Oscillations - Class 11 Physics

Oscillations

Overview: Study of periodic motion, Simple Harmonic Motion (SHM), pendulums, and resonance.

1. Periodic Motion

Motion that repeats itself at regular intervals of time (Period T). Frequency f = 1/T.

Simple Harmonic Motion (SHM)

Restoring force is directly proportional to displacement and acts towards mean position.

F = -kx

Equation of SHM:

x(t) = A cos(ωt + φ)

Where A = Amplitude, ω = Angular Frequency, φ = Phase constant.

2. Dynamics of SHM

Velocity

v = -Aω sin(ωt + φ) = ω √(A² - x²)

Acceleration

a = -Aω² cos(ωt + φ) = -ω² x

Energy in SHM

Total Energy is conserved.

E = K + U = ½ m ω² A²

Kinetic Energy K = ½ m v²
Potential Energy U = ½ k x²

3. Oscillating Systems

Simple Pendulum

For small amplitudes:

T = 2π √(L/g)

Spring-Mass System

T = 2π √(m/k)

4. Damped & Forced Oscillations

Damped Oscillations

Amplitude decreases with time due to dissipative forces (friction/drag).

A(t) = A₀ e^-bt/2m

Forced Oscillations & Resonance

Oscillation driven by external periodic force. Amplitude becomes maximum when natural frequency matches driving frequency (Resonance).

Numericals - Oscillations

Numericals

SHM Equation

Q1. x(t) = 5 cos(2πt + π/4). Find Amplitude and Period.

Compare with A cos(ωt + φ)

A = 5 units

ω = 2π → 2π/T = 2π

T = 1 second

Velocity Calculation

Q2. Particle executes SHM with amplitude 0.1m and period 6s. Max velocity?

v_max = Aω

ω = 2π/T = 2π/6 = π/3

v_max = 0.1 × π/3

v_max ≈ 0.105 m/s

Spring Mass System

Q3. Mass of 2kg attached to spring of k=200 N/m. Find Time period.

T = 2π √(m/k)

T = 2π √(2/200)

T = 2π √(1/100) = 2π / 10

T = 0.2π ≈ 0.628 s

Acceleration

Q4. In SHM, acc at displacement 5cm is 20 cm/s². Find time period.

|a| = ω² x

20 = ω² (5)

ω² = 4 → ω = 2

ω = 2π/T → T = 2π/2 = π sec

Simple Pendulum

Q5. Length of seconds pendulum? (T=2s, g=9.8)

T = 2π √(L/g)

T² = 4π² L/g

L = g T² / 4π²

L = 9.8 × 4 / 39.44 ≈ 1 meter

Energy in SHM

Q6. Amplitude = 2cm, Total Energy = 40J. What is Displacement when KE = 30J?

Total E = 40. KE = 30. PE = 10 J.

U = ½ k x², E = ½ k A²

U/E = x²/A² = 10/40 = 1/4

x/A = 1/2

x = A/2 = 1 cm

Springs in Series

Q7. Two springs k1=200, k2=200 connected in series. Find equivalent k.

1/k_eq = 1/k1 + 1/k2

1/k_eq = 1/200 + 1/200 = 2/200 = 1/100

k_eq = 100 N/m

Pendulum on Moon

Q8. T on earth is 2s. What is T on moon? (g_moon = g/6)

T ∝ 1/√g

T_m / T_e = √(g_e / g_m) = √6

T_m = 2 × 2.45 = 4.9 s

Clock runs slower.

Max Acceleration

Q9. A=0.1m, f=50Hz. Find max acc.

a_max = ω² A

ω = 2πf = 100π

a_max = (100π)² × 0.1

a_max = 10000 π² × 0.1 = 1000 π² m/s²

Phase Difference

Q10. x1 = A sin(ωt), x2 = A cos(ωt). Phase difference?

cos(ωt) = sin(ωt + π/2)

φ2 - φ1 = π/2

Phase diff = 90°

Formulas & Facts - Oscillations

Equations & Formulas

Concept	Formula
SHM Equation	x = A cos(ωt + φ)
Velocity	v = ω √(A² - x²)
Acceleration	a = -ω² x
Time Period (Pendulum)	T = 2π √(l/g)
Time Period (Spring)	T = 2π √(m/k)
Angular Frequency	ω = 2πf = 2π/T
Total Energy	E = ½ m ω² A²
Kinetic Energy	K = ½ m ω² (A² - x²)
Potential Energy	U = ½ m ω² x²
Spring Series	1/k = 1/k1 + 1/k2

50 NEET Facts

Key points for Oscillations.

