Simple harmonic motion

A particle of mass $m$ is attached to one end of a mass-less spring of force constant $k$, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time $t=0$ with an initial velocity $u_0$. When the speed of the particle is $0.5u_0$, it collides elastically with a rigid wall. After this collision,

A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at $t=0$. It then executes simple harmonic motion with angular frequency $\omega=\dfrac\pi3$ rad/s. Simultaneously at $t=0$, a small pebble is projected with speed $v$ from point $P$ at an angle of 45° as shown in the figure. Point $P$ is at a horizontal distance of 10 m from $O$. If the pebble hits the block at $t=1$ s, the value of $v$ is (take $g=10\ \mathrm{m/s^2}$)

If the total energy of the particle is E, it will perform periodic motion only if

For periodic motion of small amplitude A, the time period T of this particle is proportional to

The $x$-$t$ graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at $t=4/3$ s is

The centre of mass of the disk undergoes simple harmonic motion with angular frequency $\omega$ equal to

Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II and indicate your answer by darkening appropriate bubbles in the 4 × 4 matrix given in the ORS.