In an oscillating spring-mass system,a spring is connected to a box filled with sand. As the box oscillates,sand leaks slowly out of the box vertically so that the average frequency $\omega(t)$ and average amplitude $A(t)$ of the system change with time $t$. Which one of the following options schematically depicts these changes correctly?

$A$ simple harmonic oscillator consists of a particle of mass $m$ and an ideal spring with spring constant $k$. The particle oscillates with a time period $T$. The spring is cut into two equal parts. If one part oscillates with the same particle, the new time period will be

$A$ load of mass $m$ falls from a height $h$ onto a scale pan hung from a spring as shown in the figure. If the spring constant is $k$,the mass of the scale pan is zero,and the mass $m$ does not bounce relative to the pan,then the amplitude of vibration is

$A$ $5\; kg$ 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.

Inside a lift, a spring (force constant $k = 1000 \ N/m$) and a block $(mass = 1 \ kg)$ are both in a state of rest. Now, the lift suddenly starts moving upwards with an acceleration $a = g$. Find the maximum total compression in the spring in centimeters. $(g = 10 \ m/s^2)$

$A$ mass $m$ is suspended separately by two springs of spring constants $k_1$ and $k_2$. The time periods of oscillations in the two cases are $T_1$ and $T_2$ respectively. If the same mass $m$ is suspended by connecting the two springs in parallel (as shown in the figure),then the time period of the oscillation is $T$. The correct relation is: