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

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. $2\mathrm{\pi }\frac{\mathrm{Mg}<mspace\; width="1em"></mspace>\mathrm{sin\theta }}{2\mathrm{K}}$         
4. $2\mathrm{\pi }{\left(\frac{2\mathrm{Mg}}{\mathrm{K}}\right)}^{1/2}$

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 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 the force constant is \(k''\). Then \(k':k''\) is:
1. \(1:9\)
2. \(1:11\)
3. \(1:14\)
4. \(1:6\)