The displacement-time graph of a particle executing SHM is shown in the figure. Its displacement equation will be: (Time period = 2 second)

1. $\mathrm{x}=10$ $\sin \left(\mathrm{\pi t}+\frac{\mathrm{\pi }}{6}\right)$

2. $\mathrm{x}=10$ $\sin \left(\mathrm{\pi t}\right)$

3. $\mathrm{x}=10$ $\cos \left(\mathrm{\pi t}\right)$

4. $x=5$ $\sin \left(\mathrm{\pi t}+\frac{\mathrm{\pi }}{6}\right)$

The figure depicts four \((\mathrm{x-t})\) plots for the linear motion of a particle.

(a)

(b)
(c)
(d)

Which of the following is true​​​​​​?

The velocity-time diagram of a harmonic oscillator is shown in the figure given below. The frequency of oscillation will be:
                

1. 25 Hz
2. 50 Hz
3. 12.25 Hz
4. 33.3 Hz

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)

Which of the following functions of time represent/s periodic motion? [ω is any positive constant]

(a) sinωt + cosωt

(b) sinωt + cos2ωt + sin4ωt

(c) e^–ωt

(d) log(ωt)

(e) sin²ωt

1.  (a)

2.  (a), (b) and (d)

3.  (a), (b) and (e)

4.  All

Which of the following statements is/are correct.

1. (a), (b) and (d)
2. (b) and (d)
3. (a), (c) and (d)
4. All are correct

For a particle executing SHM the displacement \(x \) is given by, \(A\cos \omega t.\)  Identify the graph which represents the variation of potential energy (P.E.) as a function of time \(t\) and displacement \(x.\)

   
1. I, III
2. II, IV
3. II, III
4. I, IV

The variation of acceleration, \(a~\) of a particle executing SHM with displacement \(x~\) is:

1.		2.
3.		4.