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

The equation of motion of a particle is x =$a \cos {(\alpha t)}^2$. The motion is:

1. periodic but not oscillatory

2. periodic and oscillatory

3. oscillatory but not periodic

4. neither periodic nor oscillatory

A particle executing SHM has a maximum speed of \(30\) cm/s and a maximum acceleration of \(60\) cm/s².  The period of oscillation is:
1. \(\pi \) s
2. \(\frac{\pi }{2}\) s
3. \(2\pi\) s
4. \(\frac{\pi }{4}\) s

When a mass \(m\) is connected individually to two springs \(S_1\) and \(S_2,\) the oscillation frequencies are \(\nu_1\) and \(\nu_2.\) If the same mass is attached to the two springs as shown in the figure, the oscillation frequency would be: ${}_{}$

         
1. \(v_2+v_2\)

2. \(\sqrt{v_1^2+v_2^2}\)

3. \(\frac{1}{v_1}+\frac{1}{v_1}^{-1}\)

4. \(\sqrt{v_1^2-v_2^2}\)

The following statements are given for a simple harmonic oscillator.
 

Choose the correct alternatives:

1. (a, b, d)

2. (a, c)

3. (b, d)

4. (c, d)

The (displacement-time) graph of a particle executing SHM is shown in the figure. Then,
     

Which of the statement/s given above is/are true?

A body is performing SHM, then its:

(a) average total energy per cycle is equal to its maximum kinetic energy.

(b) average kinetic energy per cycle is equal to half of its maximum kinetic energy.

(c) mean velocity for a complete cycle is equal to $\frac{2}{\mathrm{\pi }}$ times of its maximum velocity.

(d) root mean square velocity is $\frac{1}{\sqrt{2}}$ times of its maximum velocity.

Choose the correct alternatives:

1. (a, b, d)

2. (a, c)

3. (b, d)

4. (b, c, d)

The rotation of the earth about its axis is:

(a) periodic motion

(b) simple harmonic motion

(c) periodic but not simple harmonic motion

(d) non-periodic motion

Choose the correct alternatives:

1. (a, b, d)

2. (a, c)

3. (b, d)

4. (c, d)

The motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower point is
(a) simple harmonic motion
(b) non-periodic motion
(c) periodic motion
(d) periodic but not SHM
Choose the correct alternatives:
1. (a), (c)
2. (a), (d)
3. (c), (d)
4. (b), (c)

Four pendulums A, B, C, and D are suspended from the same elastic support as shown in the figure. A and C are of the same length, while B is smaller than A and D is large than A. If A is given a transverse displacement,

1. D will vibrate with maximum amplitude

2. C will vibrate with maximum amplitude

3. B will vibrate with maximum amplitude

4. All four will oscillate with equal amplitude