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.  $\mathrm{T}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g}}}$

2.  $\mathrm{T}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g}<mspace\; width="1em"></mspace>\mathrm{cos\theta }}}$

3.  $\mathrm{T}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g}<mspace\; width="1em"></mspace>\mathrm{sin\theta }}}$

4.  $\mathrm{T}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g}<mspace\; width="1em"></mspace>\mathrm{tan\theta }}}$

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

1.  \(1.256\ \text{s}\)

2.  \(12.56 \ \text{s}\)

3.  \(0.1256 \ \text{s}\)

4.  \(125.6 \ \text{s}\)

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 $\mathrm{a}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>-\frac{\mathrm{\pi }}{4}\sin \left(\frac{\mathrm{\pi }}{4}\mathrm{t}<mspace\; width="1em"></mspace>-<mspace\; width="1em"></mspace>\frac{\mathrm{\pi }}{6}\right)$.  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 $\mathrm{x}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\mathrm{Acos}\frac{\mathrm{\pi }}{2}\mathrm{t}$. 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 $\mathrm{t}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{4}{3}$ is:

1.  $\frac{\sqrt{3}}{32}{\mathrm{\pi }}^3<mspace\; width="1em"></mspace>\mathrm{cm}/{\mathrm{s}}^2$

2.  $-\frac{{\mathrm{\pi }}^2}{32}<mspace\; width="1em"></mspace>\mathrm{cm}/{\mathrm{s}}^2$

3.  $\frac{{\mathrm{\pi }}^2}{32}<mspace\; width="1em"></mspace>\mathrm{cm}/{\mathrm{s}}^2$

4.  $-\frac{\sqrt{3}}{32}{\mathrm{\pi }}^2<mspace\; width="1em"></mspace>\mathrm{cm}/{\mathrm{s}}^2$