

$y = A{e^{ - \frac{{bt}}{{2m}}}}\sin (\omega 't + \phi )$
where the symbols have their usual meanings. If a $2\ kg$ mass $(m)$ is attached to a spring of force constant $(K)$ $1250\ N/m$ , the period of the oscillation is $\left( {\pi /12} \right)s$ . The damping constant $‘b’$ has the value. ..... $kg/s$
| Column $I$ | Column $II$ |
| $(A)$ The object moves on the $\mathrm{x}$-axis under a conservative force in such a way that its "speed" and "position" satisfy $v=c_1 \sqrt{c_2-x^2}$, where $\mathrm{c}_1$ and $\mathrm{c}_2$ are positive constants. | $(p)$ The object executes a simple harmonic motion. |
| $(B)$ The object moves on the $\mathrm{x}$-axis in such a way that its velocity and its displacement from the origin satisfy $\mathrm{v}=-\mathrm{kx}$, where $\mathrm{k}$ is a positive constant. | $(q)$ The object does not change its direction. |
| $(C)$ The object is attached to one end of a massless spring of a given spring constant. The other end of the spring is attached to the ceiling of an elevator. Initially everything is at rest. The elevator starts going upwards with a constant acceleration a. The motion of the object is observed from the elevator during the period it maintains this acceleration. | $(r)$ The kinetic energy of the object keeps on decreasing. |
| $(D)$ The object is projected from the earth's surface vertically upwards with a speed $2 \sqrt{\mathrm{GM}_e / R_e}$, where, $M_e$ is the mass of the earth and $R_e$ is the radius of the earth. Neglect forces from objects other than the earth. | $(s)$ The object can change its direction only once. |