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. $\dfrac{9}{16}$