Mathematical Representation
of Simple Harmonic Motion
φ is called the phase constant (or initial phase angle) and, along with the amplitude A, is determined uniquely by the position and velocity of the particle at t = 0.
The quantity (ωt+φ) is called the phase of the motion.
Mathematical Representation
of Simple Harmonic Motion
Figure 4.17 An experimental apparatus for demonstrating simple harmonic motion. A pen attached to the oscillating object traces out a sinusoidal pattern on the moving chart paper.
Mathematical Representation 
of Simple Harmonic Motion
The period T of the motion is the time interval required for the particle to go through one full cycle of its motion
(4.36)
Mathematical Representation 
of Simple Harmonic Motion
The inverse of the period is called the frequency f of the motion. Whereas the period is the time interval per oscillation, the frequency represents the number of oscillations that the particle undergoes per unit time interval: 
(4.37)
The units of f are cycles per second, or hertz (Hz). Rearranging Equation 8.11 gives
(4.38)
Mathematical Representation
of Simple Harmonic Motion
(4.39)
(4.40)
That is, the period and frequency depend only on the mass of the particle and the force constant of the spring, and not on the parameters of the motion, such as A or φ.
Mathematical Representation
of Simple Harmonic Motion
(4.41)
(4.42)
(4.43) |
(4.44) |
Energy of the Simple
Harmonic Oscillator
(4.45)
(4.46)
Graphical representation of
simple harmonic motion. (a) 
Position versus time. (b) Velocity
versus |
time. |
(c) |
Acceleration |
versus |
|
time. Note that at any specified time the velocity is 90° out of phase with the position and the acceleration is 180° out of phase with the position. 
The Pendulum
The simple pendulum is another mechanical system that exhibits periodic motion. It consists of a particle-like bob of mass m suspended by a light string of length L that is fixed at the upper end, as shown in Figure 8.4.
The Pendulum
(4.47)