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

(a) $2\mathrm{\pi }{\left(\frac{M}{2K}\right)}^{1/2}$           (b) $2\mathrm{\pi }{\left(\frac{2\mathrm{M}}{K}\right)}^{1/2}$

(c) $2\mathrm{\pi }\frac{\mathrm{Mg} \mathrm{sin\theta }}{2\mathrm{K}}$           (d) $2\mathrm{\pi }{\left(\frac{2\mathrm{Mg}}{\mathrm{K}}\right)}^{1/2}$

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 unstretched. Then the maximum extension in the spring is -

(1) 4 Mg/K         

(2) 2 Mg/K

(3) Mg/K             

(4) Mg/2K

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 SHM with time. We infer from this graph that:

For a particle executing SHM the displacement \(x \) is given by, \(A\cos \omega t.\)  Identify the graph which represents the variation of potential energy (P.E.) as a function of time \(t\) and displacement \(x.\)

   
1. I, III
2. II, IV
3. II, III
4. I, IV

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 variation of potential energy of harmonic oscillator is as shown in figure. The spring constant is

(1) 1 $\times$10² N/m               

(2) 150 N/m

(3) 0.667 $\times$ 10² N/m       

(4) 3 $\times$ 10² N/m

A body performs S.H.M. . Its kinetic energy K varies with time t as indicated by graph

(a)    (b) 

(c)      (d) 

The displacement of a body executing SHM is given by x= A sin (2$\mathrm{\pi }$t + $\mathrm{\pi }$/3). The first time from t = 0 when the velocity is maximum is :
1. 0.33 sec
2. 0.16 sec
3. 0.25 sec
4. 0.5 sec