Physics Oscillations

In SHM, the restoring force, F is directly proportional to the negative of displacement. That is, F=- kx. This follows Hooke's law and ensures motion is oscillatory about the mean position with angular frequency,  

$\omega =\sqrt{\frac{k}{m}}$

Uniform circular motion has constant speed. But SHM requires velocity and acceleration to vary sinusoidally with displacement. SHM is the projection of uniform circular motion. But it's not the motion itself, due to differing energy and force characteristics.

x = A sin $\omega t$

$\Rightarrow v=A\omega cos\omega t$

$\Rightarrow v=\pm \omega \sqrt[]{A^2-x^2}$

$\frac{v^2}{{\omega }^2}+x^2=A^2$

Elliptical

No, only oscillations with a linear restoring force and sinusoidal motion are SHM. Non-linear oscillations, such as large pendulum swings, deviate from SHM.

Oscillations are larger, slower to-and-fro motions. Vibrations, on the other hand, are smaller, faster oscillations. Vibrations often occur in machines, strings, and structural components.

No, periodic motion isn't always oscillatory. It may not involve back-and-forth movement about an equilibrium point. Some known examples include circular motion in satellites or planets. 

A simple pendulum shows simple harmonic motion only for small angular displacements. Otherwise, its motion is oscillatory but not purely SHM.

Tension in the string and gravitational force are common forces. These forces are resolved into tangential and radial components.

For small displacements,  sinθ≈θ (in radians), the small-angle approximation simplifies pendulum motion equations and makes it to approximate simple harmonic motion.

$x=5sin\left(\pi t+\frac{\pi }{3}\right)m$

Amplitude  $=5 m$

$=5 m$

$\omega =\pi =\frac{2\pi }{T}$

$T=\frac{2\pi }{\pi }=2 s$