The displacement of a particle in a periodic motion is given by $y = 4 \cos^{2}\left(\frac{t}{2}\right) \sin(1000 t)$. This displacement may be considered as the result of the superposition of $n$ independent harmonic oscillations. Here $n$ is

Two simple harmonic motions,as shown,are at right angles. They are combined to form Lissajous figures.
$x(t) = A \sin(at + \delta)$
$y(t) = B \sin(bt)$
Identify the correct match below.

Two particles execute simple harmonic motion along the same straight line with the same amplitude and same frequency. The two particles pass one another when moving in opposite directions each time at a distance of $\frac{1}{\sqrt{2}}$ times the amplitude from their common mean position. The phase difference between the two particles is (in $^{\circ}$)

Two particles are oscillating in $SHM$ along two very close parallel paths such that they have the same mean position. The equations of $SHM$ for the two particles are $x_1 = A \sin \omega t$ and $x_2 = A \sin(\omega t + \phi)$ respectively. If the maximum distance between them is $\frac{6A}{5}$,then $\phi$ is equal to ..... $^o$.

Two simple harmonic motions are represented by the equations ${y_1} = 0.1 \sin(100\pi t + \frac{\pi}{3})$ and ${y_2} = 0.1 \cos(\pi t)$. The phase difference of the velocity of particle $1$ with respect to the velocity of particle $2$ is

Two simple harmonic motions are represented by the equations $y_{1} = 10 \sin(3 \pi t + \frac{\pi}{3})$ and $y_{2} = 5(\sin 3 \pi t + \sqrt{3} \cos 3 \pi t)$. The ratio of the amplitude of $y_{1}$ to $y_{2}$ is $x : 1$. The value of $x$ is ...... .