class 11 notes

Class 11 Physics Notes Chapter 14 (Chapter 14) – Examplar Problems (English) Book

Philoid

Philoid

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Examplar Problems (English)
Alright class, let's dive deep into Chapter 14: Oscillations. This is a crucial chapter, not just for your Class 11 understanding but also because concepts from oscillations appear frequently in various government entrance exams. We'll focus on the core ideas, especially those highlighted in the NCERT Exemplar, which often tests deeper conceptual clarity.

Chapter 14: Oscillations - Detailed Notes for Exam Preparation

1. Periodic Motion:

  • Any motion that repeats itself after a regular interval of time is called periodic motion.
  • Time Period (T): The smallest interval of time after which the motion repeats itself. Unit: second (s).
  • Frequency (ν or f): The number of repetitions (cycles) per unit time. ν = 1/T. Unit: Hertz (Hz) or s⁻¹.
  • Angular Frequency (ω): ω = 2πν = 2π/T. Unit: radian per second (rad s⁻¹).
  • Examples: Motion of planets around the sun, motion of the hands of a clock, vibrations of a guitar string.

2. Oscillatory Motion (Vibratory Motion):

  • A specific type of periodic motion where a body moves back and forth (to and fro) repeatedly about a fixed point, called the mean position or equilibrium position.
  • Key Idea: All oscillatory motions are periodic, but not all periodic motions are oscillatory (e.g., uniform circular motion is periodic but not oscillatory about a single mean point in the same way).
  • Examples: Motion of a simple pendulum, vibrations of a mass attached to a spring, vibrations of atoms in a solid.

3. Simple Harmonic Motion (SHM):

  • The simplest form of oscillatory motion.
  • Definition: An oscillatory motion is called SHM if:
    • The restoring force (F) acting on the particle is directly proportional to its displacement (x) from the mean position.
    • The force is always directed towards the mean position.
  • Mathematically: F = -kx
    • 'k' is the force constant or spring factor (Unit: N m⁻¹). It represents the stiffness.
    • The negative sign indicates that the force opposes the displacement.
  • Acceleration in SHM: Since F = ma, we have ma = -kx.
    • a = -(k/m)x
    • Since k and m are constants, acceleration (a) is directly proportional to displacement (x) and directed opposite to it.
    • We define ω² = k/m, where ω is the angular frequency of SHM.
    • So, a = -ω²x. This is the defining condition for SHM in terms of acceleration.

4. Differential Equation of SHM:

  • Acceleration is the second derivative of displacement with respect to time (a = d²x/dt²).
  • Substituting a = -ω²x, we get:
    • d²x/dt² = -ω²x
    • d²x/dt² + ω²x = 0
  • This is the second-order linear homogeneous differential equation representing SHM.

5. Displacement, Velocity, and Acceleration in SHM:

  • The general solution to the differential equation gives the displacement (x) as a function of time (t):
    • x(t) = A cos(ωt + φ) or x(t) = A sin(ωt + φ)
    • A: Amplitude (Maximum displacement from the mean position). Unit: meter (m).
    • (ωt + φ): Phase (Specifies the state of motion at time t). Unit: radian (rad).
    • φ: Initial Phase or Phase Constant or Epoch (Phase at t=0). Unit: radian (rad). It depends on the initial position and velocity.
    • ω: Angular Frequency (ω = √(k/m)).
  • Velocity (v(t)): v = dx/dt
    • If x(t) = A cos(ωt + φ), then v(t) = -Aω sin(ωt + φ)
    • Maximum velocity (at mean position, x=0): v_max = ± Aω
    • Velocity in terms of displacement: v = ± ω√(A² - x²)
  • Acceleration (a(t)): a = dv/dt = d²x/dt²
    • If x(t) = A cos(ωt + φ), then a(t) = -Aω² cos(ωt + φ) = -ω²x(t) (consistent with the definition)
    • Maximum acceleration (at extreme positions, x=±A): a_max = ± Aω²

Phase Relationships:

  • Velocity leads displacement by π/2 radians (90°).
  • Acceleration leads velocity by π/2 radians (90°).
  • Acceleration leads displacement by π radians (180°) or is in opposite phase with displacement.

Graphical Representation:

  • Displacement, velocity, and acceleration are sinusoidal functions of time.
  • If displacement is a cosine curve starting from +A, velocity is a negative sine curve starting from 0, and acceleration is a negative cosine curve starting from -Aω².

