Class 11 Physics · Chapter 13 NotesOscillations

A motion that repeats itself at regular intervals of time is called periodic motion. The smallest interval after which the motion repeats is the period T, measured in seconds. The reciprocal of the period is the frequency ν = 1/T, measured in hertz (1 Hz = 1 s⁻¹). When a body moves to and fro about a mean or equilibrium position, the motion is called oscillatory. Every oscillatory motion is periodic, but not every periodic motion is oscillatory — circular motion, for example, is periodic but not oscillatory. Oscillations and vibrations differ mainly in frequency: slow to-and-fro motion is called oscillation, while rapid motion is called vibration. Displacement in this chapter means the change with time of any physical quantity from its equilibrium value, such as the angle of a pendulum or the voltage in an AC circuit.

• Period Tsmallest time after which motion repeats; SI unit second
• Frequency ν = 1/T; SI unit hertz (Hz), where 1 Hz = 1 s⁻¹
• Oscillatory motionto-and-fro motion about a mean position
• Displacement can be distance, angle, voltage, pressure or field, depending on the system

Simple harmonic motion (SHM) is the simplest form of oscillatory motion. It occurs when the restoring force is directly proportional to displacement from the mean position and is always directed towards it. The displacement of a particle in SHM is a sinusoidal function of time: x(t) = A cos(ωt + φ). Here A is the amplitude, the maximum magnitude of displacement; (ωt + φ) is the phase, which fixes the state of motion at time t; and φ is the phase constant, the phase at t = 0. The angular frequency ω is related to the period by ω = 2π/T, so ω = 2πν. The period T does not depend on the amplitude. A useful way to picture SHM is as the projection of uniform circular motion on a diameter of the reference circle: a particle P moving uniformly on a circle of radius A has a projection P′ on any diameter that executes SHM.

• x(t) = A cos(ωt + φ)
• Amplitude Amaximum displacement from the mean position
• Angular frequency ω = 2π/T = 2πν; SI unit radian per second
• Phase = (ωt + φ); phase constant = φ, the phase at t = 0
• SHM is the projection of uniform circular motion on a diameter

Differentiating the displacement equation gives the velocity and acceleration of a particle in SHM. The velocity is v(t) = –ωA sin(ωt + φ), so its magnitude varies between zero at the extreme positions and a maximum value ωA at the mean position. The acceleration is a(t) = –ω²A cos(ωt + φ) = –ω²x(t). This shows that acceleration is proportional to displacement and is always directed towards the mean position. The velocity amplitude is ωA and the acceleration amplitude is ω²A. Compared with the displacement graph, the velocity graph is shifted in phase by π/2 and the acceleration graph by π. All three quantities — displacement, velocity and acceleration — vary sinusoidally with the same period T.

• v(t) = –ωA sin(ωt + φ); maximum speed = ωA at x = 0
• a(t) = –ω²A cos(ωt + φ) = –ω²x(t); maximum acceleration = ω²A at x = ±A
• Velocity leads displacement in phase by π/2; acceleration is opposite in phase to displacement
• Speed is zero at the extremes and maximum at the mean position

Using Newton's second law with a(t) = –ω²x(t), the force on a particle of mass m in SHM is F(t) = ma = –mω²x(t), which is written as F(t) = –kx(t) with k = mω². The force is linearly proportional to displacement and always directed towards the mean position, so it is called the restoring force. A particle oscillating under such a force is called a linear harmonic oscillator. This gives another equivalent definition of SHM: motion under a force F = –kx. From k = mω² we get ω = √(k/m) and the period T = 2π√(m/k). If additional terms proportional to x², x³, etc., are present, the oscillator is called non-linear.

• F(t) = –k x(t), where k = mω²
• ω = √(k/m) and T = 2π√(m/k)
• Restoring force is proportional to displacement and directed towards the mean position
• SHM can be defined either by x(t) = A cos(ωt + φ) or by F = –kx

The kinetic energy of a particle of mass m in SHM is K = ½mv² = ½kA² sin²(ωt + φ), and the potential energy associated with the spring force is U = ½kx² = ½kA² cos²(ωt + φ). Both energies vary periodically with time, each completing two cycles during one period of oscillation, so their period is T/2. Kinetic energy is zero at the extreme positions and maximum at the mean position, while potential energy is maximum at the extremes and zero at the mean position. The total mechanical energy E = K + U = ½kA² is constant in time, as expected for motion under a conservative force. Energy continuously changes form between kinetic and potential, but the total remains unchanged when friction is absent.

• K = ½mv² = ½kA² sin²(ωt + φ)
• U = ½kx² = ½kA² cos²(ωt + φ)
• Total energy E = K + U = ½kA², independent of time
• K and U each repeat with period T/2

A block of mass m attached to a spring of spring constant k on a frictionless surface executes SHM with angular frequency ω = √(k/m) and period T = 2π√(m/k). If two identical springs of constant k are attached to the block on opposite sides, the net force becomes –2kx, so the motion is still simple harmonic but with period T = 2π√(m/2k). A simple pendulum consists of a small bob of mass m tied to an inextensible, massless string of length L fixed at a rigid support. For small angular displacements, the restoring torque is proportional to the angular displacement, and the motion is approximately simple harmonic with period T = 2π√(L/g). The period of a simple pendulum is independent of the mass of the bob and depends only on its length and the local value of g.

• Spring–mass systemω = √(k/m), T = 2π√(m/k)
• Two springs on opposite sidesT = 2π√(m/2k)
• Simple pendulum (small angles)T = 2π√(L/g)
• Pendulum period does not depend on the mass of the bob

In real systems, friction and other dissipative forces reduce the energy of an oscillator, so its amplitude gradually decreases and it eventually comes to rest at the equilibrium position. Such motion is called damped oscillation. Because the amplitude changes with time, damped motion is not strictly simple harmonic; it is approximately simple harmonic only for time intervals much shorter than 2m/b, where b is the damping constant. An oscillator can be kept oscillating by an external periodic force; this is called a forced oscillation. When the frequency of the external force is close to the natural frequency of the oscillator, the amplitude of oscillation becomes very large. This phenomenon is called resonance, and it plays an important role in many physical systems.

• Damped oscillationamplitude decreases with time due to friction or dissipative forces
• Damped motion is approximately SHM only for times much less than 2m/b
• Forced oscillationan external periodic force maintains the oscillation
• Resonanceamplitude becomes very large when the driving frequency is close to the natural frequency