Three masses $100 \text{ g}$, $300 \text{ g}$, and $500 \text{ g}$ are suspended at the end of a spring and are in equilibrium. When the $500 \text{ g}$ mass is removed, the system oscillates with a period of $3 \text{ s}$. When the $300 \text{ g}$ mass is also removed, it will oscillate with a period of: (in $\text{ s}$)

$A$ spring-mass system (mass $m$,spring constant $k$,and natural length $\ell_{0}$) rests in equilibrium on a horizontal disc. The free end of the spring is fixed at the center of the disc. If the disc,together with the spring-mass system,rotates about its axis with an angular velocity $\omega$ (where $k >> m \omega^{2}$),the relative change in the length of the spring is best given by which option?

$A$ $5\; kg$ collar is attached to a spring of spring constant $500\; N m^{-1}$. It slides without friction over a horizontal rod. The collar is displaced from its equilibrium position by $10.0\; cm$ and released. Calculate
$(a)$ the period of oscillation.
$(b)$ the maximum speed and
$(c)$ maximum acceleration of the collar.

If $b = a - \frac{a^2}{2} + \frac{a^3}{3} - \frac{a^4}{4} + \dots$,then $b + \frac{b^2}{2!} + \frac{b^3}{3!} + \frac{b^4}{4!} + \dots \infty = $

$A$ block of mass $m$ is attached to two springs of spring constants $k_1$ and $k_2$ as shown in the figure. The block is displaced by $x$ towards the right and released. The velocity of the block when it is at $x/2$ will be

When a mass $M$ is attached to a spring of force constant $k$,the spring stretches by $l$. If the mass oscillates with amplitude $l$,what will be the maximum potential energy stored in the spring?