6. Energy in SHM:

  • Potential Energy (U): Energy stored due to position. It's the work done to displace the particle against the restoring force.
    • U = ∫(-F)dx = ∫(kx)dx = ½ kx²
    • Using x = A cos(ωt + φ) and k = mω², U(t) = ½ mω²A² cos²(ωt + φ)
    • Maximum at extremes (x=±A): U_max = ½ kA²
    • Minimum at mean position (x=0): U_min = 0
  • Kinetic Energy (K): Energy due to motion.
    • K = ½ mv²
    • Using v = -Aω sin(ωt + φ), K(t) = ½ m A²ω² sin²(ωt + φ)
    • Maximum at mean position (x=0): K_max = ½ m(Aω)² = ½ kA²
    • Minimum at extremes (x=±A): K_min = 0
  • Total Mechanical Energy (E): Sum of Kinetic and Potential Energy.
    • E = K + U = ½ mω²A² sin²(ωt + φ) + ½ mω²A² cos²(ωt + φ)
    • E = ½ mω²A² [sin²(ωt + φ) + cos²(ωt + φ)]
    • E = ½ kA² = ½ mω²A²
    • Key Point: The total mechanical energy in SHM is constant (conserved) if there are no dissipative forces (like friction or air resistance). It is proportional to the square of the amplitude (A²) and the square of the angular frequency (ω²).
  • Frequency of Energy Oscillation: Both K and U oscillate with time. Since sin²θ and cos²θ have a frequency twice that of sinθ or cosθ, the frequency of energy oscillation (K or U) is 2ν (twice the frequency of displacement oscillation). The time period of energy oscillation is T/2.

7. Examples of SHM:

  • (a) Oscillations of a Loaded Spring:

    • Horizontal: T = 2π√(m/k)
    • Vertical: When a mass 'm' is suspended, the spring extends by 'l' such that mg = kl. If pulled down further by 'y', the net restoring force is F = -k(l+y) + mg = -kl - ky + mg = -ky (since mg=kl). Motion is SHM.
      • T = 2π√(m/k) (Same as horizontal)
    • Series Combination: 1/k_s = 1/k₁ + 1/k₂ => T = 2π√(m(1/k₁ + 1/k₂))
    • Parallel Combination: k_p = k₁ + k₂ => T = 2π√(m/(k₁ + k₂))

  • (b) Simple Pendulum:

    • A point mass (bob) suspended by a massless, inextensible string from rigid support.
    • For small angular displacement (θ), the restoring force is F = -mg sinθ.
    • For small θ (in radians), sinθ ≈ θ. Also, θ = x/L (where x is arc length, L is length of pendulum).
    • F ≈ -mgθ = -(mg/L)x
    • This is in the form F = -Kx, where the effective force constant K = mg/L.
    • Time Period T = 2π√(m/K) = 2π√(m / (mg/L)) = 2π√(L/g)
    • Important: Time period is independent of the mass and amplitude (for small amplitudes). It depends only on the length (L) and acceleration due to gravity (g).
    • Seconds Pendulum: A pendulum whose time period is exactly 2 seconds. Its length is approximately 99.3 cm (≈ 1m) at sea level.

8. Damped Oscillations:

  • In real systems, dissipative forces (friction, air resistance) are present. These forces oppose the motion and dissipate energy, usually proportional to the velocity (F_damping = -bv, where 'b' is the damping constant).
  • The amplitude of oscillation decreases exponentially with time: A(t) = A₀e^(-bt/2m), where A₀ is the initial amplitude.
  • The total energy also decreases exponentially: E(t) = ½ k [A(t)]² = ½ k A₀² e^(-bt/m) = E₀ e^(-bt/m).
  • The angular frequency of damped oscillation (ω') is slightly less than the natural frequency (ω = √(k/m)): ω' = √(k/m - (b/2m)²).
  • Motion is still periodic (approximately) but not strictly SHM.
  • Types of Damping:
    • Underdamped: Oscillations occur with decreasing amplitude (b/2m < √(k/m)).
    • Critically Damped: System returns to equilibrium as quickly as possible without oscillating (b/2m = √(k/m)).
    • Overdamped: System returns to equilibrium slowly without oscillating (b/2m > √(k/m)).

9. Forced Oscillations and Resonance:

  • When a system is subjected to an external periodic force (driving force) with frequency ω_d (driving frequency).
  • Initially, the system oscillates with its natural frequency (ω), but eventually, it settles into oscillating with the driving frequency (ω_d). These are forced oscillations.
  • The amplitude of forced oscillation depends on the difference between ω_d and ω, and also on the damping factor 'b'.
  • Resonance: When the driving frequency (ω_d) matches the natural frequency (ω) of the oscillator (ω_d ≈ ω), the amplitude of oscillation becomes very large (theoretically infinite in the absence of damping). This phenomenon is called resonance.
  • The amplitude at resonance is limited by damping. Lower damping leads to a larger resonance amplitude and a sharper resonance peak.
  • Quality Factor (Q): A measure of the sharpness of resonance. Q ≈ ω₀ / (Δω), where ω₀ is the resonant frequency and Δω is the bandwidth (frequency range over which power drops to half its maximum value). Also related to energy loss per cycle: Q = 2π (Energy Stored / Energy dissipated per cycle). High Q means low damping and sharp resonance.

