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1.

When two displacements represented by y₁=asin(ωt) and y₂=bcos(ωt) are superimposed,the motion is -
(1) not a simple harmonic
(2) simple harmonic with amplitude a/b
(3) simple harmonic with amplitude $\sqrt{a^{2 }+ b^2}$
(4) simple harmonic with amplitude (a+b)/2

2. A body performs simple harmonic motion about \(x=0\) with an amplitude a and a time period \(T\). The speed of the body at \(x= \frac{a}{2}\) will be:
1. \(\frac{\pi a\sqrt{3}}{2T}\)
2. \(\frac{\pi a}{T}\)
3. \(\frac{3\pi^2 a}{T}\)
4. \(\frac{\pi a\sqrt{3}}{T}\)

3. Which one of the following equations of motion represents simple harmonic motion? (where \(k,k_0,k_1~\text{and}~a\) are all positive.)
1. Acceleration \(=-k_0x+k_1x^2\)
2. Acceleration \(=-k(x+a)\)
3. Acceleration \(=k(x+a)\)
4. Acceleration \(=kx\)

4.

A point performs simple harmonic oscillation of period T and the equation of motion is given by x= a sin $\left(\omega t +\pi /6\right)$.After the elapse of what fraction of the time period the velocity of the point will be equal to half to its maximum velocity?
(1)$\frac{T}{8}$
(2) $\frac{T}{6}$
(3) $\frac{T}{3}$
(4) $\frac{T}{12}$

5.

The displacement equation of a particle is $x=3\sin 2t+4\cos 2t$ The amplitude and maximum velocity will be respectively
(a) 5, 10      
(b) 3, 2
(c) 4, 2       
(d) 3, 4

6.

For a particle executing simple harmonic motion, the kinetic energy K is given by $k=k_{o }{\cos }^2 \omega t$. The maximum value of potential energy is
(a) $K_o$                     
(b) Zero
(c) $\frac{K_o}{2}$                 
(d) Not obtainable

7.

A simple pendulum is set up in a trolley which moves to the right with an acceleration a on a horizontal plane. Then the thread of the pendulum in the mean position makes an angle $\mathrm{\theta }$ with the vertical
(1) ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{g}}\right)$ in the forward direction
(2) ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{g}}\right)$ in the upward direction 
(3) ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{g}}\right)$ in the backward direction
(4) ${\tan }^{-1}\left(\frac{g}{a}\right)$ in the forward directions 

8.

The bob of a pendulum of length l is pulled aside from its equilibrium position through an angle $\theta$ and then released. The bob will then pass through its equilibrium position with a speed v, where v equals
(1) $\sqrt{2\mathrm{gl}(1-\mathrm{sin\theta })}$
(2) $\sqrt{2\mathrm{gl}(1+\mathrm{cos\theta })}$
(3) $\sqrt{2\mathrm{gl}(1-\mathrm{cos\theta })}$
(4) $\sqrt{2\mathrm{gl}(1+\mathrm{sin\theta })}$

9.

The kinetic energy of a particle executing S.H.M. is 16 J when it is in its mean position. If the amplitude of oscillations is 25 cm and the mass of the particle is 5.12 kg, the time period of its oscillation is -
(1) $\frac{\mathrm{\pi }}{5}sec$       
(2) $2\mathrm{\pi } \sec$
(3) $20\mathrm{\pi } \sec$   
(4) $5\mathrm{\pi } \sec$

10. A pendulum has time period \(T\). If it is taken on to another planet having acceleration due to gravity half and mass \(9\) times that of the earth, then its time period on the other planet will be:

1.	\(\sqrt{T} \)	2.	\(T \)
3.	\({T}^{1 / 3} \)	4.	\(\sqrt{2} {T}\)

11.

The time period of a simple pendulum of length L as measured in an elevator descending with acceleration $\frac{\mathrm{g}}{3}$ is
(1) $2\mathrm{\pi }\sqrt{\frac{3\mathrm{L}}{\mathrm{g}}}$
(2) $\mathrm{\pi }\sqrt{\left(\frac{3\mathrm{L}}{\mathrm{g}}\right)}$
(3) $2\mathrm{\pi }\sqrt{\left(\frac{3\mathrm{L}}{2\mathrm{g}}\right)}$
(4) $2\mathrm{\pi }\sqrt{\frac{2\mathrm{L}}{3\mathrm{g}}}$

12.

The displacement of a particle varies with time as $x=12\sin wt-16 {\sin }^3 wt$ (in cm). If its motion is S.H.M., then its maximum acceleration is -
(a) $12{\omega }^2$      
(b) $36{\omega }^2$
(c) $144{\omega }^2$       
(d)  $\sqrt{192}{\omega }^2$

13.

A cylindrical piston of mass M slides smoothly inside a long cylinder closed at one end, enclosing a certain mass of gas. The cylinder is kept with its axis horizontal. If the piston is disturbed from its equilibrium position, it oscillates simple harmonically. The period of oscillation will be
 

(1) $T=2\pi \sqrt{\left(\frac{Mh}{PA}\right)}$
(2) $T=2\pi \sqrt{\left(\frac{MA}{Ph}\right)}$
(3) $T=2\pi \sqrt{\left(\frac{M}{PAh}\right)}$
(4) $T=2\pi \sqrt{MPhA}$

14.

An ideal spring with spring-constant K is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring initially unstretched. Then the maximum extension in the spring is -
(1) 4 Mg/K         
(2) 2 Mg/K
(3) Mg/K             
(4) Mg/2K

15.

Two pendulums have time periods T and 5T/4. They start SHM at the same time from the mean position. What will be the phase difference between them after the bigger pendulum completed one oscillation?
1. $45°$
2. $90°$
3. $60°$
4. $30°$