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Oscillations

Simple Harmonic Motion

Such type of vibratory motion in which restoring force is directly proportional to the displacement from the mean position and it always acts towards the mean position.

• Vibratory motion is that in which a body moves to and fro about a fixed position along same path. Examples:
  • Motion of simple pendulum.
  • Motion of molecules of a solid.
• Simple harmonic motion (S.H.M) is a special type of vibratory motion in which:

$$ a\propto -x\quad and\quad a\quad\text{is directed towards mean position.} $$

• Restoring force is always directed towards mean position hence assigned negative sign.
• Periodic motion is that which repeats itself after equal time intervals.
• Vibration is one complete round trip of a body about its mean position.
• Time period is defined as time taken by vibrating body to complete its one vibration and denoted by \(T\).
• Frequency is number of vibrations per second and denoted by \(f\) so \(f\) =\(\frac{1}{T}\). Its unit is \(\text{Hz}\). Other units are \(vibration/s\), \(cycle/s\), \(rev/s\).
• Amplitude is maximum distance from mean position.
• Angular frequency is \(\omega=2\pi/T\rightarrow\omega=2\pi f\)
• Phase is the angle which specifies the displacement and direction of motion of the point executing S.H.M i.e. phase = \(\theta\) = \(\omega t\)
• Initial angle at \(t = 0\) is called phase constant and denoted by \(\varphi\)
• If phase constant \(\phi=-90^{o}\), then displacement \(x=x_o \sin (\omega t+90^{o})= x_o \cos\omega t\), and simple harmonic oscillator starts its SHM from positive extreme position.

Horizontal Mass-Spring System

• For spring, Hooke's law states that: $$ strain\propto stress\space\text{(within elastic limit)} $$ $$ F=kx $$ Where \(k\)=\(\frac{F}{x}\) is called spring constant or force constant. If a spring is cut into two equal parts then spring constant of each spring is doubled.

• Mass attached to spring has S.H.M, $$ a=-\frac{k}{m}x $$.

• For spring mass system doing S.H.M, $$ \omega=\sqrt\frac{k}{m} $$ $$ a\propto -x $$

• Mass spring system has S.H.M and we can trace its waveform (pictorial) display between time and displacement for S.H.M by following relation: $$ x=x_o\sin\bigg(\frac{2\pi}{T}\bigg)t $$

Spring in Series

The resultant of spring constant in case of the series combination is $$ \frac{1}{k}=\frac{1}{k_1}+\frac{1}{k_2}+..... $$

Spring in Parallel

The resultant of spring constant in case of the parallel combination is $$ k=k_1 + k_2+..... $$

This behaviour of springs resembles with capacitances in series and in parallel combinations.

• Time period of single mass attached to spring is given as:\(\quad\) \(T=2\pi\sqrt\frac{m}{k}\)

$$ T\propto\sqrt{m}\qquad T\propto\frac{1}{\sqrt{k}} $$

Its displacement is given as: \(\quad x=x_o\sin\omega t\).

• Instantaneous velocity of mass \(m\) attached to a spring is given as:

$$ v_{ins}=\sqrt{\frac{k}{m}(x^{2}_o -x^{2})} $$

$$ or\space v_{ins}=\sqrt\frac{k}{m}x_o\sqrt{1-\frac{x^ {2} }{x^ {2}_o}} $$

$$ or\space v_{ins}= v_{max}\sqrt{1-\frac{x^ {2} }{x^ {2}_o}} $$

• Maximum speed of mass attached to spring is given as:

$$ v_{max}= v_o= x_o\sqrt\frac{k}{m} $$

• Instantaneous velocity of spring-mass system doing S.H.M is proportional to constant of proportionality being the maximum velocity.

$$ \sqrt{1-\frac{x^ {2}}{x^ {2}_o}} $$

• In case of vertical spring,\(\space F=mg=kx\)

$$ \frac{m}{k}=\frac{x}{g}\quad(Here\space x\space is\space elongation) $$

• Then time period will be \(\qquad T=2\pi\sqrt\frac{x}{g}\)

Do you know?

In one complete vibration, a body covers a distance equal to four times of the amplitude.

Motion of Projection of a Body Moving in a Circle on Diameter

• Motion of projection of a body moving in a circle, on the diameter with constant speed is S.H.M.
• Its acceleration is given as: \(\qquad a= -\omega^{2}x\)
• Time period of projection is given as: \(\qquad T=\frac{2\pi}{\omega}\)
• Speed of projection is given as:

$$ v=\omega\sqrt{{r^ {2}}-{x^ {2}}}\space where\space r=radius\space of\space the\space circle=amplitude\space of\space S.H.M $$

• Projection speeds up when moving towards the centre of the circle.
• Projection slows down when moving away from the centre of the circle.
• If speed \(\omega\) of body in circular motion is not constant then projection does not have S.H.M but has vibratory motion, which is non-S.H.M.

