The displacement of a particle along the x-axis is given by $x=a{\sin }^2$ $\omega t$. The motion of the particle corresponds to:

1.	simple harmonic motion of frequency ω / π.
2.	simple harmonic motion of frequency  3 ω / 2 π.
3.	non-simple harmonic motion.
4.	simple harmonic motion of frequency ω / 2 π.

 

The period of oscillation of a mass \(M\) suspended from a spring of negligible mass is \(T.\) If along with it another mass \(M\) is also suspended, the period of oscillation will now be:
1. \(T\)
2. \(T/\sqrt{2}\)
3. \(2T\)
4. \(\sqrt{2} T\)

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

Which one of the following equations of motion represents simple harmonic motion? (where \(k\), \(k_0\), \(k_1\) and  α  are all positive.)
1. Acceleration = -$k_{0}x+k_1{x}^2$\(k_0\)  
2. Acceleration = -$k\left(x+a\right)$
3. Acceleration = k$\left(x+a\right)$
4. Acceleration = kx

Two simple harmonic motions of angular frequency 100 rad s ^-1 and 1000 rad $s^{-1}$ have the same displacement amplitude. The ratio of their maximum acceleration will be:
1. 1:10
2. 1:10²
3. 1:10³                               
4. 1:10⁴

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) $\frac{T}{12}$

An SHM has an amplitude \(a\) and  a time period \(T.\) The maximum velocity will be:
1. \({4a \over T}\)       
2.  \({2a \over T}\)
3. \({2 \pi \over T}\)
4. \({2a \pi \over T}\)  

Two particles P and Q start from origin and execute Simple Harmonic Motion along X-axis with same amplitude but with periods 3 seconds and 6 seconds respectively. The ratio of the velocities of P and Q when they meet is -

(1)   1 : 2                 

(2)  2 : 1

(3)   2 : 3                 

(4)  3 : 2

The angular velocities of three bodies in simple harmonic motion are ${\omega }_1,{\omega }_2,{\omega }_3$ with their respective amplitudes as $A_1,A_2,A_3$. If all the three bodies have same mass and maximum velocity, then

(a) $A_1{\omega }_1=A_2{\omega }_2=A_3{\omega }_3$        (b)  $A_1{{\omega }_1}^2=A_2{{\omega }_2}^2=A_3{A_3}^2$

(b) ${A_1}^2{\omega }_1={A_2}^2{\omega }_2={A_3}^2{\omega }_3$   (d) ${A_1}^2{{\omega }_1}^2={A_2}^2{{\omega }_2}^2=A^2$  

The amplitude of a particle executing SHM is 4 cm. At the mean position the speed of the particle is 16 cm/sec. The distance of the particle from the mean position at which the speed of the particle becomes $8\sqrt{3}\mathrm{cm}/\sec$ will be

(1)         $2\sqrt{3}cm$             

(2)  $\sqrt{3}cm$

(3)         1 cm                   

(4)  2 cm