The uniform stick of mass m length \(\text L\) is pivoted at the centre. In the equilibrium position shown in the figure, the identical light springs have their natural length. If the stick is turned through a small angle $\theta$, it executes SHM. The frequency of the motion is:

1. \(\dfrac{1}{2 \pi} \sqrt{\dfrac{6 K}{m}} \)

2. \(\dfrac{1}{2 \pi} \sqrt{\dfrac{3 K}{2 m}} \)

3. \(\dfrac{1}{2 \pi} \sqrt{\dfrac{3 K}{m}} \)

4. None of these

A simple pendulum is oscillating without damping. When the displacement of the bob is less than maximum, its acceleration vector \(\vec a\) is correctly shown in: 

1.		2.
3.		4.

There is a simple pendulum hanging from the ceiling of a lift. When the lift is stand still, the time period of the pendulum is T. If the resultant acceleration becomes g/4, then the new time period of the pendulum is 

1. 0.8 T

2. 0.25 T

3. 2 T

4. 4 T

­­A man measures the period of a simple pendulum inside a stationary lift and finds it to be T sec. If the lift accelerates upwards with an acceleration $\raisebox{1ex}{$\mathrm{g}$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ , then the period of the pendulum will be

1. T

2. $\frac{\mathrm{T}}{4}$

3. $\frac{2\mathrm{T}}{\sqrt{5}}$

4. $2\mathrm{T}\sqrt{5}$

The bob of a pendulum of length l is pulled aside from its equilibrium position through an angle $\theta$ and then released. The bob will then pass through its equilibrium position with a speed v, where v equals

1. $\sqrt{2\mathrm{gl}(1-\mathrm{sin\theta })}$

2. $\sqrt{2\mathrm{gl}(1+\mathrm{cos\theta })}$

3. $\sqrt{2\mathrm{gl}(1-\mathrm{cos\theta })}$

4. $\sqrt{2\mathrm{gl}(1+\mathrm{sin\theta })}$

If the length of a pendulum is made \(9\) times and the mass of the bob is made \(4\) times, then the value of time period will become:
1. \(3T\)
2. \(\dfrac{3}{2}{T}\)
3. \(4{T}\)
4. \(2{T}\)

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 \(20\text{ m/s}^2\) at a distance of \(5\text{ m}\) from the mean position. The time period of oscillation is:
1. \(2\pi \text{ s}\)
2. \(\pi \text{ s}\)
3. \(2 \text{ s}\)
4. \(1 \text{ s}\)