Oscillations Class 11 Physics Notes

• Period T = 2π/ω; frequency f = 1/T = ω/2π.
• Velocity:
• $v=\pm \omega \sqrt{A^2-x^2}$
• — maximum at mean position, zero at extremes.
• Acceleration is maximum at extremes, zero at mean position.

• Total energy is constant and proportional to A²:
• $E=\frac{1}{2}m{\omega }^2{A}^2$
• .
• Kinetic energy:
• $K=\frac{1}{2}m{\omega }^2(A^2-x^2)$
• .
• Potential energy:
• $U=\frac{1}{2}m{\omega }^2{x}^2$
• .
• Kinetic energy is maximum at the mean position; potential energy is maximum at the extremes.

• Damped oscillations lose energy to friction/drag; amplitude decays exponentially over time.
• Forced oscillations occur when a periodic driving force is applied; amplitude depends on the driving frequency.
• Resonance is a sharp rise in amplitude when the driving frequency equals the natural frequency (e.g. a swing pushed at its natural rhythm).

Example: A pendulum of length $2\text{m}$ swings with small amplitude at a place where $g=9.8{\text{m/s}}^2$. Find its period.

Solution: $T=2\pi \sqrt{\frac{L}{g}}=2\pi \sqrt{\frac{2}{9.8}}\approx 2\times 3.14\times 0.45\approx 2.8\text{s}$.

Exercises (NCERT Q13.1–Q13.18)

• 13.1) Which of the following examples represent periodic motion? (a) A swimmer completing one (return) trip from one bank of a river to the other and back. (b) A freely suspended bar magnet displaced from its N-S direction and released. (c) A hydrogen molecule rotating about its centre of mass. (d) An arrow released from a bow.
• 13.2) Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion? (a) the rotation of earth about its axis. (b) motion of an oscillating mercury column in a U-tube. (c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lowermost point. (d) general vibrations of a polyatomic molecule about its equilibrium position.
• 13.3) Fig. 13.18 depicts four x-t plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion)?
• 13.4) Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (ω is any positive constant): (a) sin ωt − cos ωt (b) sin³ ωt (c) 3 cos (π/4 − 2ωt) (d) cos ωt + cos 3ωt + cos 5ωt (e) exp(−ω²t²) (f) 1 + ωt + ω²t²
• 13.5) A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is (a) at the end A, (b) at the end B, (c) at the mid-point of AB going towards A, (d) at 2 cm away from B going towards A, (e) at 3 cm away from A going towards B, and (f) at 4 cm away from B going towards A.
• 13.6) Which of the following relationships between the acceleration a and the displacement x of a particle involve simple harmonic motion? (a) a = 0.7x (b) a = −200x² (c) a = −10x (d) a = 100x³
• 13.7) The motion of a particle executing simple harmonic motion is described by the displacement function, x(t) = A cos(ωt + φ). If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is π s⁻¹. If instead of the cosine function, we choose the sine function to describe the SHM: x = B sin(ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions.
• 13.8) A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this balance, when displaced and released, oscillates with a period of 0.6 s. What is the weight of the body?
• 13.9) A spring having with a spring constant 1200 N m⁻¹ is mounted on a horizontal table as shown in Fig. 13.19. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released. Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.
• 13.10) In Exercise 13.9, let us take the position of mass when the spring is unstretched as x = 0, and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for the oscillating mass if at the moment we start the stopwatch (t = 0), the mass is (a) at the mean position, (b) at the maximum stretched position, and (c) at the maximum compressed position. In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?
• 13.11) Figures 13.20 correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure. Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.
• 13.12) Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (t = 0) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: (x is in cm and t is in s). (a) x = −2 sin (3t + π/3) (b) x = cos (π/6 − t) (c) x = 3 sin (2πt + π/4) (d) x = 2 cos πt
• 13.13) Figure 13.21(a) shows a spring of force constant k clamped rigidly at one end and a mass m attached to its free end. A force F applied at the free end stretches the spring. Figure 13.21(b) shows the same spring with both ends free and attached to a mass m at either end. Each end of the spring in Fig. 13.21(b) is stretched by the same force F. (a) What is the maximum extension of the spring in the two cases? (b) If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case?
• 13.14) The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is its maximum speed?
• 13.15) The acceleration due to gravity on the surface of moon is 1.7 m s⁻². What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is 3.5 s? (g on the surface of earth is 9.8 m s⁻²)
• 13.16) A simple pendulum of length l and having a bob of mass M is suspended in a car. The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?
• 13.17) A cylindrical piece of cork of density of base area A and height h floats in a liquid of density ρ_l. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period T = 2π√(hρ/ρ_l g), where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).
• 13.18) One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.

