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Simple Harmonic Motion


One of the most important regular motions encountered in science and technology is oscillatory motion. Oscillatory (or vibrational ) motion is any motion that repeats itself periodically, i.e. goes back and forth over the same path, making each complete trip or cycle in an equal interval of time. Some examples include a simple pendulum swinging back and forth and a mass moving up and down when suspended from the end of a spring. Other examples are a vibrating guitar string, air molecules in a sound wave, ionic centers in solids, and many kinds of machines.


             By definition, a particle is said to be in simple harmonic motion if its displacement x from the center point

of the oscillations can be expressed as


                                                 x(t)=Acos (ω0t+φ) .


where ω is the angular frequency of the oscillation and t is the elapsed time, A is the amplitude of Oscillations and φ is phase angle.


DynamicsIn a linear mass-spring system, the physical basis for this kind of motion is that the restoring force F exerted

on a mass m that has been displaced a distance x from equilibrium must be proportional to –x.

This relationship may be written


                                                 F = - kx                                 (1)


where k is a constant called spring constant that characterizes the stiffness of the spring. A large value of k would indicate that the spring is difficult to stretch or compress. In the case of a simple pendulum, there is no spring, and k is replaced by the quantity (mg/L), where m is the mass of the pendulum bob, g is the acceleration due to gravity, L is the length of the pendulum, and x represents the (small) lateral displacement of the bob. Eq. (1) can be generalized to represent other physical situations. For example, the displacement might be given in terms of an angle, in which case the restoring variable would be a torque.


Using Newton’s second law, F = md2x/dt2, we can write Eq. (1) as a differential equation,




                                                 d2x/dt2   = - (k/m)x                 (2)



This equation has as a possible solution the sinusoidal oscillation x = A cosω0t, which you can verify by direct substitution in Eq. (2). Here A is the amplitude and ω0 = √ (k/m) is the circular frequency. The frequency depends on physical characteristics of the system. For example, ω0 = √ (k/m) for a linear mass-spring system and ω0 = √ (g/L) for small oscillations of a simple pendulum. Since the period T = 2π /ω0, we have T = 2π √ (m/k) for the mass-spring system and T = 2π √ (L/g) for the simple pendulum, respectively.



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