Vardaan Learning Institute

A particle executes SHM described by $x(t) = 5 \cos\left(2\pi t + \frac{\pi}{4}\right)$ in SI units. Find amplitude $A$, frequency $f$, time period $T$, and initial phase $\phi_0$.

Derive expressions for kinetic energy $K = \frac{1}{2} m \omega^2 (A^2 - x^2)$ and potential energy $U = \frac{1}{2} m \omega^2 x^2$ in SHM, and show total energy $E = \frac{1}{2} m \omega^2 A^2$ is constant.

Derive the time period formula of a simple pendulum $T = 2\pi\sqrt{\frac{L}{g}}$ for small angular displacements.

A mass $m$ suspended from a spring of constant $k$ oscillates with period $T$. If the spring is cut into two equal halves and same mass is attached to one half, find new time period $T'$.