A particle of mass $m$ is attached to one end of a massless 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 $0.5$$u_0$, it collides elastically with a rigid wall. After this collision,

Two independent harmonic oscillators of equal masses oscillate about the origin with angular frequencies ${\mathrm{\omega }}_1$ and ${\mathrm{\omega }}_2$ and have total energies $E_1$ and $E_2$, respectively. The variations of their momenta $p$ with positions $x$ are shown in figures. If $\frac{a}{b}=n^2$ and $\frac{a}{R}=n$, then the correct equation(s) is (are),

A block with mass $\mathrm{M}$ is connected by a massless spring with stiffness constant $\mathrm{k}$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude $\mathrm{A}$ about an equilibrium position${\mathrm{x}}_0$. Consider two cases :

$\left(\mathrm{i}\right)$when the block is at ${\mathrm{x}}_0$ ; and

$\left(\mathrm{ii}\right)$ when the block is at $\mathrm{x} = {\mathrm{x}}_0 + \mathrm{A}.$In both the cases, a particle with mass $\mathrm{m} (< \mathrm{M})$ is softly placed on the block after which they stick to each other.

Which of the following statement(s) is(are) true about the motion after the mass $\mathrm{m}$ is placed on the mass $\mathrm{M}$?

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0 \mathrm{N} {\mathrm{m}}^{-1}$ and the mass of the block is$2.0 \mathrm{kg}$. Ignore the mass of the spring. Initially, the spring is in an un-stretched condition. Another block of mass $1.0 \mathrm{kg}$ moving with a speed of $2.0 \mathrm{m} {\mathrm{s}}^{–1}$ collides elastically with the first block. The collision is such that the $2.0 \mathrm{kg}$ block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its un-stretched position for the first time after the collision is ____.

The displacement of an object attached to a spring and executing simple harmonic motion is given by $x=\left(2\times {10}^{-2}\cos \pi t\right) \mathrm{m}$. The time at which the maximum speed first occurs is$:$

Two springs of force constants $k_1$ and $k_2$ are connected to a mass $m$ as shown. The frequency of oscillation of mass is $ƒ$. If both $k_1$ and $k_2$ are made four times their original values, the frequency of oscillation becomes,