Oscillations and Waves

The equation of motion of a particle is given by $x=a\sin(50t+\pi/3)$ cm. The particle will come to rest at time $t_1$ and it will have zero acceleration at time $t_2$. The $t_1$ and $t_2$ respectively are ______.

The velocity of a particle executing simple harmonic motion along $x$-axis is described as $v^2=50-x^2$, where $x$ represents displacement. If the time period of motion is $\frac{x}{7}$ s, the value of $x$ is ______.

A transverse wave on a string is described by $y = 3\sin(36t + 0.018x + \pi/4)$, where $x$, $y$ are in cm and $t$ in seconds. The least distance between the two successive crests in the wave is ____ cm. (Nearest integer) ($\pi = 3.14$)

Match List - I with List - II. List - I: A. $\sin^2\omega t$; B. $\sin^3(2\omega t)$; C. $\sin(\omega t)+\cos(\pi\omega t)$; D. $\cos\omega t+\cos2\omega t$. List - II: I. Periodic with time period $T=\frac{\pi}{\omega}$ but not simple harmonic motion (SHM); II. Periodic with time period $T=\frac{2\pi}{\omega}$ but Not SHM; III. Periodic with time period $T=\frac{\pi}{\omega}$ and SHM; IV. Non-periodic. Choose the correct answer from the options given below :

A particle is executing simple harmonic motion. Its amplitude is $A$ and time period is 5 sec. The time required by it to move from $x=A$ to $x=\frac{A}{\sqrt2}$ is ________ sec.

The frequency of oscillation of a mass $m$ suspended by a spring is $v_1$. If the length of the spring is cut to half, the same mass oscillates with frequency $v_2$. The value of $v_2/v_1$ is ______.

A closed organ and an open organ tube are filled by two different gases having same bulk modulus but different densities $\rho_1$ and $\rho_2$, respectively. The frequency of $9^{\text{th}}$ harmonic of closed tube is identical with $4^{\text{th}}$ harmonic of open tube. If the length of the closed tube is 10 cm and the density ratio of the gases is $\rho_1 : \rho_2 = 1 : 16$, then the length of the open tube is:

A light hollow cube of side length 10 cm and mass 10 g, is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is $y\pi \times 10^{-2}$ s, where the value of $y$ is (Acceleration due to gravity, $g = 10\ \mathrm{m/s^2}$, density of water $= 10^3\ \mathrm{kg/m^3}$)

The equation of a transverse wave travelling along a string is $y(x, t) = 4.0\sin[20 \times 10^{-3}x + 600t]$ mm, where $x$ is in mm and $t$ is in second. The velocity of the wave is: