain why every simple harmonic motion is a periodic motion whereas the converse need not be true.

Given an one dimensional system with total energy $E=\frac{{p_x}^2}{2m}+V\left(x\right)=$ constant, where $p_x$ is the $x$ component of the linear momentum and $V\left(x\right)$ is the potential energy of the system. Show that total time derivative of energy gives us force $F_x=-\frac{d}{dx}V\left(x\right)$. Verify Hooke’s law by choosing potential energy $V\left(x\right)=\frac{1}{2}k{x}^2$.

Consider a simple pendulum of length $l=0.9 \mathrm{m}$ which is properly placed on a trolley rolling down on a inclined plane which is at $\theta =45°$ with the horizontal. Assuming that the inclined plane is frictionless, calculate the time period of oscillation of the simple pendulum.

(Answer approximately to the nearest integer)