The speed $v$ of a particle moving along a straight line, when it is at a distance ($x$) from a fixed point of the line is given by $v^2=108-9x^2$ (all quantities are in CGS units). Which of the following is/are true?

A horizontal plank has a rectangular block placed on it. The plank starts oscillating vertically and simple harmonically with an amplitude of 40 cm. The block just loses contact with the plank when the latter is at momentary rest. Then–

An object of mass $m$ is performing simple harmonic motion on a smooth horizontal surface as shown in the figure. Just as the oscillating object reaches its extreme position, another object of mass $2 \mathrm{m}$ is dropped on to oscillating object, which sticks to it. For this situation mark out the correct statement(s).

The given Fig. (a) shows a spring of force constant $k$ fixed at one end and carrying a mass $m$ at the other end placed on a horizontal frictionless surface. The spring is stretched by a force $F.$ Figure (b) shows the same spring with both ends free and a mass $m$ fixed at each free end. Each of the spring is stretched by the same force $F.$ The mass in case (a) and the masses in case (b) are then released. Which of the following statements are true?

Two blocks connected by a spring rest on a smooth horizontal plane as shown in the given figure. A constant force $F$ starts acting on block $m_2$ as shown in the figure. Which of the following statements are not correct?

A cylindrical block of density $d$ stays fully immersed in a beaker filled with two immiscible liquids of different densities $d_1$ and $d_2.$ The block is in equilibrium with half of it in liquid $1$ and the other half in liquid $2$ as shown in the given figure. If the block is given a displacement downwards and released, then neglecting friction study the following statements.