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

A body is placed on a horizontal platform that is undergoing vertical SHM. If the amplitude of oscillation is 40 cm, then the least period of oscillation for which an object placed over the platform is not detached from it is:

1.  1.256 s

2.  12.56 s

3.  0.1256 s

4.  125.6 s

If a pendulum giving correct time on the ground at a certain place is moved to the top of a tower 320 m high, then the loss in time (in seconds) measured by the pendulum clock in one week is:

1.  15.12

2.  7.72

3.  3.78

4.  30.24

Instantaneous acceleration (in ${\mathrm{ms}}^{-2}$) of a particle executing S.H.M. is given by $\mathrm{a} = -\frac{\mathrm{\pi }}{4}\sin \left(\frac{\mathrm{\pi }}{4}\mathrm{t} - \frac{\mathrm{\pi }}{6}\right)$.  The maximum speed of the particle will occur first time at

1.  1.75 s

2.  1.4 s

3.  1.2 s

4.  0.67 s

A particle moves according to the equation $\mathrm{x} = \mathrm{Acos}\frac{\mathrm{\pi }}{2}\mathrm{t}$. Distance covered by it in the time interval of t =0 to t =3 s is: (symbols have their usual meanings) 

1.  A

2.  4A

3.  3A

4.  2A

A body of mass m hanging with the help of three springs, each of spring constant k as shown. If the mass is slightly displaced and released, then the system will oscillate with the time period

1.  $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{3\mathrm{k}}}$

2.  $2\mathrm{\pi }\sqrt{\frac{2\mathrm{m}}{3\mathrm{k}}}$

3.  $2\mathrm{\pi }\sqrt{\frac{3\mathrm{m}}{2\mathrm{k}}}$

4.  $2\mathrm{\pi }\sqrt{\frac{3\mathrm{m}}{\mathrm{k}}}$

A block of mass m attached to a spring of constant k oscillates on a smooth surface. The other end of the spring is fixed to a wall. When spring is at its natural length, the speed of the block is v. Displacement of the block from its mean position before coming to instantaneous rest is:

1.  $\sqrt{\frac{\mathrm{mv}}{\mathrm{k}}}$

2.  $\mathrm{v}\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$

3.  $\mathrm{m}\sqrt{\frac{\mathrm{v}}{\mathrm{k}}}$

4.  $\frac{1}{\mathrm{k}}\sqrt{\frac{\mathrm{m}}{\mathrm{v}}}$

A particle under SHM has a maximum speed of 30 cm/s and a maximum acceleration of 60 cm/s? The time period of oscillation (in second) is

1.  $\frac{\mathrm{\pi }}{2}$

2.  2$\mathrm{\pi }$

3.  $\mathrm{\pi }$

4.  4$\mathrm{\pi }$

The x-t graph of a particle undergoing SHM is shown below. The acceleration of the particle at $\mathrm{t} = \frac{4}{3}$ is:

1.  $\frac{\sqrt{3}}{32}{\mathrm{\pi }}^3 \mathrm{cm}/{\mathrm{s}}^2$

2.  $-\frac{{\mathrm{\pi }}^2}{32} \mathrm{cm}/{\mathrm{s}}^2$

3.  $\frac{{\mathrm{\pi }}^2}{32} \mathrm{cm}/{\mathrm{s}}^2$

4.  $-\frac{\sqrt{3}}{32}{\mathrm{\pi }}^2 \mathrm{cm}/{\mathrm{s}}^2$