Oscillations

Oscillations Class 11th Topics (NCERT Syllabus)

1. Introductions

In daily life we see different motions some non-repetitive like rectilinear or projectile motion, and some periodic like circular or planetary motion. A special kind of periodic motion is oscillatory motion, where an object moves to and fro about a mean position (e.g. pendulum, swing, vibrating strings). Oscillatory motion is fundamental in physics and is described using period, frequency, amplitude, displacement, and phase.

2. Periodic and Oscillatory Motion

Periodic Motion: A motion that repeats itself at regular intervals of time (e.g., revolution of Earth around the Sun, motion of a planet).

Oscillatory Motion: A type of periodic motion in which the object moves to and fro about a mean position (e.g., pendulum, swing, vibrating string).

• Periodic and Oscillatory Motion
• Period and Angular Frequency

3. Simple Harmonic Motion

Simple Harmonic Motion is a type of oscillatory motion in which the restoring force (or acceleration) acting on the particle is directly proportional to its displacement from the mean position and is always directed towards the mean position.

$
a \propto-x \quad \text { or } \quad a=-\omega^2 x
$

Equations of SHM:
Displacement: $x(t)=A \sin (\omega t+\phi)$
Velocity: $v(t)=\omega A \cos (\omega t+\phi)$
Acceleration: $a(t)=-\omega^2 A \sin (\omega t+\phi)$

• Simple Harmonic Motion (S.H.M.) And Its Equation

4. Simple Harmonic Motion and Uniform Circular Motion

A simple harmonic motion (SHM) can be understood by relating it to a uniform circular motion (UCM).

Uniform Circular Motion

When a particle moves with constant speed along a circular path, its motion is called uniform circular motion. The angular velocity of the particle is constant in this case.

• Simple Harmonic Motion And Uniform Circular Motion

5. Velocity and Acceleration in Simple Harmonic Motion

Displacement in SHM
The displacement of a particle in SHM at any time $t$ is given by:

$
x(t)=A \sin (\omega t+\phi)
$

Velocity in SHM
Velocity is the time derivative of displacement:

$
v(t)=\frac{d x}{d t}=A \omega \cos (\omega t+\phi)
$

Maximum velocity:

$
v_{\max }=\omega A
$

Velocity is maximum at the mean position ( $x=0$ ) and zero at extreme positions ( $x= \pm A$ ).

Acceleration in SHM
Acceleration is the time derivative of velocity (or second derivative of displacement):

$
a(t)=\frac{d v}{d t}=-\omega^2 A \sin (\omega t+\phi)
$

or,

$
a=-\omega^2 x
$

Maximum acceleration:

$
a_{\max }=\omega^2 A
$

Force Law in SHM (Short)

$
F=-m \omega^2 x
$

6. Force Law for Simple Harmonic Motion

Force $\propto$ displacement, opposite in direction (restoring).
Max force: $F_{\text {max }}=m \omega^2 A$.
Same as Hooke's law: $F=-k x$, with $k=m \omega^2$.

7. Energy in Simple Harmonic Motion

1. Kinetic Energy (KE):

$
K E=\frac{1}{2} m v^2=\frac{1}{2} m \omega^2\left(A^2-x^2\right)
$

Maximum at mean position ( $x=0$ ).
Zero at extremes $(x= \pm A)$.
2. Potential Energy (PE):

$
P E=\frac{1}{2} k x^2=\frac{1}{2} m \omega^2 x^2
$

Maximum at extremes.
Zero at mean position.
3. Total Energy (E):

$
E=K E+P E=\frac{1}{2} m \omega^2 A^2
$

Constant at all positions.
Proportional to square of amplitude $\left(A^2\right)$.

• Energy in Simple Harmonic Motion

8. The Simple Pendulum

A simple pendulum consists of a small heavy bob suspended from a fixed support by a light, inextensible string. When the bob is displaced slightly from its mean position and released, it oscillates to and fro under the action of gravity.

For a small angular displacement $\theta$, the restoring force is:

$
F=-m g \sin \theta \approx-\frac{m g}{l} x
$

which is of the form $F \propto-x$. Hence, the motion of the pendulum is Simple Harmonic Motion for small oscillations.

The time period of oscillation is given by:

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

• Oscillation Of Pendulum

Important Formula of Oscillation

1. General SHM Equations

Displacement:

$
x(t)=A \sin (\omega t+\phi) \quad \text { or } \quad x(t)=A \cos (\omega t+\phi)
$

Velocity:

$
v=\frac{d x}{d t}=\omega A \cos (\omega t+\phi)
$

Acceleration:

$
a=\frac{d v}{d t}=-\omega^2 x
$

Force:

$
F=-m \omega^2 x \quad(\text { Hooke's law form })
$

2. Angular Frequency, Time Period, Frequency

Angular frequency:

$
\omega=2 \pi f=\frac{2 \pi}{T}
$

Time period:

$
T=\frac{2 \pi}{\omega}
$

Frequency:

$
f=\frac{1}{T}
$

3. Energy in SHM

Kinetic Energy:

$
K E=\frac{1}{2} m \omega^2\left(A^2-x^2\right)
$

Potential Energy:

$
P E=\frac{1}{2} m \omega^2 x^2
$

Total Energy:

$
E=\frac{1}{2} m \omega^2 A^2 \quad \text { (constant) }
$

Oscillation Real Life Example

1. Pendulum of a clock – swings to and fro about its mean position.
2. Swing in a park – oscillates back and forth when pushed.
3. Vibrating strings of a guitar/violin – produce sound by oscillating.
4. Drum membrane or speaker diaphragm – vibrates to create sound waves.

Weightage of Oscillations in Exams

NCERT Notes Subject Wise Link:

• NCERT notes Class 11 Maths
• NCERT notes Class 11 Physics
• NCERT notes Class 11 Chemistry
• NCERT notes Class 11 Biology

NCERT Solutions Subject wise link:

• NCERT solutions for class 11 Physics.
• NCERT solutions for class 11 Chemistry.
• NCERT solutions for class 11 Mathematics.
• NCERT solutions for class 11 Biology.

NCERT Exemplar Solutions Subject wise link:

• NCERT exemplar solutions for class 11 Physics.
• NCERT exemplar solutions for class 11 Chemistry.
• NCERT exemplar solutions for class 11 Mathematics.
• NCERT exemplar solutions for class 11 Biology.