All the surfaces are smooth and springs are ideal. If a block of mass \(m\) is given the velocity \(v_0\) in the right direction, then the time period of the block shown in the figure will be:

                       
1. \(\frac{12l}{v_0}\)
2. \(\frac{2l}{v_0}+ \frac{3\pi}{2}\sqrt{\frac{m}{k}}\)
3. \(\frac{4l}{v_0}+ \frac{3\pi}{2}\sqrt{\frac{m}{k}}\)
4. \( \frac{\pi}{2}\sqrt{\frac{m}{k}}\)

The displacement \( x\) of a particle varies with time \(t\) as \(x = A sin\left (\frac{2\pi t}{T} +\frac{\pi}{3} \right)\)$\mathrm{}$. The time taken by the particle to reach from \(x = \frac{A}{2} \) to \(x = -\frac{A}{2} \) will be:

A force, \(F=-Kx^{n}\) acts on a particle, where \(K\) is a positive constant. The value of \(n\) for which motion can be oscillatory is:
1. \(4\)
2. \(6\)
3. \(3\)
4. \(2\)

When a periodic force \(\vec{F_1}\) acts on a particle, the particle oscillates according to the equation \(x=A\sin\omega t\). Under the effect of another periodic force \(\vec{F_2}\), the particle oscillates according to the equation \(y=B\sin(\omega t+\frac{\pi}{2})\). The amplitude of oscillation when the force (\(\vec{F_1}+\vec{F_2}\)) acts are:

The total mechanical energy of a linear harmonic oscillator is \(600~\text J.\) At the mean position, its potential energy is \(100~\text J.\) The minimum potential energy of the oscillator is: 
1. \(50~\text J\)
2. \(500~\text J\)
3. \(0\) 
4. \(100~\text J\)

Simple harmonic motion is an example of:

A particle is performing SHM with amplitude \(A\) and angular velocity \(\omega.\) The ratio of the magnitude of maximum velocity to maximum acceleration is:
1. \(\omega\)
2. \(\dfrac{1}{\omega }\)
3. \(\omega^{2} \)
4. \(A\omega\)

The acceleration-time graph of a particle undergoing SHM is shown in the figure. Then,