Key Points for Exams (Exemplar Focus):

  • Distinguish carefully between periodic, oscillatory, and simple harmonic motion.
  • SHM is defined by F ∝ -x or a ∝ -x.
  • Understand the phase relationships between x, v, and a. Be able to interpret graphs.
  • Energy conservation in ideal SHM (E = K + U = constant = ½ kA²). Know how K and U vary with time and position. Frequency of energy oscillation is 2ν.
  • Derivations for T of spring-mass systems and simple pendulum (especially the small angle approximation for pendulum).
  • Effect of damping on amplitude, energy, and frequency.
  • Concept of resonance: condition (ω_d ≈ ω), effect of damping on amplitude and sharpness (Q-factor).

Multiple Choice Questions (MCQs)

  1. Which of the following relationships between acceleration 'a' and displacement 'x' of a particle involves Simple Harmonic Motion?
    (a) a = 0.7 x
    (b) a = -200 x²
    (c) a = -10 x
    (d) a = 100 x³

  2. The displacement of a particle executing SHM is given by x = 4 cos(πt + π/4) cm. The frequency of oscillation is:
    (a) 1 Hz
    (b) 0.5 Hz
    (c) π Hz
    (d) 2 Hz

  3. A particle executes SHM with an amplitude A. At what displacement from the mean position is its kinetic energy equal to its potential energy?
    (a) A/2
    (b) A/√2
    (c) A
    (d) A/4

  4. The total energy of a particle executing SHM is E. What is its kinetic energy when the displacement is half the amplitude?
    (a) E/4
    (b) E/2
    (c) 3E/4
    (d) E

  5. A simple pendulum has a time period T₁ when on the Earth's surface, and T₂ when taken to a height R (equal to the radius of Earth) above the Earth's surface. What is the value of T₂/T₁? (g on surface = GM/R²)
    (a) 1
    (b) √2
    (c) 4
    (d) 2

  6. A mass 'm' attached to a spring oscillates with a period of 2 s. If the mass is increased by 2 kg, the period increases by 1 s. The initial mass 'm' is:
    (a) 1 kg
    (b) 1.6 kg
    (c) 2 kg
    (d) 2.4 kg

  7. In forced oscillations, a particle oscillates with the:
    (a) Natural frequency of the particle
    (b) Frequency of the driving force
    (c) Frequency which is the average of natural and driving frequency
    (d) Frequency which is the difference between natural and driving frequency

  8. For a particle executing SHM, which of the following statements is NOT correct?
    (a) The total energy of the particle always remains the same.
    (b) The restoring force is always directed towards a fixed point.
    (c) The restoring force is maximum at the extreme positions.
    (d) The acceleration of the particle is maximum at the equilibrium position.

  9. In damped oscillations, the damping force is generally proportional to:
    (a) Displacement
    (b) Velocity
    (c) Acceleration
    (d) Amplitude

  10. Resonance occurs when:
    (a) Damping is zero
    (b) Applied frequency is much larger than natural frequency
    (c) Applied frequency is equal or very close to the natural frequency
    (d) Amplitude of oscillation is minimum

Answers to MCQs:

  1. (c) [Condition for SHM is a ∝ -x]
  2. (b) [Comparing with x = A cos(ωt + φ), ω = π. Since ω = 2πν, ν = ω/2π = π/2π = 0.5 Hz]
  3. (b) [K = U => ½ mω²(A² - x²) = ½ mω²x² => A² - x² = x² => A² = 2x² => x = ± A/√2]
  4. (c) [E = ½ kA². When x = A/2, U = ½ k(A/2)² = ½ kA²/4 = E/4. Since K + U = E, K = E - U = E - E/4 = 3E/4]
  5. (d) [T = 2π√(L/g). g' at height R = GM/(R+R)² = GM/(2R)² = (GM/R²)/4 = g/4. So g' = g/4. T₂/T₁ = √(g/g') = √(g/(g/4)) = √4 = 2]
  6. (b) [T = 2π√(m/k). T₁ = 2 = 2π√(m/k). T₂ = 3 = 2π√((m+2)/k). Dividing T₂/T₁: 3/2 = √((m+2)/m). Squaring: 9/4 = (m+2)/m => 9m = 4m + 8 => 5m = 8 => m = 8/5 = 1.6 kg]
  7. (b) [In steady state, forced oscillations occur at the driving frequency.]
  8. (d) [Acceleration a = -ω²x. It is maximum when |x| is maximum (extreme positions) and zero at the equilibrium position (x=0).]
  9. (b) [Damping force is typically modelled as F_damping = -bv]
  10. (c) [Resonance is the phenomenon of large amplitude oscillations when driving frequency matches natural frequency.]

Make sure you understand the reasoning behind each MCQ answer. Go through the Exemplar problems for this chapter thoroughly, as they often involve graphical analysis and conceptual twists. Good luck with your preparation!

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