(i) Oscillatory motion are periodic motions that repeat itself after definite interval of time and occur about a mean position due to property of inertia and elasticity of oscillating body. e.g., vibration of strings of musical instrument, and motion of swing.
(ii) The revolution of earth around the sun and the motion of hands of a clock is a periodic motion but not oscillatory.
(iii) Harmonic oscillation is that oscillation which can be expressed in terms of single harmonic function (i.e., sine function or cosine function).
Non-harmonic oscillation cannot be expressed in terms of single harmonic function.
(iv) The oscillation of a physical system results from two basic properties of the system namely elasticity and inertia.
(v) The time period of a simple harmonic oscillator depend upon its inertial and elastic properties as follows.
T = \(2 \pi \sqrt{\frac{\text { Inertia factor }}{\text { Elastic factor }}}\) 
(vi) Peridoc functions are those functions which are used to represent periodic motion. A functionJ(t) is said to be periodic, ifJ(t) = J(t + T) = J(t + 2T)
Sine and Cosine functions are the periodic function.

(i) Simple harmonic motion is the projection ofa uniform circular motion on any diameter of a circle of reference x = a cos \(\theta=a \cos \omega t\) .

(ii) Velocity v = \(\omega \sqrt{a^{2}-y^{2}}\) 
acceleration a = \(-\omega^{2} y\) .
(iii) The particle velocity in S.H.M is in phase with acceleration when particle is moving from extreme position to mean position and is in opposite phase with acceleration when particle is moving from mean position to extreme position.
(iv) Average value of total energy of a particle in S.H.M in one complete oscillation \(E_{a v}=\frac{1}{2} m w^{2} a^{2}\) .
(v) The frequency of PE and K.E is double than that of S.H.M while the frequency of total energy of particle in S.H.M is zero.
(vi) T = \(2 \pi \sqrt{\frac{l}{g}}\) = 2
T = \(2 \pi \sqrt{\frac{2 l}{g}}\) 
= \(\sqrt{2} \times 2 \pi \sqrt{\frac{l}{g}}\) 
= \(\sqrt{2} \times 2=2.828 \mathrm{sec}\).
(vii) The periodic time T will decrease because in the standing position, the location of CM of the girl shifts upwards. Due to which the effective length of the pendulum l becomes less. As T \(T \propto \sqrt{l}\). Therefore T decreases.

(i) In fig (a) two springs are in parallel combination, Net restoring force is sum of restoring force produced in k₁ and k₂, for displacement y.
F = F₁ + F₂ = - (k₁ + k₂)y
therefore, spring constant of the combination is k = k₁ + k₂
(ii) Fig (b) is series combination of springs, therefore restoring force is same in each spring but both experience different extensions.
F = -k₁y₁ = -k₂Y₂ 
y = y₁ + Y₂
\(\Rightarrow \quad f=-\left(\frac{k_{1} k_{2}}{k_{1}+k_{2}}\right) y\) 
there spring constant of the combination is
\(k=\frac{k_{1} k_{2}}{k_{1}+k_{2}}\) 
(iii) Fig (c) is parallel combination of spring, therefore
Total restoring force F = -(k₁ + k₂ )y
and spring constant K = k₁ + k₂
(iv) In fig (a)  v = \(\frac{1}{2 \pi} \sqrt{\frac{k}{m}}\) 
= \(\frac{1}{2 \pi} \sqrt{\frac{k_{1}+k_{2}}{m}}\) 
In fig (b)  v = \(\frac{1}{2 \pi} \sqrt{\frac{k_{1} k_{2}}{m\left(k_{1}+k_{2}\right)}}\) 
In fig (c) v = \(\frac{1}{2 \pi} \sqrt{\frac{k_{1}+k_{2}}{m}}\) 

(i) When a simple harmonic system oscillates with a constant amplitude which does not change with time, its oscillations are called undamped simple harmonic oscillations.
(ii) When a system of simple harmonic oscillates with decreasing amplitude with time its oscillation are called damped simple harmonic oscillations. The dissipative forces or damping forces are active in the system which are generally the frictional or viscous forces. For example, the oscillations of the bob of a simple pendulum in air or medium are damped oscillations.
(iii) A system capable of oscillating is said to be executing free oscillations if it vibrates with its own natural frequency without the help of any external periodic force. The natural frequency of vibration of system depends upon 
(i) inertia facstor (m)
(ii) elastic properties and dimension of the system
(k) i.e., V₀ = \(\frac{1}{2 \pi} \sqrt{\frac{k}{m}}\)  
(iv) When a body oscillates with the help of an external periodic with a frequency different from the natural frequency of the body, its oscillations are called forced oscillations.
(v) Sound board of all stringed musical instruments like violin, sitar, etc. execute forced oscillations.
(vi) Phenomenon of increase in amplitude of oscillation when the frequency of the driving force is close to the natural frequency of the oscillator is called resonance. For example, the loud sound is heard due to resonant oscillation when the frequency of oscillations of the air column becomes equal to the frequency of the vibrating tuning fork.
(vii) No, because damping force depends upon velocity and is more when the system moves fast and is less when system moves slow.

(i) Angular frequency
\(w=\sqrt{\frac{k}{m}}\) 
= \(\sqrt{\frac{800}{2}}\) 
= 20 rad/s
(ii) Maximum extension in the spring from mean position is x
then mg + ma = kx
\(\Rightarrow \quad x=\frac{m g+m a}{k}\) 
= \(\frac{2(10+10)}{800}\) = 5 cm
Extension of the spring when it is stretched to equilibrium line x'
mg = kx'
\(\Rightarrow \quad x^{\prime}=\frac{m g}{k}\) = 25 cm
therefore amplitude,
A = x - x' = 2.5 cm.
(iii) If downward direction is taken as postive at t = 0, x = A
using x = A sin\((\omega t+\phi)\) 
\(\Rightarrow \quad \phi=\frac{\pi}{2}\) 
(iv) y = a sin\(\left(\omega t+\phi_{0}\right)\) , At t = 0, y = \(\frac{a}{2}\) 
\(\therefore \quad \frac{a}{2}=a \sin \left(\omega \times 0+\phi_{0}\right)\) 
= \(a \sin \phi_{0} \text { or } \sin \phi_{0}=\frac{1}{2}=\sin \frac{\pi}{6}\) 
or \(\phi_{0}=\frac{\pi}{6}\) 
(v) acceleration \(a=-\omega^{2} y\) 
= \(\frac{-4 \pi^{2}}{T^{2}} \times \frac{A}{2}\) 
= \(\frac{-2 \pi^{2} A}{T^{2}}\)