A particle performs simple harmonic motion with amplitude \(A\). Its speed is tripled at the instant that it is at a distance \(\frac{2A}{3}\) from the equilibrium position. The new amplitude of the motion is: 
1. \( \frac{A}{3} \sqrt{41} \)
2. \(3 \mathrm{A} \)
3. \(A \sqrt{3} \)
4. \(\frac{7 A}{3}\)

The displacement-time \((S \text-t)\) graph of a particle executing simple harmonic motion (SHM) is provided (the sketch is schematic and not to scale).

Which of the following statements are true for this motion?

Choose the correct option from the options given below:

When a particle executes SHM, the nature of graphical representation of velocity as a function of displacement is:
1. circular
2. elliptical
3. parabolic
4. straight line

Point A moves with a uniform speed along the circumference of a circle of radius \(0.36\) m and covers \(30^\circ\) in \(0.1\) s. The perpendicular projection 'P' from 'A' on the diameter MN represents the simple harmonic motion of 'P'. The restoration force per unit mass when P touches M will be :

  
1. \(100~\mathrm{N}\)
2. \(0.49~\mathrm{N}\)
3. \(50~\mathrm{N}\)
4. \(9.87~\mathrm{N}\)

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance (\(R/2\)) from the earth's center, where '\(R\)' is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period :
1. \(\frac{2 \pi R}{g} \)
2. \(\frac{\mathrm{g}}{2 \pi \mathrm{R}} \)
3. \(\frac{1}{2 \pi} \sqrt{\frac{g}{R}} \)
4. \(2 \pi \sqrt{\frac{R}{g}} \)