• The motion that repeats itself is called periodic motion.
• The period \(T\) is the time required for one complete oscillation or cycle. It is related to the frequency \(f\) by,
  \(
  T=\frac{1}{f}
  \)
  The frequency \(f\) of periodic or oscillatory motion is the number of oscillations per unit time. In the SI, it is measured in hertz :
  \(
  1 \text { hertz }=1 \mathrm{~Hz}=1 \text { oscillation per second }=1 \mathrm{~s}^{-1}
  \)
• In simple harmonic motion (SHM), the displacement \(x(t)\) of a particle from its equilibrium position is given by,
  \(
  x(t)=A \cos (\omega t+\phi) \quad \text { (displacement), }
  \)
  in which \(A\) is the amplitude of the displacement, the quantity \((\omega t+\phi)\) is the phase of the motion, and \(\phi\) is the phase constant. The angular frequency \(\omega\) is related to the period and frequency of the motion by,
  \(
  \omega=\frac{2 \pi}{T}=2 \pi f \quad \text { (angular frequency). }
  \)
• Simple harmonic motion can also be viewed as the projection of uniform circular motion on the diameter of the circle in which the latter motion occurs.
• The particle velocity and acceleration during SHM as functions of time are given by,
  \(
  \begin{array}{rlr}
  v(t) & =-\omega A \sin (\omega t+\phi) & \text { (velocity), } \\
  a(t) & =-\omega^2 A \cos (\omega t+\phi) \\
  & =-\omega^2 x(t) & \text { (acceleration), }
  \end{array}
  \)
  Thus we see that both velocity and acceleration of a body executing simple harmonic motion are periodic functions, having the velocity amplitude \(v_{\mathrm{m}}=\omega A\) and acceleration amplitude \(a_{\mathrm{m}}=\omega^2 A\), respectively.
• The force acting in a simple harmonic motion is proportional to the displacement and is always directed towards the centre of motion.
• A particle executing simple harmonic motion has, at any time, kinetic energy \(K=1 / 2 m v^2\) and potential energy \(U=1 / 2 k x^2\). If no friction is present the mechanical energy of the system, \(E=K+U\) always remains constant even though \(K\) and \(U\) change with time.
• A particle of mass \(m\) oscillating under the influence of Hooke’s law restoring force given by \(F=-k x\) exhibits simple harmonic motion with
  \(
  \begin{array}{ll}
  \omega=\sqrt{\frac{k}{m}} & \text { (angular frequency) } \\
  T=2 \pi \sqrt{\frac{m}{k}} & \text { (period) }
  \end{array}
  \)
  Such a system is also called a linear oscillator.
• The motion of a simple pendulum swinging through small angles is approximately simple harmonic. The period of oscillation is given by,
  \(
  T=2 \pi \sqrt{\frac{L}{g}}
  \)
• The mechanical energy in a real oscillating system decreases during oscillations because external forces, such as drag, inhibit the oscillations and transfer mechanical energy to thermal energy. The real oscillator and its motion are then said to be damped. If the damping force is given by \(F_d=-b v\), where \(v\) is the velocity of the oscillator and \(b\) is a damping constant, then the displacement of the oscillator is given by,
  \(
  x(t)=A_0 e^{-b t / 2 m} \cos \left(\omega^{\prime} t+\phi\right)
  \)
  where \(\omega^{\prime}\), the angular frequency of the damped oscillator, is given by
  \(
  \omega^{\prime}=\sqrt{\frac{k}{m}-\frac{b^2}{4 m^2}}
  \)
  If the damping constant is small then \(\omega^{\prime} \approx \omega\), where \(\omega\) is the angular frequency of the undamped oscillator. The mechanical energy \(E\) of the damped oscillator is given by
  \(
  E(t)=\frac{1}{2} k A_0^2 e^{-b t / m}
  \)
• If an external force with angular frequency \(\omega_d\) acts on an oscillating system with natural angular frequency \(\omega\), the system oscillates with angular frequency \(\omega_{d}\). The amplitude of oscillations is greatest when
  \(
  \omega_d=\omega
  \)
  a condition called resonance.

Points to Remember

1. The period \(T\) is the least time after which motion repeats itself. Thus, motion repeats itself after \(n T\) where \(n\) is an integer.
2. Every periodic motion is not simple harmonic motion. Only that periodic motion governed by the force law \(F=-k x\) is simple harmonic.
3. Circular motion can arise due to an inverse-square law force (as in planetary motion) as well as due to simple harmonic force in two dimensions equal to: \(-m \omega^2 r\). In the latter case, the phases of motion, in two perpendicular directions ( \(x\) and \(y\) ) must differ by \(\pi / 2\). Thus, for example, a particle subject to a force \(-m \omega^2 r\) with the initial position \((0\), A) and velocity \((\omega A, 0)\) will move uniformly in a circle of radius \(A\).
4. For linear simple harmonic motion with a given \(\omega\), two initial conditions are necessary and sufficient to determine the motion completely. The initial conditions may be (i) initial position and initial velocity or (ii) amplitude and phase or (iii) energy and phase.
5. From point 4 above, given amplitude or energy, the phase of motion is determined by the initial position or initial velocity.
6. A combination of two simple harmonic motions with arbitrary amplitudes and phases is not necessarily periodic. It is periodic only if the frequency of one motion is an integral multiple of the other’s frequency. However, a periodic motion can always be expressed as a sum of infinite number of harmonic motions with appropriate amplitudes.
7. The period of SHM does not depend on amplitude or energy or the phase constant. Contrast this with the periods of planetary orbits under gravitation (Kepler’s third law).
8. The motion of a simple pendulum is simple harmonic for small angular displacement.
9. For motion of a particle to be simple harmonic, its displacement \(x\) must be expressible in either of the following forms :
   \(
   \begin{aligned}
   & x=A \cos \omega t+B \sin \omega t \\
   & x=A \cos (\omega t+\alpha), x=B \sin (\omega t+\beta)
   \end{aligned}
   \)
   The three forms are completely equivalent (any one can be expressed in terms of any other two forms). Thus, damped simple harmonic motion [Eq. \((14.31)]\) is not strictly simple harmonic. It is approximately so only for time intervals much less than \(2 \mathrm{~m} / b\) where \(b\) is the damping constant.
10. In forced oscillations, the steady-state motion of the particle (after the forced oscillations die out) is simple harmonic motion whose frequency is the frequency of the driving frequency \(\omega_d\), not the natural frequency \(\omega\) of the particle.
11. In the ideal case of zero damping, the amplitude of simple harmonic motion at resonance is infinite. Since all real systems have some damping, however small, this situation is never observed.
12. Under forced oscillation, the phase of the harmonic motion of the particle differs from the phase of the driving force.