NCERT Solutions for Class 11 Physics Chapter 14 Oscillations

Ans : (b) and (c) 
(a) The swimmer's motion is not periodic. The motion of the swimmer between the banks of a river is back and forth. However, it does not have a definite period. his 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. This is 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 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. 

Ans : (b) and (c) are SHMs 
(a) and (d) are periodic, but not SHMs 
(a) During its rotation about its 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 CJ-tube is simple harmonic. This is because the mercury moves to and fro on the same path, about the fixed position, with a certain 
period of time. 
(c) The ball moves to and fro about the lowermost point of the bowl when released. Also, the ball comes back to ts 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. 

Ans : (b) and (d) are periodic 
(a) It is not a periodic motion. his 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 . 

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

Ans : (a) Zero, Positive, Positive 
(b) Zero, Negative, Negative 
(c) Negative, Zero, Zero 
(d) Negative, Negative, Negative 
(e) Zero, Positive, Positive 
(f) Negative, Negative, Negative 
The given situation is shown in the following figure. Points A and B are the two end points, with AB = 10 cm. O is the midpoint of the path. 

A particle is in linear simple harmonic motion between the endpoints 


(a) At the extreme point A, the particle is at rest momentarily. Hence, its velocity is zero at this point. Its acceleration is positive as it is directed along AO. 
Force is also positive in this case as the particle is directed rightward. 


(b) At the extreme point B, the particle is at rest momentarily. Hence, its velocity is zero at this point. Its acceleration is negative as it is directed along 3. Force is also negative in this case as the particle is directed leftward. 


(c) 

The particle is executing a simple harmonic motion. O is the mean position of the particle. Its velocity at the mean position O is the maximum. The value for velocity is 
negative as the particle is directed leftward. The acceleration and force of a particle executing SHM is zero at the mean position. 


(d)  

The particle is moving toward point O from the end B. This direction of motion is opposite to the conventional positive direction, which is from A to B, Hence, the particle's velocity and acceleration, and the force on it are all negative. 


(e) 

The particle is moving toward point O from the end A. This direction of motion is from A to B, which is the conventional positive direction. Hence, the values for velocity, 
acceleration, and force are all positive. 


(f)

This case is similar to the one given in (d).