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. \(2\pi \left( \frac{2Mg}{K}\right)^{\frac{1}{2}}\)

A mass m is suspended from two springs of spring constant $k_1$ $and$ $k_2$ as shown in the figure below. The time period of vertical oscillations of the mass will be
                

1. $2\mathrm{\pi }\sqrt{\left(\frac{{\mathrm{k}}_1+{\mathrm{k}}_2}{\mathrm{m}}\right)}$

2. $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\left({\mathrm{k}}_1+{\mathrm{k}}_2\right)}}$

3. $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}\left({\mathrm{k}}_1{\mathrm{k}}_2\right)}{\left({\mathrm{k}}_1+{\mathrm{k}}_2\right)}}$

4. $2\mathrm{\pi }\sqrt{\frac{\mathrm{m}\left({\mathrm{k}}_1+{\mathrm{k}}_2\right)}{\left({\mathrm{k}}_1{\mathrm{k}}_2\right)}}$

Two springs, of force constants k₁ and k₂ are connected to a mass m as shown in the figure. The frequency of oscillation of the mass is f. If both k₁ and k₂ are made four times their original values, the frequency of oscillation will become:
        

All the surfaces are smooth and the system, given below, is oscillating with an amplitude \(\mathrm{A}.\) What is the extension of spring having spring constant \(\mathrm{k_1},\) when the block is at the extreme position?
             

When a mass is suspended separately by two different springs, in successive order, then the time period of oscillations is \(t _1\) and \(t_2\) respectively. If it is connected by both springs as shown in the figure below, then the time period of oscillation becomes \(t_0.\) The correct relation between \(t_0,\) \(t_1\) & \(t_2\) is:

1. ${{\mathrm{t}}_0}^2={{\mathrm{t}}_1}^2+{{\mathrm{t}}_2}^2$

2. ${{\mathrm{t}}_0}^{-2}={{\mathrm{t}}_1}^{-2}+{{\mathrm{t}}_2}^{-2}$

3. ${{\mathrm{t}}_0}^{-1}={{\mathrm{t}}_1}^{-1}+{{\mathrm{t}}_2}^{-1}$

4. ${\mathrm{t}}_0={\mathrm{t}}_1+{\mathrm{t}}_2$

A mass of 30 g is attached with two springs having spring constant 100 N/m and 200 N/m and other ends of springs are attached to rigid walls as shown in the given figure. The angular frequency of oscillation will be
                                

1.  $\frac{100}{2\mathrm{\pi }}$ $\mathrm{rad}/\mathrm{s}$

2.  $\frac{100}{\mathrm{\pi }}$ $\mathrm{rad}/\mathrm{s}$

3.  100 rad/s

4.  200$\mathrm{\pi }$ rad/s

A spring is having a spring constant k. It is cut into two parts A and B whose lengths are in the ratio of m:1. The spring constant of part A will be

1. $\frac{k}{m}$

2. $\frac{k}{m+1}$

3. k

4. $\frac{k(m+1)}{m}$