1. Introduction: The Heartbeat of the Universe | Class 11 Physics Chapter 13 Oscillations NCERT solutions

Today we are starting a beautiful chapter. Up until now, we have mostly studied objects moving in straight lines (like a car on a highway) or objects moving in circles (like a satellite around the Earth). But nature has a third favorite way of moving: Oscillation.

Think about it. The universe is constantly vibrating. The atoms in your chair are vibrating right now. The sound of my voice reaches you because air molecules are oscillating back and forth. Even the alternating current (AC) that powers these lights oscillates 50 times a second. From the immense swinging of a pulsar star to the tiny beat of a mosquito’s wing, oscillation is the fundamental language of energy transfer. In this chapter, we are going to learn the grammar of this language.

We will start by distinguishing between simply doing something “again and again” and actually “vibrating.” Then, we will look at the simplest possible vibration (Simple Harmonic Motion) and analyze it mathematically. Finally, we will see how this math applies to pendulums and springs.

2. Fundamental Terminology

Before we can write equations, we need to agree on our vocabulary. Let’s define the specific terms that describe a vibrating system. Imagine a child on a swing to visualize these.

2.1 Period and Frequency

Time is the essence of oscillation.

• Period (T): The time taken to complete one full cycle.
  
  Example: If the swing starts from the back, goes forward, and comes back to the starting point, the time taken is ‘T’. Its SI unit is Seconds (s).
• Frequency (ν or f): The number of oscillations completed in one second. It tells us how “fast” the vibration is. It is the reciprocal of the period. Its SI unit is Hertz (Hz).
  
  Formula: ν = 1 / T
• Angular Frequency (ω): This is a mathematical tool we borrow from circular motion to make our Sine and Cosine equations work smoothly. It tells us how many radians of phase change occur per second.
  
  Formula: ω = 2πν = 2π / T
  
  Unit: Radians per second (rad/s).

2.2 Displacement (x or y)

In mechanics, displacement usually means “distance from origin”. In oscillations, it means distance from the Mean Position (Equilibrium point) at any specific instant.

• At the Mean position (Equilibrium), x = 0.

• At the Extreme positions (Maximum stretch), x = ±A (Amplitude).

Note: Displacement is a vector. Direction matters! Positive x means right/up, negative x means left/down.

3. Simple Harmonic Motion (SHM)

Not all vibrations are simple. If you hit a drum, the membrane vibrates in a complicated, chaotic pattern. In Physics, we always start with the simplest ideal case to build our understanding. We call this Simple Harmonic Motion (SHM).

3.1 SHM as a Projection of Uniform Circular Motion

This is a brilliant geometric trick that makes the math easy. If you watch a particle moving in a circle at constant speed, and you look at its shadow on a wall, the shadow moves back and forth in Simple Harmonic Motion!

Imagine a particle P moving in a circle of radius A with angular speed ω.

The projection of P on the X-axis is:

x = A cos(θ)

Since θ = ωt + φ (where φ is the starting angle), we get the master equation of SHM:

x(t) = A cos(ωt + φ)

Where:

• A (Amplitude): The maximum displacement from the center. It represents the “radius” of the reference circle.

• (ωt + φ) (Phase): The total angle at time t. It tells us the current status of the vibration (where the particle is and which way it is going).

• φ (Phase Constant or Epoch): The initial angle at t=0. It determines where the particle started.

  – If φ = 0, particle started at Extreme Positive (+A).

  – If φ = -π/2, particle started at Mean Position (0) moving right.

4. Velocity and Acceleration in SHM

Now that we have the equation for position x(t), we can use calculus to find how fast it moves (velocity) and how much it speeds up (acceleration).

4.1 Velocity (v)

Velocity is the rate of change of displacement (v = dx/dt).

Differentiating x = A cos(ωt + φ):

v(t) = -Aω sin(ωt + φ)

Using the trigonometry identity sin²θ + cos²θ = 1, we can rewrite velocity in terms of position ‘x’:

v = ω √(A² – x²)

Analysis:

• At Mean Position (x=0): The particle is zipping through the center. It has maximum momentum here. Speed is MAXIMUM.

v_max = ± Aω

• At Extreme Position (x=±A): The particle stops momentarily to turn around. Speed is ZERO.

v = 0

4.2 Acceleration (a)

Acceleration is the rate of change of velocity (a = dv/dt).

Differentiating v(t):

a(t) = -Aω² cos(ωt + φ)

Look closely! A cos(ωt + φ) is just ‘x’. So, we get a very simple relation:

a = -ω² x

Analysis:

• At Mean Position (x=0): Acceleration is ZERO. There is no force acting on the body at equilibrium.

