Which of the following relationships between the acceleration '\(a\)' and the displacement '\(x\)' of a particle involves simple harmonic motion?

1.	\(a =   0 . 7 x\)	2.	\(a =   - 200 x^{2} \)
3.	\(a =   - 10 x\)	4.	\(a =   100   x^{3}\)

A spring having a spring constant of 1200 N/m is mounted on a horizontal table as shown in the figure. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released. The frequency of oscillations will be:
    

Identify the correct definition:

A particle is executing SHM with time period T. If the time period of its total mechanical energy is T', then $\frac{T^{\text{'}}}{T}$ will be:

Which of the following examples represent simple harmonic motion?

The rotation of the earth about its axis is:

A particle executing SHM crosses points A and B with the same velocity. Having taken 3 s in passing from A to
B, it returns to B after another 3 s. The time period of the SHM will be:

1. 15 s

2. 6 s

3. 12 s

4. 9 s

In a simple harmonic oscillation, the graph of acceleration against displacement for one complete oscillation will be:
1. an ellipse
2. a circle
3. a parabola
4. a straight line

A particle of mass \(m\) and charge \(\text-q\) moves diametrically through a uniformly charged sphere of radius \(R\) with total charge \(Q\). The angular frequency of the particle's simple harmonic motion, if its amplitude \(<R\), is given by:
1. \(\sqrt{\dfrac{qQ}{4 \pi \varepsilon_0 ~mR} }\)
2. \(\sqrt{\dfrac{qQ}{4 \pi \varepsilon_0 ~mR^2} }\)
3. \(\sqrt{\dfrac{qQ}{4 \pi \varepsilon_0 ~mR^3}}\)
4. \( \sqrt{\dfrac{m}{4 \pi \varepsilon_0 ~qQ} }\)

The figure shows the circular motion of a particle. The radius of the circle, the period, the sense of revolution, and the initial position are indicated in the figure. The simple harmonic motion of the x-projection of the radius vector of the rotating particle P will be:

1. $x\left(t\right)=B$ $\sin \left(\frac{2\mathrm{\pi t}}{30}\right)$

2. $x\left(t\right)=B$ $\cos \left(\frac{\mathrm{\pi t}}{15}\right)$

3. $x\left(t\right)=B$ $\sin \left(\frac{\mathrm{\pi t}}{15}+\frac{\mathrm{\pi }}{2}\right)$

4. $x\left(t\right)=B$ $\cos \left(\frac{\mathrm{\pi t}}{15}+\frac{\mathrm{\pi }}{2}\right)$