Oscillations

Class 11 NCERT Physics

NCERT

1   Which of the following examples represent periodic motion?$\\$ (a) A swimmer completing one (return) trip from one bank of a river to the other and back.$\\$ (b) A freely suspended bar magnet displaced from its N-S direction and released.$\\$ (c) A hydrogen molecule rotating about its center of mass.$\\$ (d) An arrow released from a bow.

Solution :

(a) The swimmer's motion is not periodic. The motion of the swimmer between the banks of the river is to and fro. However, it does not have a definite period. This is because the time taken by the swimmer during his back and forth journey may not be the same.$\\$ (b) The motion of a freely-suspended magnet, if displaced from its $N-S$ direction and released, is periodic because the magnet oscillates about its position with a definite period of time.$\\$ (c) When a hydrogen molecule rotates about its centre of mass, it comes to the same position again and again after an equal interval of time. Such a motion is periodic.$\\$ (d) An arrow released from a bow moves only in the forward direction. It does not come backward. Hence, this motion is not a periodic.

2   Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?$\\$ (a) the rotation of earth about its axis.$\\$ (b) motion of an oscillating mercury column in a U-tube.$\\$ (c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point.$\\$ (d) general vibrations of a polyatomic molecule about its equilibrium position.

Solution :

(a)During its rotation about the axis, earth comes to the same position again and again in equal intervals of time. Hence, it is a periodic motion. However, this motion is not simple harmonic. This is because earth does not have a to and fro motion about its axis.$\\$ (b)An oscillating mercury column in a U-tube is a simple harmonic motion because the mercury moves to and from on the same path, about the fixed position, with a certain period of time.$\\$ (c) the ball moves to and from about the lowermost point of the bowl when released. Also, the

ball comes back to its initial position in the same period of time, again and again. Hence, its motion is periodic as well as simple harmonic.$\\$ (d) A polyatomic molecule has many natural frequencies of oscillation. Its vibration is the superposition of individual simple harmonic motions of a number of different molecules. Hence, it is not simple harmonic, but periodic.

3   Figure depicts four $x-t$ plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion)?

Solution :

(a) It is not a periodic motion. This represents a unidirectional, linear uniform motion. There is no repetition of motion in this case.$\\$ (b) In this case, the motion of the particle repeats itself after $2 s$ . Hence, it is a periodic motion, having a period of $2 s.$$\\$ (c) It is not a periodic motion. This is because the particle repeats the motion in one position only. For a periodic motion, the entire motion of the particle must be repeated in equal intervals of time.$\\$ (d) In this case, the motion of the particle repeats itself after $2 s$. Hence, it is a periodic motion, having a period of $2 s$.

4   Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion ($\omega $ is any positive constant):$\\$ (a)$\sin \omega t-\cos \omega t$$\\$ (b)$\sin^3 \omega t$$\\$ (c)$3 \cos \omega t+\cos 3 \omega t+\cos 5 \omega t$$\\$ (d)$\cos \omega t + \cos 3 \omega t+\cos 5 \omega t $$\\$ (e)exp$(-\omega^2 t^2)$$\\$ (f)$1+\omega t+\omega^2 t^2$

Solution :

(a) SHM$\\$ The given function is:$\\$ $\sin \omega t -\cos omega t\\ =\sqrt{2}[\dfrac{1}{\sqrt{2}}\sin \omega t -\dfrac{1}{\sqrt{2}}\cos \omega t]\\ =\sqrt{2}[\sin \omega t*\cos \dfrac{\pi}{4}\\ -\cos \omega t* \sin \dfrac{\pi}{4}]\\ =\sqrt{2}\sin(\omega t-\dfrac{\pi}{4})$$\\$ This function represents SHM as it can be written in the form:$\\$ $a \sin(\omega t+\Phi)$$\\$ Its period is: $2\pi / \omega$$\\$ (b) Periodic but not SHM$\\$ The given function is:$\\$ $\sin^3 \omega t=1/4[3 \sin \omega t -\sin 3 \omega t]$ The terms $\sin \omega t$ and $\sin \omega t$ individually represent simple harmonic motion (SHM). However, the superposition of two SHM is periodic and not simple harmonic.$\\$ This function represents simple harmonic motion because it can be written in the form:$\\$$ a \cos(\omega t+\Phi)$ Its period is :$ 2\pi / 2 \omega =\pi / \omega$$\\$ (d) Periodic, but not SHM$\\$ The given function is $\cos \omega t + \cos 3 \omega t + \cos 5 \omega t $. Each individual cosine function represents SHM. However, the superposition of three simple harmonic motions is periodic, but not simple harmonic.$\\$ (e) Non-periodic motion$\\$ The given function exp $(-\omega^2 t^2 )$ is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.$\\$ (f) The given function $1+ \omega t +\omega^2 t^2$ is non-periodic.

5   A particle is in linear simple harmonic motion between two points, A and B, $10 cm$ apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is$\\$ (a) at the end A,$\\$ (b) at the end B,$\\$ (c) at the mid-point of AB going towards A,$\\$ (d) at $2 cm$ away from B going towards A,$\\$ (e) at $3 cm$ away from A going towards B, and$\\$ (f) at $4 cm$ away from B going towards A.

Solution :

From above figure, where A and B represent the two extreme positions of a SHM. For velocity, the direction from A to B is taken to b positive. The acceleration and the force, along AP are taken as positive and also BP are taken as negative.$\\$ (a) At the end A, the particle executing SHM is momentarily at rest being its extreme position of motion. Therefore, its velocity is zero. Acceleration is positive because it is directed along AP, Force is also Positive since the force is directed along AP.$\\$ (b) At the end B, velocity is zero. Here, acceleration and force are negative as they are directed along BP.$\\$ (c) At the mid point of AB going towards A, the particle is at its mean position P, with a tendency to move along PA. Hence, velocity is positive. Both acceleration and force are zero.$\\$ (d) At $2 cm$ away from B going towards A, the particle is at Q, with a tendency to move along QP, which is negative direction. Therefore, velocity, acceleration and force all are positive.$\\$ (e) At $3 cm$ away from A going towards B, the particle is at R, with a tendency to move along RP, which is positive direction. Here, velocity, acceleration all are positive.$\\$ (f) At $4 cm$ away from A going towards A, the particles is at S, with a tendency to move along SA, which is negative direction. Therefore, velocity is negative but acceleration is directed towards mean position, along SP. Hence it is positive and also force is positive similarly.