Oscillations: NEET notes

1. Periodic and oscillatory motion

NCERT §13.1, §13.2

• Any motion that comes back to the same state again and again, at equal time intervals, is periodic: a planet in its orbit, a ball bounced between palm and floor, an insect that climbs a ramp, drops off and starts again.
• An oscillatory motion is a periodic motion to and fro about a mean position, like a swing, a cradle, the pendulum of a wall clock or a boat bobbing on a river.
• At the mean position (the equilibrium position) no net force acts, so a body left there at rest stays there. Push it a little away and a force appears that pulls it back; that is what sets up the oscillation.
• Every oscillatory motion is periodic, but a periodic motion need not be oscillatory. Uniform circular motion repeats itself but has no back-and-forth about a mean point.
• Oscillation and vibration are the same thing. By habit a slow one is called an oscillation (a swaying branch) and a fast one a vibration (a sitar string).
• The height-time graph of a ball bounced between hand and floor is built from pieces of parabolas, so the motion is periodic without being simple harmonic.
• Simple harmonic motion (SHM) is the simplest oscillation: the force on the body is proportional to its displacement from the mean position and always points back towards that position.
• Real oscillators come to rest at equilibrium because friction and other dissipative effects damp them; an external periodic push can keep them going.
• A material medium behaves like a very large number of coupled oscillators, and their collective oscillation travels through it as a wave.

2. Period, frequency and displacement

NCERT §13.2.1, §13.2.2

• The period T is the least time after which the motion repeats; its SI unit is the second. Very fast or very slow motions use handier units: a quartz crystal vibrates with periods of microseconds, Mercury's orbital period is 88 earth days, and Halley's comet returns every 76 years.
• Frequency ν = 1/T is the number of repetitions per unit time. Its unit s⁻¹ is called the hertz (Hz), after Heinrich Rudolph Hertz: 1 Hz = one oscillation per second. A frequency need not be a whole number.
• Worked example: a heart that beats 75 times a minute has ν = 75/60 s = 1.25 Hz and T = 1/1.25 Hz = 0.8 s.
• In this chapter displacement means the change with time of whatever quantity is oscillating: the position of a block on a spring measured from equilibrium, the angle of a pendulum from the vertical, the voltage across a capacitor in an AC circuit, the pressure change in a sound wave, or the electric and magnetic fields in a light wave.
• A displacement variable can be positive or negative, and its zero is chosen for convenience, usually at the equilibrium position.
• f(t) = A cos ωt takes the same value whenever ωt grows by a whole multiple of 2π, so it is periodic with T = 2π/ω. A sin ωt has the same period.
• A sin ωt + B cos ωt is periodic with the same period and equals D sin(ωt + φ), where D = √(A² + B²) and φ = tan⁻¹(B/A).
• Fourier's theorem: any periodic function can be written as a sum of sine and cosine functions of different periods with suitable coefficients.
• A sum of periodic terms repeats after the least time that is a whole multiple of every term's period, so sin ωt + cos 2ωt + sin 4ωt has period 2π/ω. Functions such as e^(−ωt) or log(ωt) never take a value again and are not periodic.

3. Simple harmonic motion: amplitude, phase and angular frequency

NCERT §13.3

• A particle moving back and forth between x = −A and x = +A is in SHM when its displacement is x(t) = A cos(ωt + φ) with A, ω and φ constant, that is, when displacement is a sinusoidal function of time. SHM is not just any periodic motion.
• Amplitude A is the magnitude of the greatest displacement from the mean position; it can always be taken as positive.
• Phase is the time-dependent argument (ωt + φ). Given the amplitude, the phase fixes the particle's state of motion, its position and velocity, at time t.
• The phase constant (phase angle) φ is the value of the phase at t = 0. If A is known, φ follows from the displacement at t = 0.
• Angular frequency ω is linked to the period by ω = 2π/T = 2πν; its SI unit is the radian per second. It follows from x(t) = x(t + T), because cosine first repeats when its argument grows by 2π.
• Two SHMs can share ω and φ but differ in amplitude, share A and ω but differ in phase constant, or share A and φ but differ in ω; halving the period doubles the frequency.
• The particle moves fastest as it passes x = 0 and stops momentarily at the extremes x = ±A. The period stays the same whatever instant is chosen as t = 0.
• sin ωt − cos ωt = √2 sin(ωt − π/4) is SHM with period 2π/ω, amplitude √2 and phase angle −π/4 (or 7π/4).
• sin² ωt = ½ − ½ cos 2ωt is periodic with period π/ω; it is a harmonic motion about the point ½, not about zero.
• An angle written without a unit is in radians: sin(15) is the sine of 15 radians, and a value in degrees must carry the degree sign.

