Qualitative Ideas Of Free, Forced And Damped Oscillations, Concept Of Lissajou Figure In Superposition Of SHM (JEE LEVEL)

Q.1

A point moves in the plane XY according to the law x = a sin wt and y = b cos wt, where a, b and w are positive constants. Find the trajectory equation y(x) of a point and the direction of its motion along this trajectory,

(a) \displaystyle \frac{{{{x}^{2}}}}{{{{a}^{2}}}}+\frac{{{{y}^{2}}}}{{{{b}^{2}}}}=1    

(b)   \displaystyle \frac{{{{x}^{2}}}}{{{{a}^{2}}}}-\frac{{{{y}^{2}}}}{{{{b}^{2}}}}=1       

(c)  \displaystyle \frac{{{{x}^{2}}}}{{{{a}^{2}}}}+\frac{{{{y}^{2}}}}{{{{b}^{2}}}}=2        

(d)  None

Ans : (a)

Q.2

The number of harmonic components in the oscillation represented by y = 4 cos² 2t sin 4t and their corresponding angular frequencies are:

(a)    there; 2 rad/s, 4 rad/s, 8 rad/s                     

(b)    two; 2 rad/s, 4 rad/s

(c)   two;4 rad/s, 8 rad/s       

(d)   two; 2 rad/s, 8 rad/s

Ans.  (c)

Q.3

A particle of mass m is attached to a spring (of spring constant k) and has a natural angular frequency w₀. An external force F(t) proportional to cos wt (w ¹ w₀) is applied to the oscillator. The time displacement of the oscillator will be proportional to:

(a)          {{\sin }^{{-1}}}(\lambda /d)             

(b)          {{\sin }^{{-1}}}(\lambda /2d)          

(c)           {{\sin }^{{-1}}}(\lambda /3d)          

(d)          {{\sin }^{{-1}}}(\lambda /4d)

Ans. (b)

Q.4

When a damped harmonic oscillator completes 100 oscillations, its amplitude is reduced to 1/3 of its initial value. What will be its amplitude when it completes 200 oscillations:

(a)  1/5                   (b)  2/3                   (c)         1/6                  (d)      1/9

Ans.  (d)

Q.5

A particle, with restoring force proportional to displacemtn and resisting force proportional to velocity is subjected to a force F sin w If the amplitude of the particle is maximum for w = w₁ and the energy of the particle is maximum for w = w_2, then

(a)      w₁ = w₀ and w₂ ≠ w₀                                                

(b)      w₁ = w₀ and w₂ = w₀            

(c)      w₁ ≠ w₀ and w₂ = w₀                                                

(d)      w₁ ≠ w₀ and w₂ ≠ w₀

Ans.  (c)

Q.6

A point moves in the plane XY according to the law x = a sin wt and y = b cos wt, where a, b and w are positive constants the acceleration of the point as a function of its radius vector r relative to the origin of coordinates must be

(a)  \displaystyle \vec{a}=\omega \vec{r}                                 

(b)  \displaystyle \vec{a}={{\omega }^{2}}\vec{r}                

(c)    \displaystyle \vec{a}=2{{\omega }^{2}}\vec{r}                        

(d)   None

Ans :(b)

Q.7

Find the trajectory equation y(x) of a point if it moves according to the following laws

         x = a sin wt    and    y = a sin 2 wt

(a)  \displaystyle y=\frac{{2x\sqrt{{{{a}^{2}}+{{x}^{2}}}}}}{a}     

(b) \displaystyle y=\frac{{4x\sqrt{{{{a}^{2}}-{{x}^{2}}}}}}{a}                

(c) \displaystyle y=\frac{{2x\sqrt{{{{a}^{2}}-{{x}^{2}}}}}}{a}         

(d)     None

Ans : (c)

Q.8

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

Ans.  (a)

Q.9

Lissajous figure obtained by combining

x = a sin w t   and   \displaystyle \pi {{\cos }^{{1}}}\frac{4}{5}

will be:

(a)  An ellipse            (b)  A straight line           (c)     A circle             (d)    A parabola

Ans.  (a)

Q.10

The phase difference between the two simple harmonic oscillations

  \displaystyle \frac{1}{2},\,\,\frac{1}{2}   and        \displaystyle \frac{1}{2},\,\,\frac{1}{2} is:

(a)          \displaystyle \frac{1}{2},\,\,\frac{1}{2}

(b)          \displaystyle \frac{1}{2},\,\,\frac{1}{2}

(c)           \displaystyle A=\hat{i}+4\hat{j}2k\,\,and\,\,B=3\hat{i}5\hat{j}+k 

(d)          \displaystyle 4\hat{i}\hat{j}k

Ans.  (c)