Answers at a glance

1. 1
   Q13.1 → (b) and (c) represent periodic motion.
2. 2
   Q13.2 → SHM: (b) and (c); periodic but not SHM: (a) and (d).
3. 3
   Q13.3 → Plots (b) and (d) are periodic, each with period 2 s.
4. 4
   Q13.4 → SHM: (a) 2π/ω, (c) π/ω; periodic not SHM: (b) 2π/ω, (d) 2π/ω; non-periodic: (e), (f).
5. 5
   Q13.5 → (a) 0, +, + (b) 0, −, − (c) −, 0, 0 (d) −, −, − (e) +, +, + (f) −, −, − (velocity, acceleration, force).
6. 6
   Q13.6 → Only (c) a = −10x represents simple harmonic motion.
7. 7
   Q13.7 → Cosine form: A = √2 cm, φ = −π/4; sine form: B = √2 cm, α = π/4.
8. 8
   Q13.8 → The body weighs about 219 N (mass ≈ 22.4 kg).
9. 9
   Q13.9 → (i) 3.18 Hz (ii) 8 m s⁻² (iii) 0.4 m s⁻¹.
10. 10
    Q13.10 → (a) x = 2 sin 20t (b) x = 2 cos 20t (c) x = −2 cos 20t; same amplitude and frequency, different initial phase.
11. 11
    Q13.11 → First: x = −3 sin πt cm; second: x = −2 cos(πt/2) m.
12. 12
    Q13.12 → (a) A = 2 cm, φ = 5π/6, ω = 3 (b) A = 1 cm, φ = −π/6, ω = 1 (c) A = 3 cm, φ = 3π/4, ω = 2π (d) A = 2 cm, φ = 0, ω = π.
13. 13
    Q13.13 → (a) Extension is F/k in both cases (b) T₁ = 2π√(m/k), T₂ = 2π√(m/2k).
14. 14
    Q13.14 → Maximum speed = 1.67 m s⁻¹ ≈ 100 m min⁻¹.
15. 15
    Q13.15 → The period on the moon is 8.4 s.
16. 16
    Q13.16 → T = 2π√(l/√(g² + v⁴/R²)).
17. 17
    Q13.17 → Restoring force = −Aρ_l g x gives SHM with T = 2π√(hρ/ρ_l g).
18. 18
    Q13.18 → Restoring force = −2Aρg h gives SHM with period T = 2π√(l/2g).

MCQ Questions

1 Mark Questions

1. 1
   Define simple harmonic motion.
2. 2
   What is a restoring force?
3. 3
   State the equation x = A sin(ωt + φ) and define each symbol.
4. 4
   Define the phase of SHM.

2 Mark Questions

1. 1
   A particle executes SHM with x = 0.05 sin(4πt) m. Find its amplitude and time period. (Ans: A = 0.05 m, T = 0.5 s)
2. 2
   A mass of 0.5 kg is attached to a spring of spring constant 200 N/m. Find its period of oscillation. (Ans: T = 2π√(m/k) ≈ 0.31 s)
3. 3
   At a displacement x the velocity of a SHM particle is half its maximum. Find x in terms of amplitude A. (Ans: x = (√3/2)A)

3 Mark Questions

1. 1
   Show that the motion of a mass on a spring is SHM and derive the expression for its time period. (Ans: T = 2π√(m/k))
2. 2
   Derive the expressions for the kinetic and potential energies of a particle executing SHM and show that the total energy is conserved.
3. 3
   Explain what is meant by the time period of a simple pendulum and derive its expression assuming small oscillations. (Ans: T = 2π√(l/g))

5 Mark Questions

1. 1
   Starting from the restoring force relation, derive the differential equation x + ω²x = 0 and show that its solution represents SHM. Hence obtain the expressions for velocity, acceleration and total energy in terms of displacement.
2. 2
   Define damped oscillations and resonance. Discuss the energy of SHM graphically and explain the difference between free, damped and forced oscillations with an example of resonance.