The displacement of a particle is represented by the equation \(y= 3 \cos \left(\frac{\pi}{4}-\omega t \right)\). The motion of the particle is:

1.	simple harmonic with period \(\frac{2\pi}{\omega}\).
2.	simple harmonic with period \(\frac{\pi}{\omega}\).
3.	periodic but not simple harmonic.
4.	non-periodic.

The displacement of a particle is represented by the equation y = ${\sin }^3$$\omega t$. The motion is

1. non-periodic

2. periodic but not simple harmonic

3. simple harmonic with period 2$\pi /\omega$

4. simple harmonic with period $\pi /\omega$

The relations between acceleration and displacement of four particles are given below. Which one of the particles is executing simple harmonic motion?
1. \(a_1 = +2x\)
2. \(a_1= +2x^2\)
3. \(a_1= -2x^2\)
4. \(a_1 = -2x\)

A particle is acted simultaneously by mutually perpendicular simple harmonic motion x=acos$\omega t$ and y = asin$\omega t$. The trajectory of motion of the particle will be:

1. an ellipse

2. a parabola

3. a circle

4. a straight line

The displacement of a particle varies with time according to the relation, \(y=a~\text{sin} \omega t+b~\text{cos} \omega t.\)

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

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