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. $\sqrt{3}\mathrm{T}$

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

If the displacement equation of a particle be represented by $y=A\sin Pt+ B\cos Pt$ , the particle executes

1.         A uniform circular motion

2.         A uniform elliptical motion

3.         A S.H.M.

4         A rectilinear motion

A particle with restoring force proportional to displacement and resisting force proportional to velocity is subjected to a force $\mathrm{Fsin\omega t}$ . If the amplitude of the particle is maximum for $\mathrm{\omega }={\mathrm{\omega }}_1$  and the energy of the particle is maximum for $\mathrm{\omega }={\mathrm{\omega }}_2$, then (where ${\omega }_0$ is natural frequency of oscillation of particle)

1. ${\mathrm{\omega }}_1={\mathrm{\omega }}_0$ and ${\mathrm{\omega }}_2\ne {\mathrm{\omega }}_0$

2. ${\mathrm{\omega }}_1={\mathrm{\omega }}_0$ and ${\mathrm{\omega }}_2={\mathrm{\omega }}_0$

3. ${\mathrm{\omega }}_1\ne {\mathrm{\omega }}_0$ and ${\mathrm{\omega }}_2={\mathrm{\omega }}_0$

4. ${\mathrm{\omega }}_1\ne {\mathrm{\omega }}_0$ and ${\mathrm{\omega }}_2\ne {\mathrm{\omega }}_0$

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

1.   8           

2.  – 4

3.   4          

4.   $4\sqrt{2}$

A S.H.M. is represented by $x=5\sqrt{2}\left(\sin 2\mathrm{\pi t}+\cos 2\mathrm{\pi t}\right).$ The amplitude of the S.H.M. is

1.   10 cm         

2.  20 cm

3.   $5\sqrt{2}$ cm     

4.  50 cm

 Amplitude of a wave is represented by

$A=\frac{c}{a+b-c}$

Then resonance will occur when

1.     $b=-c/2$              

2.     b = 0 and a =  c

3.     $b=-a/2$              

4.     None of these

The displacement of a particle varies with time as $x=12\sin wt-16 {\sin }^3 wt$ (in cm). If its motion is S.H.M., then its maximum acceleration is -

1. $12{\omega }^2$      

2. $36{\omega }^2$

3. $144{\omega }^2$       

4. $\sqrt{192}{\omega }^2$

A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy is $U\left(x\right)=k{\left[x\right]}^3$ , where k is a positive constant. If the amplitude of oscillation is a, then its time period T is -

1.  Proportional to  $\frac{1}{\sqrt{a}}$ 

2.  Independent of a

3.  Proportional to $\sqrt{a}$   

4.  Proportional to  $a^{3/2}$