Simple Pendulum

It consists of a heavy point mass suspended from a rigid support by means of almost weightless and inextensible string.

• Galileo invented simple pendulum.
• Motion of simple pendulum is S.H.M if there is no damping.
• Damping force reduces the amplitude of simple pendulum continuously and finally its motion is stopped.
• In absence of damping force, restoring force on simple pendulum is given as; \(F_r= -mg\sin\theta\), and for small amplitude oscillations \(F_r = -mg\theta\).
• Equation of acceleration of simple pendulum for small amplitude is; \(a = -\frac{g}{l}x\)

Thus \(\omega =\sqrt\frac{g}{l}\) for simple pendulum and does not depend on mass like the mass-spring system does.

• Time period and frequency of simple pendulum are given as;

$$ T=2\pi\sqrt\frac{l}{g}\quad and\quad f=\frac{1}{2\pi}\sqrt\frac{g}{l} $$

• If amplitude of simple pendulum is not small then, it has non-S.H.M as \(a= -g\sin h\) and we know that \(\sin\theta=\theta\) only when \(\theta\) is small.
• Pendulum Suspended in a Lift: If the pendulum is suspended in a lift ascending up with uniform acceleration \('a'\) then its time-period is

$$ T=2\pi\sqrt\frac{l}{g+a} $$

• If the pendulum is suspended in a lift descending down with acceleration \('a'\) then

$$ T=2\pi\sqrt\frac{l}{g-a} $$

Do you know?

The time period of simple pendulum is independent of its mass and its amplitude.

Kinds of Pendulum

1. Simple pendulum
2. Compound pendulum or physical pendulum
3. Torsion Pendulum

A second pendulum has following characteristics:

1. Time period\(= 2s\)
2. Frequency\(= 0.5 Hz\)
3. Length\(= 0.99\;\text{or}\;1m\)

Energy Conservation in S.H.M

• Its K.E is given as

\[ K.E_{inst}= \frac{1}{2}k x^{2}_o\bigg(1-\frac{x^ {2}}{x^ {2}_o}\bigg) \]

\[ K.E_{max}=\frac{1}{2}k x^{2}_o \quad \text{It is at mean position} \]

\[ K.E_{min}=0 \qquad\text{It is at extreme position} \]

\[ K.E_{inst}= K.E_{max}\bigg(1-\frac{x^ {2}}{x^ {2}_o}\bigg) \]

• Its P.E is given as

\[ P.E_{inst}=\frac{1}{2}kx^{2} \]

\[ P.E_{max}=\frac{1}{2}k x^{2}_o \quad\text{It is at extreme position} \]

\[ P.E_{min}= 0 \qquad\text{It is at mean position} \]

• Total energy of system =\(\frac{1}{2}k x^{2}_o\) energy remain conserve in S.H.M. In one vibration K.E attains its maximum value twice.

Free & Forced Oscillations

• Oscillations of a system is called free vibration if it oscillates without the interference of an external force.
• Frequency of free oscillation is called natural frequency of the system.
• When a system performs oscillation in the presence of external periodic force, its vibration is called forced oscillation.
• A physical system under going forced vibration is known as driven harmonic oscillator.

Damped Oscillation

Such oscillations in which the amplitude decreases steadily with time are called as damped oscillations.

• In shock absorber of a car critical damping is applied.

Resonance

Phenomenon of increase in amplitude of a body (capable of vibrating) under the action of a periodic force whose time period is equal to natural time period of body.

\[ OR \]

Specific response of a system to external periodic force whose time period is equal to natural time period of a body.

\[ OR \]

Process in which one body transfers its vibrations to nearby body whose natural time period is agreeable to it.

Do you know?

Damping is a process where energy is dissipated from the oscillating system.

• For tuning circuit of T.V or radio or mobile phone, elctrical resonance takes place at following frequency:

$$ f=\frac{1}{2\pi\sqrt{LC}} $$

• Magnetic resonance imaging (MRI) is a resonance phenomenon using radio frequency waves. It is less damaging than X-rays imaging process.
• Suspension bridge may break down due to vibration with increased amplitude caused by resonance.
• We get tired on walking briskly because of forced oscillations fed into our legs for resonance.
• Loose parts of car produce noise at specific speed due to resonance.

Do you know?

A microwave oven generates high frequency waves, which heats up water and fat molecules, by large energy absorption, and hence food is cooked.

Sharpness of Resonance

• Amplitude of vibration decreases with damping force.
• Amplitude of vibration remain constant with undamped force.
• Smaller the damped force, sharper is the resonance and vice versa.

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