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) $T\sqrt{\frac{\rho }{\rho -1}}$

The period of oscillation of a simple pendulum of length \(\mathrm{L}\) 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. $2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{gtan\theta }}}$

One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to massless spring of spring constant K. A mass m hangs freely from the free end of the spring. The area of cross-section and Young's modulus of the wire is A and Y respectively. If the mass is slightly pulled down and released, it will oscillate with a time period T equal to -

(1)    $2\mathrm{\pi }\left(\frac{\mathrm{m}}{\mathrm{K}}\right)$                   

(2)     $2\mathrm{\pi }{\left\{\frac{\left(\mathrm{YA}+\mathrm{KL}\right)\mathrm{m}}{\mathrm{YAK}}\right\}}^{1/2}$

(3)    $2\mathrm{\pi }\frac{\mathrm{mYA}}{\mathrm{KL}}$               

(4)      $2\mathrm{\pi }\frac{\mathrm{mL}}{\mathrm{YA}}$

On a smooth inclined plane, a body of mass \(M\) is attached between two springs. The other ends of the springs are fixed to firm supports. If each spring has force constant \(K\), the period of oscillation of the body (assuming the springs as massless) will be:
                
1. \(2\pi \left( \frac{M}{2K}\right)^{\frac{1}{2}}\)
2. \(2\pi \left( \frac{2M}{K}\right)^{\frac{1}{2}}\)
3. \(2\pi \left(\frac{Mgsin\theta}{2K}\right)\)
4. \(2\pi \left( \frac{2Mg}{K}\right)^{\frac{1}{2}}\)

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 un-stretched. Then the maximum extension in the spring will be:
1. 4 Mg/K 
2. 2 Mg/K
3. Mg/K 
4. Mg/2K

The displacement y of a particle executing periodic motion is given by $y=4{\cos }^2\left(t/2\right)\sin \left(1000t\right).$ This expression may be considered to be a result of the superposition of  ........... independent harmonic motions

1. Two         

2. Three

3. Four         

4. Five

A particle of mass m is attached to three identical springs A, B and C each of force constant k a shown in figure. If the particle of mass m is pushed slightly against the spring A and released then the time period of oscillations is -

(a) $2\mathrm{\pi }\sqrt{\frac{2\mathrm{m}}{\mathrm{k}}}$          (b) $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{2\mathrm{k}}}$

(c) $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$            (d) $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{3\mathrm{k}}}$

The graph shows the variation of displacement of a particle executing S.H.M. with time. We infer from this graph that -

(1) The force is zero at time T/8

(2) The velocity is maximum at time T/4

(3) The acceleration is maximum at time T

(4) The P.E. is equal to total energy at time T/4