A pendulum is hung from the roof of a sufficiently high building and is moving freely to and fro like a simple harmonic oscillator. The acceleration of the bob of the pendulum is 20 m/s² at a distance of 5 m from the mean position. The time period of oscillation is:

A particle executes linear simple harmonic motion with an amplitude of of 3 cm. When the particle is at 2 cm from the mean position, the magnitude of its velocity  is equal to that of its acceleration. Then, its time period in seconds is 

1. $\sqrt{\frac{5}{\pi }}$

2.$\sqrt{\frac{5}{2\mathrm{\pi }}}$

3. $\frac{4\mathrm{\pi }}{\sqrt{5}}$

4. $\frac{2\mathrm{\pi }}{\sqrt{3}}$

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. \(1:6\)

A particle executes linear simple harmonic motion with amplitude of \(3~\text{cm}\). When the particle is at \(2~\text{cm}\) from the mean position, the magnitude of its velocity is equal to that of its acceleration. Then its time period in seconds is:
1. \(\frac{\sqrt5}{2\pi}\)
2. \(\frac{4\pi}{\sqrt5}\)
3. \(\frac{4\pi}{\sqrt3}\)
4. \(\frac{\sqrt5}{\pi}\)

A body of mass \(m\) is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass \(m\) is slightly pulled down and released, it oscillates with a time period of \(3~\text{s}\). When the mass \(m\) is increased by \(1~\text{kg}\), the time period of oscillations becomes \(5~\text{s}\). The value of \(m\) in \(\text{kg}\) is:
1. \(\dfrac{3}{4}\)

2. \(\dfrac{4}{3}\)

3. \(\dfrac{16}{9}\)

4. \(\dfrac{9}{16}\)