• At Extreme Position (x=±A): Acceleration is MAXIMUM. The spring is stretched to the max, pulling back with maximum force.

a_max = −± ω² A

5. Energy in Simple Harmonic Motion

In an ideal SHM (assuming no friction or air resistance), energy is strictly conserved. It acts like a shapeshifter, constantly converting between Kinetic Energy (motion) and Potential Energy (strain).

5.1 Potential Energy (U)

The energy stored in the system due to displacement (like a stretched spring).

U = ½ k x²

Substituting x(t):

U = ½ k A² cos²(ωt + φ)

5.2 Kinetic Energy (K)

The energy due to the velocity of the particle.

K = ½ m v²

Substituting v(t):

K = ½ k A² sin²(ωt + φ)

5.3 Total Energy (E)

Total Energy E = K + U

E = ½ k A² (sin²θ + cos²θ)

Since sin²θ + cos²θ = 1, we arrive at a beautiful result:

E = ½ k A² = ½ m ω² A²

6. Systems Executing SHM

The theory is great, but let’s apply it to real-world mechanical systems. We will look at the two most common examples: the Spring and the Pendulum.

6.1 Oscillations of a Spring

Horizontal Spring: A mass ‘m’ attached to a spring of constant ‘k’ on a frictionless floor.

Restoring Force F = -kx.

Comparing with F = -mω² x, we get ω = √(k/m).

Time Period:

T = 2π √(m/k)

Vertical Spring: What happens if we hang the spring vertically? Gravity pulls the mass down, stretching the spring to a new equilibrium point. However, mathematically, the time period remains the same. Gravity only shifts the mean position; it does not change the restoring nature (stiffness) of the spring.

6.2 The Simple Pendulum

A simple pendulum consists of a heavy point mass (bob) suspended by a massless, inextensible string of length ‘L’. This is a classic example of SHM, but with a catch: it is only SHM for small angles.

Derivation of Period:

When displaced by a small angle θ, the restoring force along the arc is F = -mg sin(θ).

The Small Angle Approximation: For angles less than about 10 degrees, sin(θ) ≈ θ (in radians).

Also, arc length x = Lθ, so θ = x/L.

Therefore, F ≈ -mg(x/L) = – (mg/L) x.

This looks exactly like F = -kx! Here, the effective spring constant k = mg/L.

Using T = 2π √(m/k):

T = 2π √(L/g)

Seconds Pendulum: A pendulum whose time period is exactly 2 seconds (1 sec to swing one way, 1 sec to come back). On Earth, its length is approximately 1 meter.

7. Damped and Forced Oscillations

In the idealized world of physics problems, pendulums swing forever. In the real world, they eventually stop. Why? Because of friction at the pivot and air resistance. This leads us to more realistic scenarios.

7.1 Damped Oscillations

Real systems experience resistive forces that dissipate energy. This is called Damping.

The Damping Force: It is usually proportional to velocity but opposite in direction (like air drag).

F_d = -b v

(where ‘b’ is the damping constant).

Equation of Motion: ma = -kx – bv

The Result: The system still oscillates, but the Amplitude ‘A’ is no longer constant. It decreases exponentially with time.

A(t) = A₀ e^-bt/2m

7.2 Forced Oscillations and Resonance

If damping kills oscillations, how do we keep a swing going? We push it! This is Forced Oscillation. We apply an external periodic force F(t) = F₀ cos(ω_d t).

• Natural Frequency (ω₀): The frequency at which the system wants to oscillate naturally (based on its k and m).
• Driving Frequency (ω_d): The frequency of the external force you apply.

8. Comprehensive Practice Set

To master Oscillations, you need to be comfortable with the math. Let’s solve some diverse problems, step-by-step.

Part A: Conceptual Questions

1. Q: Can the path of a particle in SHM be a straight line?
   
   A: Yes, linear SHM (like a spring-mass on a table) is always along a straight line. Circular paths are not SHM (though SHM can be a projection of it).
2. Q: At what point in the path is the acceleration maximum?
   
   A: Acceleration is proportional to displacement ($a = -\omega^2 x$). It is maximum at the extreme positions ($x = \pm A$) where the restoring force is strongest.
3. Q: Why does a pendulum clock run slow in summer?
   
   A: In summer, due to thermal expansion, the length ‘L’ of the metal pendulum rod increases. Since T ∝ √L, the time period increases. The clock takes longer to complete a tick, so it runs slow (loses time).

Part B: Numerical Problems (Detailed Solutions)