Physics Ch14 Oscillations Notes Class 11

Oscillations is the study of back-and-forth or repetitive motion of objects around a fixed point. Examples are the swinging of a pendulum, the vibration of a guitar string, or a spring moving up and down. In this chapter, you will learn how such motions can be described mathematically, what factors control their speed and time period, and why they repeat regularly. Understand Oscillations chapter clearly with CBSE Syllabus Class 11 Physics.

By the end of this chapter, you will understand how oscillations are not only part of daily life but also form the basis for waves, sound, and many physical phenomena.

Oscillations Summary

Oscillations are a fundamental part of physics and nature. Many systems in the world exhibit periodic motion, the swing of a pendulum, vibrations of a guitar string, the motion of a piston, or even the beating of the human heart. Understanding oscillations helps us analyze natural phenomena, design scientific instruments, and explain wave motion.

Periodic and Oscillatory Motion

Motion that repeats itself after equal intervals of time is called periodic motion.

When a body or particle moves back and forth around a fixed position (known as the equilibrium position), such motion is called oscillatory motion.

For example:

• The motion of a pendulum,
• A loaded spring moving up and down,
• Vibrations of atoms in solids.

Oscillatory motion is always periodic, but not all periodic motions are oscillatory. For instance, the motion of the Earth around the Sun is periodic but not oscillatory, because it does not occur about an equilibrium position.

Simple Harmonic Motion (SHM)

The simplest type of oscillatory motion is called Simple Harmonic Motion (SHM). A particle performs Simple Harmonic Motion when the restoring force (or acceleration) is directly proportional to the displacement from the equilibrium position and always directed towards it.

Mathematically:

If the displacement of a particle from equilibrium is xx, then

F = −kx  

a = −ω²x

where ω is called the angular frequency.

This negative sign shows that the acceleration is always opposite to displacement, restoring the particle back to equilibrium.

Characteristics of SHM

1. Equilibrium position: The point about which oscillations take place.
2. Restoring force: A force proportional to displacement and directed towards equilibrium.
3. Periodicity: The motion repeats itself in equal time intervals.
4. Displacement, velocity, and acceleration: These vary sinusoidally with time.

Mathematical Representation of SHM 

The displacement in SHM is expressed as:

x(t) = A sin⁡(ωt+ϕ)

or

x(t) = A cos⁡(ωt+ϕ)

• A: Amplitude (maximum displacement)
• ω: Angular frequency (ω = 2π / T = 2πf)
• T: Time period (time for one oscillation)
• f: Frequency (oscillations per second)
• ϕ: Phase constant (decides initial position)

Examples of Simple Harmonic Motion

(a) Mass attached to a Spring (Spring-Mass System)

• A mass mm attached to a spring of spring constant kk oscillates about equilibrium.
  
  
• Restoring force:

F = − k x

• Equation of motion:

m d²x/dt² = − kx

• Time period:

T = 2 π √(m/k)

(b) Simple Pendulum

• A pendulum consists of a bob of mass mm attached to a light inextensible string of length L.
  
  
• Restoring force due to gravity:

F = − m g sin⁡θ ≈ −mg(x/L) (for small angles)

• Time period:

T = 2 π √L/g

Thus, the time period depends only on the length of the pendulum and acceleration due to gravity.

Velocity and Acceleration in SHM

From displacement:

x(t) = A sin⁡(ωt+ϕ)

• Velocity:

v = dx/dt = ω A cos⁡(ωt+ϕ)

• Acceleration:

a = dv/dt = −ω² A sin⁡(ωt+ϕ)=−ω²x

Thus, acceleration is directly proportional to displacement but opposite in direction.

Phase in SHM

The phase represents the state of oscillation at any time. It is given by:

θ = ωt + ϕ 

The displacement depends on whether the motion is represented by sine or cosine function.

• If θ = 0, the particle is at equilibrium.
• If θ = π/2, the particle is at maximum displacement.
• For x = A cos(ωt + ϕ):

θ = 0 → particle at maximum displacement

Phase decides whether the particle is at equilibrium, moving forward, or backward.

Energy in SHM 

An oscillating body possesses both potential energy (PE) and kinetic energy (KE).

• Kinetic Energy:

KE = ½ mv²  = ½ mω² (A²−x²)

• Potential Energy:

PE = ½ kx² = ½ mω²x²

• Total Energy:

E = KE + PE = ½ kA² = ½ mω²A²

This energy remains constant, but KE and PE keep interchanging as the particle mov

Function of Circular Motion

SHM can be understood as a projection of uniform circular motion (UCM) on a diameter of the circle.

• Imagine a particle moving in a circle with constant angular speed ω\omega.
• Its projection on the diameter performs SHM with the same angular frequency.

This connection helps visualize SHM easily.

Angular SHM 

In rotational systems, SHM occurs when restoring torque is proportional to angular displacement.

τ = −Cθ

Where:
C = torsional constant

Used in torsion pendulum

Damped Oscillations

In reality, oscillations gradually decrease in amplitude due to resistive forces like friction or air resistance. Such oscillations are called damped oscillations.

• Light damping: Oscillations gradually die out but motion remains oscillatory.
• Critical damping: Oscillator returns to equilibrium in minimum time without oscillating.
• Heavy damping: Oscillator slowly returns to equilibrium without oscillating.

Forced Oscillations and Resonance

When an external periodic force is applied to an oscillator, it is called a forced oscillation.

• At low frequencies, the system follows the external force sluggishly.
• At high frequencies, it cannot respond quickly.
• When the frequency of external force matches the natural frequency of the oscillator, the amplitude becomes very large. This condition is called resonance.

Resonance has useful applications (musical instruments, radio tuning) but can also be destructive (bridge collapse due to oscillations).

Conclusion

That’s the summary of oscillation for you. By learning about Simple Harmonic Motion, energy changes, damping, and resonance, you now have a clear idea of how repetitive motions work and why they are so important.

This chapter also sets the foundation for understanding waves and sound, which are just extended forms of oscillatory motion. If this summary notes have helped you in any manner, do share it with your friends. 

FAQs

Q1. What is the time period and frequency in oscillations?
Ans. Time period (T) is the time taken for one complete oscillation. Frequency (f) is the number of oscillations per second and is given by f = 1/T.

Q2. What is angular frequency?
Ans. Angular frequency is the rate of change of phase of the oscillation, given by ω = 2 π f = 2 π / T.

Q3. What is the relation between acceleration and displacement in SHM?
Ans. In SHM, acceleration is proportional to displacement but in the opposite direction: a=−ω²x.

Q4. What is meant by free, damped, and forced oscillations?
Ans. Free oscillations occur without external resistance, damped oscillations lose energy due to resistive forces and forced oscillations occur under an external periodic force.

Q5. What is resonance in oscillations?
Ans. Resonance is the phenomenon in which a system oscillates with maximum amplitude when the frequency of an external periodic force matches its natural frequency.