A particle is executing a simple harmonic motion. Its maximum acceleration is $\alpha$ and maximum velocity is $\beta$. Then its time period of vibration will be:
1. $\frac{{\beta }^2}{{\alpha }^2}$
2. $\frac{\beta }{\alpha }$
3. $\frac{{\beta }^2}{\alpha }$
4. $\frac{2\pi \beta }{\alpha }$

When two displacements represented by y₁ =asin($\omega$t) and y₂ =bcos($\omega$t) are superimposed, the motion is:
1. not simple harmonic.
2. simple harmonic with amplitude $\frac{a}{b}$.
3. simple harmonic with amplitude $\sqrt{a^2+b^2}$.
4. simple harmonic with amplitude $\frac{\left(a+b\right)}{2}$.

A particle is executing SHM along a straight line. Its velocities at distances x₁ and x₂ from the mean position are v₁ and v₂, respectively. Its time period is:
1. \(2 \pi \sqrt{\frac{x_{1}^{2}+x_{2}^{2}}{v_{1}^{2}+v_{2}^{2}}}~\)
2. \(2 \pi \sqrt{\frac{\mathrm{x}_{2}^{2}-\mathrm{x}_{1}^{2}}{\mathrm{v}_{1}^{2}-\mathrm{v}_{2}^{2}}}\)
3. \(2 \pi \sqrt{\frac{v_{1}^{2}+v_{2}^{2}}{x_{1}^{2}+x_{2}^{2}}}\)
4. \(2 \pi \sqrt{\frac{v_{1}^{2}-v_{2}^{2}}{x_{1}^{2}-x_{2}^{2}}}\)

The oscillation of a body on a smooth horizontal surface is represented by the equation, \(X=A \text{cos}(\omega t)\),
where \(X=\) displacement at time \(t,\) \(\omega=\) frequency of oscillation.
Which one of the following graphs correctly shows the variation of acceleration, \(a\) with time, \(t?\)
(\(T=\) time period)

1.		2.
3.		4.

The equation of a simple harmonic wave is given by \(y=3\sin \frac{\pi}{2}(50t-x)\) where \(x \) and \(y\) are in meters and \(t\) is in seconds. The ratio of maximum particle velocity to the wave velocity is:
1. \(\frac{3\pi}{2}\)
2. \(3\pi\)
3. \(\frac{2\pi}{3}\)
4. \(2\pi\)

A particle of mass \(m\) is released from rest and follows a parabolic path as shown. Assuming that the displacement of the mass from the origin is small, which graph correctly depicts the position of the particle as a function of time?
           

1.		2.
3.		4.

Two particles are oscillating along two close parallel straight lines side by side, with the same frequency and amplitudes. They pass each other, moving in opposite directions when their displacement is half of the amplitude. The mean positions of the two particles lie in a straight line perpendicular to the paths of the two particles. The phase difference is:

The displacement of a particle along the x-axis is given by, x = asin²$\mathrm{\omega }$t. The motion of the particle corresponds to: