Physics Oscillations

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  $=5 m$

$=5 m$

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

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

Resonance occurs when the frequency of an external periodic force matches the natural frequency of a system. From that, physicists know that resonance causes the amplitude of oscillations to increase significantly. This can be beneficial in devices, such as musical instruments, but dangerous in structures like bridges.

The phase in SHM tells us the position and direction of motion of the particle at a specific instant. It determines the state of oscillation and includes both displacement and time information.