A body mass m is attached to the lower end of a spring whose upper end is fixed. The spring has neglible mass. When the mass m is slightly pulled down and released, it oscillates with a time period of 3s. When the mass m is increased by 1 kg, the time period of oscillations becomes 5s. The value of m in kg is-

(1) $\frac{3}{4}$               

(2)$\frac{4}{3}$

(3) $\frac{16}{9}$               

(4) $\frac{9}{16}$

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

A spring of force constant \(k\) is cut into lengths of ratio \(1:2:3\). They are connected in series and the new force constant is \(k'\). Then they are connected in parallel and force constant is \(k''\). Then \(k':k''\) is:
1. \(1:9\)
2. \(1:11\)
3. \(1:14\)
4. \(1:6\)

A mass of \(2.0\) kg is put on a flat pan attached to a vertical spring fixed on the ground as shown in the figure. The mass of the spring and the pan is negligible. When pressed slightly and released, the mass executes a simple harmonic motion. The spring constant is \(200\) N/m. What should be the minimum amplitude of the motion, so that the mass gets detached from the pan? 
(Take \(g=10\) m/s²) 
                

Two springs of force constants k and 2k are connected to a mass m as shown below. The frequency of oscillation of the mass is:

(1) $\frac{1}{2\mathrm{\pi }}\sqrt{k/m}$

(2) $\frac{1}{2\mathrm{\pi }}\sqrt{2k/m}$

(3) $\frac{1}{2\mathrm{\pi }}\sqrt{\frac{3k}{m}}$

(4) $\frac{1}{2\mathrm{\pi }}\sqrt{\frac{m}{k}}$

An object suspended from a spring exhibits oscillations of period T. Now, the spring is cut in two halves and the same object is suspended with two halves as shown in the figure. The new time period of oscillation will become :

1. $\frac{\mathrm{T}}{2\sqrt{2}}$

2. $\frac{T}{2}$

3. $\frac{T}{\sqrt{2}}$

4. 2T

What will be the force constant of the spring system shown in the figure?

1. $\frac{k_1}{2}+k_2$

2. ${\left[\frac{1}{2k_1}+\frac{1}{k_2}\right]}^{-1}$

3. $\frac{1}{2k_1}+\frac{1}{k_2}$

4. ${\left[\frac{2}{k_1}+\frac{1}{k_2}\right]}^{-1}$