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

(a) $2\mathrm{\pi }\sqrt{\frac{3\mathrm{L}}{\mathrm{g}}}$

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

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

(d) $2\mathrm{\pi }\sqrt{\frac{2\mathrm{L}}{3\mathrm{g}}}$

If a body is released into a tunnel dug across the diameter of earth, it executes simple harmonic motion with time period

(1) $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{{\mathrm{R}}_{\mathrm{e}}}{\mathrm{g}}}$

(2) $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{2{\mathrm{R}}_{\mathrm{e}}}{\mathrm{g}}}$

(3) $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{{\mathrm{R}}_{\mathrm{e}}}{2\mathrm{g}}}$

(4) T=2 seconds

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 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 -

(a)      $12{\omega }^2$       (b)        $36{\omega }^2$

(c)    $144{\omega }^2$       (d)  $\sqrt{192}{\omega }^2$

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 -

(a)   Proportional to  $\frac{1}{\sqrt{a}}$  (b)         Independent of a

(c)   Proportional to $\sqrt{a}$    (d)         Proportional to  $a^{3/2}$

A cylindrical piston of mass M slides smoothly inside a long cylinder closed at one end, enclosing a certain mass of gas. The cylinder is kept with its axis horizontal. If the piston is disturbed from its equilibrium position, it oscillates simple harmonically. The period of oscillation will be

(1) $T=2\pi \sqrt{\left(\frac{Mh}{PA}\right)}$

(2) $T=2\pi \sqrt{\left(\frac{MA}{Ph}\right)}$

(3) $T=2\pi \sqrt{\left(\frac{M}{PAh}\right)}$

(4) $T=2\pi \sqrt{MPhA}$

The metallic bob of a simple pendulum has the relative density $\rho$. The time period of this pendulum is T. If the metallic bob is immersed in water, then the new time period is given by

(1) $T\frac{\rho -1}{\rho }$                 

(2)     $T\frac{\rho }{\rho -1}$

(3) $T\sqrt{\frac{\rho -1}{\rho }}$               

(4) $T\sqrt{\frac{\rho }{\rho -1}}$