A particle is executing a simple harmonic motion. Its maximum acceleration is a and maximum velocity is b. Then, its time period of vibration will be

1. $\frac{2\mathrm{\pi \beta }}{\mathrm{\alpha }}$

2. $\frac{{\mathrm{\beta }}^2}{{\mathrm{\alpha }}^2}$

3. $\frac{\mathrm{\alpha }}{\mathrm{\beta }}$

4. $\frac{{\mathrm{\beta }}^2}{\mathrm{\alpha }}$

A particle executes linear simple harmonic motion with an amplitude of 3 cm. When the particle is at 2cm from the mean position, the magnitude of its velocity is equal to that of its acceleration. Then its time period in seconds is:

1. $\frac{\sqrt{5}}{2\mathrm{\pi }}$

2. $\frac{4\mathrm{\pi }}{\sqrt{5}}$

3. $\frac{2\mathrm{\pi }}{\sqrt{3}}$

4. $\frac{\sqrt{5}}{\mathrm{\pi }}$

The oscillation of a body on a smooth horizontal surface is represented by the equation.

          $\mathrm{x}=\mathrm{A}<mspace\; width="1em"></mspace>\cos \left(\mathrm{\omega t}\right)$

where x=displacement at time t, $\mathrm{\omega }$=frequency of oscillation.

Which one of the following graph shows correctly the variation of acceleration 'a' with 't'?

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 position of the two particles lies on a straight line perpendicular to the paths of the two particles. The phase difference is -

1. $\frac{\mathrm{\pi }}{6}$

2. 0

3. $\frac{2\mathrm{\pi }}{3}$

4. $\mathrm{\pi }$

The period of oscillation of a mass M suspended from a spring of negligible mass is T. If along with it another mass M is also suspended, the period of oscillation will now be

1. $\sqrt{2}T$

2. T

3. $\frac{T}{\sqrt{2}}$

4. 2T

The displacement of a particle along the x-axis is given by x= ${\mathrm{asin}}^2\mathrm{\omega t}$. The motion of the particle corresponds to

1. simple harmonic motion of frequency $\frac{\mathrm{\omega }}{2\mathrm{\pi }}$

2. simple harmonic motion of frequency $\frac{\mathrm{\omega }}{\mathrm{\pi }}$

3. simple harmonic motion of frequency $\frac{3\mathrm{\omega }}{2\mathrm{\pi }}$

4. non simple harmonic motion.

A simple pendulum performs simple harmonic motion about x=0 with an amplitude a and time period T. The speed of the pendulum at x=a/2 will be

1. $\frac{\mathrm{\pi a}}{\mathrm{T}}$

2. $\frac{3{\mathrm{\pi }}^2\mathrm{a}}{\mathrm{T}}$

3. $\frac{\mathrm{\pi a}\sqrt{3}}{\mathrm{T}}$

4. $\frac{\mathrm{\pi a}\sqrt{3}}{2\mathrm{T}}$

Two simple harmonic motions of angular frequency 100 and 1000 rad ${\mathrm{s}}^{-1}$ have the same displacement amplitude. The ratio of their maximum acceleration is -

1. $1:{10}^4$

2. $1:10$

3. $1:{10}^2$

4. $1:{10}^3$

The particle executing simple harmonic motion has a kinetic energy ${\mathrm{K}}_0{\cos }^2\mathrm{\omega t}$. The maximum values of the potential energy and are the total energy respectively.

1. ${\mathrm{K}}_0/2<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>{\mathrm{K}}_0$

2. ${\mathrm{K}}_0<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>2{\mathrm{K}}_0$

3. ${\mathrm{K}}_0<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>{\mathrm{K}}_0$

4. $0<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>2{\mathrm{K}}_0$

The circular motion of a particle with constant speed is

1. periodic and simple harmonic

2. simple harmonic but not periodic

3. neither periodic nor simple harmonic

4. periodic but not simple harmonic