Class 11 Physics Chapter 14 - Oscillations

• Periodic Motion - Periodic motions, processes, or events are those that repeat themselves at regular intervals.

• Oscillatory motion - Oscillatory motion is defined as the movement of a body around a fixed point after regular intervals of time. The mean position or equilibrium position is the fixed position around which the body oscillates.

• Simple harmonic motion - Simple harmonic motion is a sort of periodic oscillatory motion in which the particle I oscillates down a straight line and the particle's acceleration is always directed towards a fixed point on the line. The particle's movement from the fixed point determines the magnitude of acceleration.

• Characteristics of SHM - \[\mathrm{x=A\left ( \sin \omega t + \phi \right ) }\] gives the displacement x in SHM at time t, where the three constants A, and differentiate one SHM from another. A cosine function may also be used to define an SHM:

\[\mathrm{x=A\left ( \cos  \omega t + \delta  \right ) }\]

At any point in time, the displacement of an oscillating particle equals the change in its position vector. "Displacement Amplitude" or "Simple Amplitude" refers to the highest magnitude of displacement in an Oscillatory Motion on each side of its mean position.

As a result, Amplitude A = x max.

\[T=2\pi \sqrt{\frac{m}{k}}\] where, k is the force constant (or spring factor) of spring.

• Frequency - Frequency is defined as the number of Oscillations per second. It is measured in seconds per second or Hertz per second. Amplitude has no bearing on Frequency or Time Period.

Frequency(v)= \[\frac{1}{Time Period}\left ( \frac{1}{T} \right )\]

• Phrase - The amount \[\mathrm{\left (   \omega t + \phi  \right ) }\] is known as the SHM Phase at time t, and it defines the state of motion at that moment. The amount is the Phase at time f = 0 and is referred to as the SHM's Phase Constant, Starting Phase, or Epoch. In the cosine or sine function, the phase constant is the time-independent term.

If displacement of SHM is given by

\[\mathrm{x = A\sin \left (  \omega t\pm \phi  \right ), then }\]

Particle Velocity \[\mathrm{v=\frac{\mathrm{d} x}{\mathrm{d} t} = \omega A\cos \left ( \omega t\pm \phi  \right )}\]

\[\mathrm{v=\omega A\sqrt{1-\sin ^{2}\left ( \omega t\pm \phi  \right )}=\omega A\sqrt{1-\left ( \frac{x^{2}}{A^{2}} \right )}=\omega \sqrt{A^{2}-x^{2}}}\]

Velocity Amplitude of SHM is equal to \[A\omega\]. Moreover, the particle velocity is ahead of displacement of the particle by an angle of \[\mathrm{\frac{\pi }{2}}\]

Phase Difference between displacement and velocity is \[\mathrm{\frac{\pi }{2}}\]

Acceleration of the particle executing S.H.M is given by

\[\mathrm{a= -\omega ^{2}A\sin \left ( \omega t \pm \phi  \right )=-\omega ^{2}A}\]

Phase Difference between displacement and acceleration is π

The restorative force is in charge of keeping the S.H.M. in good working order.

The Restoring Force (F) exerted on the body is given by F = -kx if the displacement (x) from the equilibrium position is modest where k is a constant of force.

• Energy in S.H.M. - The energy of a body that performs SHM alternates between Kinetic and Potential, while the overall energy remains constant. When x is moved away from the equilibrium position:

Time Period and Frequencyof a particle executing S.H.M may be expressed in the following way

\[\mathrm{\textrm{Total Period,T} =2\pi \sqrt{\frac{displacement}{acceleration}}}\]

\[\mathrm{\textrm{Total Period,}\upsilon  =\frac{1}{2\pi }\sqrt{\frac{displacement}{acceleration}}}\]

In general, the time period and frequency can be expressed as

\[\mathrm{T  =2\pi \sqrt{\frac{displacement}{acceleration}}}\]

\[\mathrm{\upsilon  =\frac{1}{2\pi }\sqrt{\frac{displacement}{acceleration}}}\]

• Springs in series - If two springs with spring constants k1 and k2 are connected in series, the combination's spring constant is given by

  \[\mathrm{\frac{1}{k}=\frac{1}{k_{1}}+\frac{1}{k_{2}}}\]

Subtopics of Oscillations

• Introduction

• Periodic and oscillatory motions

• Simple harmonic motion

• Simple harmonic motion and uniform circular motion

• Velocity and acceleration in simple harmonic motion

• Force law for simple harmonic motion

• Energy in simple harmonic motion

• Some systems executing SHM

• Damped simple harmonic motion

• Forced Oscillations and resonance