The position x (in centimeter) of a simple harmonic oscillator varies with time t (in second) as $x = 2\cos \left(0.5\mathrm{\pi t} + \frac{\mathrm{\pi }}{3}\right)$. The magnitude of the maximum acceleration of the particle in $\mathrm{cm}/{\mathrm{s}}^2$ is:

1.  $\mathrm{\pi }$/2

2.  $\mathrm{\pi }$/4

3.  ${\mathrm{\pi }}^2$/2

4.  ${\mathrm{\pi }}^2$/4

A horizontal platform is executing simple harmonic motion in the vertical direction with frequency f. A block of mass m is placed on the platform. What is the maximum amplitude of the SHM, so that the block is not detached from it?

1.  $\frac{\mathrm{mg}}{2{\mathrm{\pi }}^2{\mathrm{f}}^2}$

2.  $\frac{\mathrm{mg}}{4{\mathrm{\pi }}^2{\mathrm{f}}^2}$

2.  $\frac{\mathrm{g}}{2{\mathrm{\pi }}^2{\mathrm{f}}^2}$

4.  $\frac{\mathrm{g}}{4{\mathrm{\pi }}^2{\mathrm{f}}^2}$

A body at the end of a spring executes S.H.M. with a period ${\mathrm{T}}_1$ while the corresponding period for another spring is ${\mathrm{T}}_2$. If the period of oscillation with two springs in series is T, then:

1.  $\mathrm{T} = {\mathrm{T}}_1 + {\mathrm{T}}_2$

2.  ${\mathrm{T}}^2 = {\mathrm{T}}_1^2 + {\mathrm{T}}_2^2$

3.  $\frac{1}{\mathrm{T}} = \frac{1}{{\mathrm{T}}_1} + \frac{1}{{\mathrm{T}}_2}$

4.  $\frac{1}{{\mathrm{T}}^2} = \frac{1}{{\mathrm{T}}_1^2} + \frac{1}{{\mathrm{T}}_2^2}$

A body is executing linear S.H.M. At a position x, its potential energy is ${\mathrm{E}}_1$, and at a position y, its potential energy is ${\mathrm{E}}_2$. The potential energy at the position (x + y) is

1.  ${\mathrm{E}}_1 + {\mathrm{E}}_2$

2.  $\sqrt{{\mathrm{E}}_1^2 + {\mathrm{E}}_2^2}$

3.  ${\mathrm{E}}_1 + {\mathrm{E}}_2 + 2\sqrt{{\mathrm{E}}_1{\mathrm{E}}_2}$

4.  $\sqrt{{\mathrm{E}}_1{\mathrm{E}}_2}$

The equation of a particle executing simple harmonic motion is $y = 0.4 \sin \left(2\mathrm{\pi t} + \frac{\mathrm{\pi }}{3}\right)$ (where t is in seconds and y is in meters). The initial phase of the particle is:

1.  $2\mathrm{\pi t} + \frac{\mathrm{\pi }}{3}$

2.  $\frac{\mathrm{\pi }}{3}$

3.  $\frac{7\mathrm{\pi }}{3}$

4.  $\frac{8\mathrm{\pi }}{3}$

An ideal spring-mass system has a time period of vibration T. If the spring is cut into 4 identical parts and same mass oscillates with one of these parts, then the new time period of vibration will be

1.  $\frac{\mathrm{T}}{2}$

2.  T

3.  $\frac{\mathrm{T}}{4}$

4.  2T

The equation of a particle executing simple harmonic motion is $y=2\sqrt{2}\sin 314t.$ Displacement y from the mean position where acceleration becomes zero is: (y is in cm and t is in second) 

1.  2 cm

2.  0

3.  $\frac{1}{\sqrt{2}}$ cm

4.  $2\sqrt{2}$ cm

The displacement \((\mathrm{x})\) of an SHM varies with time \((\mathrm{t})\) as shown in the figure. The frequency of variation of potential energy is:

A simple pendulum bob is a hollow sphere full of sand suspended by means of a wire. If all the sand is drained out immediately, then the time period of the pendulum will:

A simple pendulum of length L is suspended from the ceiling of a cart which is sliding without friction on an inclined plane of inclination $\mathrm{\theta }$. The time period of the pendulum is

1.  $\mathrm{T} = 2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g}}}$

2.  $\mathrm{T} = 2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g} \mathrm{cos\theta }}}$

3.  $\mathrm{T} = 2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g} \mathrm{sin\theta }}}$

4.  $\mathrm{T} = 2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g} \mathrm{tan\theta }}}$