1. Periodic Motion Motion that repeats after fixed time period T.

2. Oscillatory Motion To and fro motion about a mean position. All oscillatory are periodic, but not vice versa (e.g., Earth orbit).

3. SHM Condition Force F ∝ -x. Restoring force proportional to displacement.

4. Amplitude (A) Maximum displacement from mean position.

5. Phase (φ) State of motion (position and direction) at t=0.

6. Velocity Phase Velocity leads displacement by π/2 (90 degrees).

7. Acceleration Phase Acceleration leads velocity by π/2. Leads displacement by π.

8. Mean Position Position where Net Force = 0. Velocity is Maximum.

9. Extreme Position Velocity = 0. Acceleration = Maximum.

10. Total Energy Conserved. Proportional to Amplitude squared (A²) and Frequency squared (f²).

11. U and K Frequency If SHM freq is f, then KE and PE oscillate with frequency 2f.

12. Seconds Pendulum Pendulum with Period T = 2 seconds. Length approx 1m on Earth.

13. Mass Independent Period of simple pendulum independent of mass of bob.

14. Shape Independent Oscillations of spring independent of g (gravity).

15. Infinite Length Pendulum T is not infinity. Max T is 84.6 minutes (Radius of Earth limit).

16. Pendulum in Lift Accelerating up: g_eff = g+a (T decreases). Accelerating down: g_eff = g-a (T increases).

17. Free Fall In satellite or free fall, g_eff = 0. T = Infinity. Pendulum doesn't oscillate.

18. Spring in Lift Period remains same (depends on m and k, not g).

19. Cutting Spring If spring cut into n equal parts, k of each part becomes nk.

20. Springs Series 1/k_eq = 1/k1 + 1/k2. System becomes softer.

21. Springs Parallel k_eq = k1 + k2. System becomes stiffer.

22. Pendulum in Liquid Apparent weight decreases. g_eff = g(1 - ρ/σ). T increases.

23. Hollow Bob Filled to Half COM lowers -> Effective Length increases -> T increases initially.

24. Resonance Driving frequency = Natural frequency. Amplitude becomes maximum.

25. Damped Oscillation Amplitude decays exponentially. Energy dissipates.

26. Forced Oscillation Body oscillates with frequency of driver, not its own natural freq.

27. Sharpness of Resonance Depends on Damping. Less damping -> Sharper resonance (higher peak).

28. Tacoma Narrows Bridge Collapsed due to resonance with wind.

29. Soldiers Marching Soldiers break step on bridge to avoid resonance.

30. U-Tube Oscillation Liquid in U-tube executes SHM. T = 2π √(h/g).

31. Tunnel through Earth Particle dropped executes SHM. T = 84.6 min.

32. Angular SHM Torque τ = -C θ. T = 2π √(I/C).

33. Physical Pendulum T = 2π √(I/mgl).

34. Floating Cylinder Executes SHM if pushed down. Restoring force is Buoyancy.

35. Phase Difference Path Diff Δφ = (2π/λ) Δx.

36. Graphical Representation Displacement-time graph is Sine/Cosine curve.

37. Velocity-Position Graph Ellipse.

38. Acceleration-Position Graph Straight line with negative slope passing through origin.

39. Restoring Force Must be conservative for undamped SHM.

40. Energy Graph Parabola for U and K. Constant line for Total E.

41. At Mean Position x=0, U=min, K=max, v=max, a=0.

42. At Extreme Position x=A, U=max, K=0, v=0, a=max.

43. Time to A/2 From mean to A/2 takes T/12.

44. Time to A/√2 From mean to A/√2 takes T/8.

45. Superposition collinear x = x1 + x2. Resultant is SHM with same freq.

46. Superposition perpendicular Lissajous figures.

47. Critical Damping Returns to equilibrium fastest without oscillating. (Car shock absorbers).

48. Overdamping Returns to equilibrium slowly without oscillating.

49. Underdamping Oscillates with decreasing amplitude.

50. Helmholtz Resonator Acoustic resonator. Detects specific frequency.

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