The displacement of a particle varies according to the relation x = 4(cosp$\mathrm{\pi }$t + sinp$\mathrm{\pi }$t). The amplitude of the particle is

1. 8

2. -4

3. 4

4. $4\sqrt{2}$

The period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination $\theta$, is given by -

1.   $2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{gcos\theta }}}$               

2.  $2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{gsin}\theta }}$

3.  $2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g} }}$                     

4. $2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{gtan\theta }}}$

An ideal spring with spring-constant K is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring initially unstretched. Then the maximum extension in the spring is -

1. 4 Mg/K         

2. 2 Mg/K

3. Mg/K             

4. Mg/2K

The graph shows the variation of displacement of a particle executing SHM with time. We infer from this graph that:

The velocity-time diagram of a harmonic oscillator is shown in the adjoining figure. The frequency of oscillation is

1. 25 Hz           

2. 50 Hz

3. 12.25 Hz       

4. 33.3 Hz

The time period of a simple pendulum of length L as measured in an elevator descending with acceleration $\frac{\mathrm{g}}{3}$ is

1. $2\mathrm{\pi }\sqrt{\frac{3\mathrm{L}}{\mathrm{g}}}$

2. $\mathrm{\pi }\sqrt{\left(\frac{3\mathrm{L}}{\mathrm{g}}\right)}$

3. $2\mathrm{\pi }\sqrt{\left(\frac{3\mathrm{L}}{2\mathrm{g}}\right)}$

4. $2\mathrm{\pi }\sqrt{\frac{2\mathrm{L}}{3\mathrm{g}}}$

The period of a simple pendulum measured inside a stationary lift is found to be T. If the lift starts accelerating upwards with acceleration of g/3 then the time period of the pendulum is

1. $\frac{\mathrm{T}}{\sqrt{3}}$

2. $\frac{\mathrm{T}}{3}$

3. $\frac{\sqrt{3}}{2}\mathrm{T}$

4. $\sqrt{3}\mathrm{T}$

A simple harmonic wave having an amplitude a and time period T is represented by the equation $y=5 \mathrm{sin\pi }\left(t+4\right)$m Then the value of amplitude (a) in (m) and time period  (T) in second are       

1.   $a=10, T=2$   

2. $a=5, T=1$

3.    $a=10, T=1$    

4. $a=5, T=2$

If the length of a pendulum is made \(9\) times and mass of the bob is made \(4\) times, then the value of time period will become:
1. \(3T\)
2. \(\dfrac{3}{2}T\)
3. \(4T\)
4. \(2T\)