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  $a^{3/2}$

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 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) is

1. $2\mathrm{\pi }{\left(\frac{M}{2K}\right)}^{1/2}$         
2. $2\mathrm{\pi }{\left(\frac{2\mathrm{M}}{K}\right)}^{1/2}$
3. $2\mathrm{\pi }\frac{\mathrm{Mg} \mathrm{sin\theta }}{2\mathrm{K}}$         
4. $2\mathrm{\pi }{\left(\frac{2\mathrm{Mg}}{\mathrm{K}}\right)}^{1/2}$

A particle of mass m is attached to a spring (of spring constant k) and has a natural angular frequency ${\omega }_0$. An external force F (t) proportional to $\cos \omega t\left(\omega \ne {\omega }_0\right)$ is applied to the oscillator. The time displacement of the oscillator will be proportional to -

1. $\frac{m}{{{\omega }^2}_0-{\omega }^2}$                 
2. $\frac{1}{m\left({{\omega }^2}_0-{\omega }^2\right)}$

3. $\frac{1}{m\left({{\omega }^2}_1+{\omega }^2\right)}$           
4. $\frac{m}{{{\omega }^2}_1+{\omega }^2}$

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 unstretched. Then the maximum extension in the spring is -

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

Three masses 700g, 500g, and 400g are suspended at the end of a spring as shown and are in equilibrium. When the 700g mass is removed, the system oscillates with a period of 3 seconds. When the 500 gm mass is also removed, it will oscillate with a period of

1. 1 s

2. 2 s

3. 3 s

4. $\sqrt{\frac{12}{5}}s$