4. SHM as the shadow of uniform circular motion

NCERT §13.4

• If a particle P goes round a circle of radius A at a steady angular speed ω, its projection P′ on any diameter moves in simple harmonic motion.
• When OP makes an angle φ with the +x axis at t = 0 and turns anticlockwise, the angle at time t is ωt + φ and the projection is x(t) = A cos(ωt + φ): radius = amplitude, angular speed = angular frequency, starting angle = phase constant.
• P is called the reference particle and its circle the reference circle.
• A ball whirled in a horizontal circle, watched edge-on or through its shadow on a wall perpendicular to the plane of the circle, appears to move to and fro along a straight line centred on the point of rotation.
• The projection on the y-axis, y = A sin(ωt + φ), is also SHM of the same amplitude but differs in phase by π/2 from the x-projection.
• A clockwise sense makes the angle decrease with time. A point that starts at 90° and turns clockwise with period T has x-projection B cos(π/2 − 2πt/T) = B sin(2πt/T). With a 30 s period this becomes B cos(πt/15 − π/2): phase constant −π/2.
• Starting at 45° and turning anticlockwise with period 4 s, the x-projection is A cos(2πt/4 + π/4): amplitude A, period 4 s, phase constant π/4.
• The link is geometric only. The force on a body in linear SHM acts along the line, towards the mean point, and grows with displacement; the centripetal force that keeps P on its circle has a constant magnitude.

5. Velocity and acceleration in SHM

NCERT §13.5

• The reference particle moves along the tangent with speed ωA. Projecting that velocity on the x-axis gives v(t) = −ωA sin(ωt + φ), which is also what dx/dt gives.
• Its centripetal acceleration ω²A points to the centre. Projecting it gives a(t) = −ω²A cos(ωt + φ) = −ω²x(t), which is also dv/dt.
• The acceleration is proportional to the displacement and opposite in sign: when x > 0, a < 0, and when x < 0, a > 0. Whatever x is, the acceleration points towards the mean position.
• With φ = 0: x = A cos ωt, v = −ωA sin ωt, a = −ω²A cos ωt. All three have the same period; relative to x, the velocity differs in phase by π/2 and the acceleration by π.
• x ranges from −A to A, v from −ωA to ωA and a from −ω²A to ω²A. The velocity amplitude is v_m = ωA and the acceleration amplitude is a_m = ω²A.
• Speed is greatest at the mean position and zero at the extremes; acceleration is zero at the mean position and greatest at the extremes.
• Eliminating time with sin² + cos² = 1 gives the speed at displacement x: v = ω√(A² − x²).
• Worked example: for x = 5 cos(2πt + π/4) in SI units, ω = 2π s⁻¹ and T = 1 s. At t = 1.5 s the phase is 3π + π/4, so x = −3.535 m, the speed is 10π × 0.707 ≈ 22 m s⁻¹ and a = −(2π)² × (−3.535) ≈ 140 m s⁻².

6. Force law for SHM: F = −kx

NCERT §13.6

• Newton's second law with a = −ω²x gives the force on a particle of mass m in SHM: F = ma = −mω²x, written F = −kx with k = mω².
• So ω = √(k/m) and the period is T = 2π√(m/k). A stiffer spring makes it faster; a heavier block makes it slower.
• The force always points towards the mean position, which is why it is called the restoring force.
• SHM can be defined in either of two equivalent ways: by the displacement x(t) = A cos(ωt + φ), or by the force law F = −kx. Differentiating the first twice gives the second; integrating the second twice gives back the first.
• A body acted on by a force exactly proportional to x is a linear harmonic oscillator. If the force also contains small terms in x², x³ and so on, it is a non-linear oscillator.
• Test for SHM: the acceleration must be a negative constant times x. a = −(constant) × x is SHM; an acceleration proportional to +x, to x² or to x³ is not.
• Worked example: a block of mass m held between two identical springs of constant k fixed on either side. Displaced by x, one spring stretches and the other compresses by x, and both push it back, so F = −2kx and T = 2π√(m/2k).
• The period of SHM does not depend on the amplitude, the energy or the phase constant; it is set only by m and k.

7. Energy in SHM

NCERT §13.7

• Kinetic energy K = ½mv² = ½mω²A² sin²(ωt + φ) = ½kA² sin²(ωt + φ). It is zero at the extremes and largest at the mean position.
• The spring force F = −kx is conservative, with potential energy U = ½kx² = ½kA² cos²(ωt + φ). It is zero at the mean position and largest at the extremes.
• Adding them, E = K + U = ½kA², because sin² + cos² = 1. The total mechanical energy stays constant in time, as it must under a conservative force.
• K and U are each periodic with period T/2, not T: each reaches its peak twice in every oscillation, since the sign of v and of x does not matter.
• In terms of displacement, U = ½kx² and K = ½k(A² − x²): two parabolas that always add up to the flat line E = ½kA².
• Both K and U are never negative. K cannot be, since it depends on v²; U is made non-negative by choosing its zero at the mean position.
• Worked example: a 1 kg block on a spring of 50 N m⁻¹, pulled 10 cm from equilibrium and released. ω = √(50/1) = 7.07 rad s⁻¹. At 5 cm from the mean position cos(7.07t) = 0.5 and v = 0.1 × 7.07 × 0.866 ≈ 0.61 m s⁻¹, so K ≈ 0.19 J, U = ½ × 50 × 0.05² = 0.0625 J and E = 0.25 J, equal to ½ × 50 × 0.1².

8. The simple pendulum

NCERT §13.8

• Galileo is said to have timed a swinging church chandelier against his pulse and found its motion periodic. A stone on an inextensible thread about 100 cm long swings with a period of about two seconds.
• The idealised simple pendulum is a small bob of mass m on a massless, inextensible string of length L, fixed at a rigid support and swinging in a vertical plane.
• The bob feels just two forces: its weight mg and the string's tension T. Split the weight into mg cos θ, acting along the string, and mg sin θ, acting at right angles to it along the arc.
• The bob moves on a circle of radius L, so it has a radial acceleration supplied by T − mg cos θ and a tangential acceleration supplied by mg sin θ. The radial force passes through the support and gives no torque about it.
• The restoring torque about the support is τ = −L(mg sin θ), negative because it acts to reduce θ. With τ = Iα, this gives α = −(mgL/I) sin θ.
• sin θ = θ − θ³/3! + θ⁵/5! − …, with θ in radians. For small θ, sin θ ≈ θ, and the two stay close even up to about 20°.
• Then α = −(mgL/I)θ, which has the form of the SHM equation for angular displacement, with ω = √(mgL/I) and T = 2π√(I/mgL).
• For a massless string I = mL², so T = 2π√(L/g). The period depends only on the length and on g; it does not depend on the mass of the bob, nor (for small swings) on the amplitude.
• A seconds pendulum has T = 2 s (one second each way). From L = gT²/4π² with g = 9.8 m s⁻², L = 9.8 × 4/(4π²) ≈ 1 m.

Must-know facts

1. Every oscillatory motion is periodic; uniform circular motion is periodic but not oscillatory.
2. ν = 1/T, ω = 2πν = 2π/T; 1 Hz = 1 s⁻¹. A heart at 75 beats per minute: 1.25 Hz, 0.8 s.
3. SHM: x = A cos(ωt + φ); A amplitude, (ωt + φ) phase, φ phase constant.
4. SHM is the projection of uniform circular motion on a diameter; radius = A, angular speed = ω.
5. v = −ωA sin(ωt + φ), a = −ω²x; v_max = ωA at the mean position, a_max = ω²A at the extremes.
6. Relative to x, the velocity differs in phase by π/2 and the acceleration by π.
7. Speed at displacement x: v = ω√(A² − x²).
8. F = −kx, k = mω², ω = √(k/m), T = 2π√(m/k).
9. Block between two identical springs k on either side: T = 2π√(m/2k).
10. E = ½kA² is constant; K and U each have period T/2.
11. At x = A/2, U = E/4 and K = 3E/4.
12. Simple pendulum, small angles: T = 2π√(L/g), independent of bob mass and amplitude.
13. Seconds pendulum (T = 2 s): L ≈ 1 m for g = 9.8 m s⁻².
14. sin² ωt has period π/ω; sin ωt − cos ωt = √2 sin(ωt − π/4) is SHM with period 2π/ω.
15. The period of SHM does not depend on amplitude, energy or phase constant.

Common traps

Formulas

Key terms

Periodic motion

Motion that repeats itself after equal intervals of time.

Oscillatory motion

Periodic back-and-forth motion about a mean (equilibrium) position.

Period

The least time after which a periodic motion repeats.

Frequency

Number of repetitions per second, the reciprocal of the period; unit hertz.

Displacement (in oscillations)

The time-varying deviation of the oscillating quantity from its mean value: a position, an angle, a voltage, a pressure.

Simple harmonic motion

Oscillation in which displacement is a sinusoidal function of time, produced by a restoring force proportional to displacement.

Amplitude

The largest magnitude of displacement from the mean position.

Phase

The argument ωt + φ that fixes position and velocity at a given time.

Phase constant

The phase at t = 0.

Angular frequency

2π times the frequency; ω in rad s⁻¹.

Reference circle

The circle whose uniformly moving point projects onto a diameter as SHM.

Restoring force

A force that always points back towards the equilibrium position.

Spring constant

The force per unit extension of a spring, k in F = −kx.

Linear harmonic oscillator

A system whose restoring force is exactly proportional to displacement.

Simple pendulum

A point bob on a massless, inextensible string swinging about a fixed support.

Seconds pendulum

A simple pendulum of period 2 s, about 1 m long on earth.